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Understanding how to apply Stokes theorem to higher dimensional spaces, non-Euclidean metrics, and with curvilinear coordinates has been a long standing goal.
A traditional answer to these questions can be found in the formalism of differential forms, as covered for example in [2], and [8]. However, both of those texts, despite their small size, are intensely scary. I also found it counter intuitive to have to express all physical quantities as forms, since there are many times when we don't have any pressing desire to integrate these.
Later I encountered Denker's straight wire treatment [1], which states that the geometric algebra formulation of Stokes theorem has the form
This is simple enough looking, but there are some important details left out. In particular the grades do not match, so there must be some sort of implied projection or dot product operations too. We also need to understand how to express the hypervolume and hypersurfaces when evaluating these integrals, especially when we want to use curvilinear coordinates.
I'd attempted to puzzle through these details previously. A collection of these attempts, to be removed from my collection of geometric algebra notes, can be found in [4]. I'd recently reviewed all of these and wrote a compact synopsis [5] of all those notes, but in the process of doing so, I realized there was a couple of fundamental problems with the approach I had used.
One detail that was that I failed to understand, was that we have a requirement for treating a infinitesimal region in the proof, then summing over such regions to express the boundary integral. Understanding that the boundary integral form and its dot product are both evaluated only at the end points of the integral region is an important detail that follows from such an argument (as used in proof of Stokes theorem for a 3D Cartesian space in [7].)
I also realized that my previous attempts could only work for the special cases where the dimension of the integration volume also equaled the dimension of the vector space. The key to resolving this issue is the concept of the tangent space, and an understanding of how to express the projection of the gradient onto the tangent space. These concepts are covered thoroughly in [6], which also introduces Stokes theorem as a special case of a more fundamental theorem for integration of geometric algebraic objects. My objective, for now, is still just to understand the generalization of Stokes theorem, and will leave the fundamental theorem of geometric calculus to later.
Now that these details are understood, the purpose of these notes is to detail the Geometric algebra form of Stokes theorem, covering its generalization to higher dimensional spaces and non-Euclidean metrics (i.e. especially those used for special relativity and electromagnetism), and understanding how to properly deal with curvilinear coordinates. This generalization has the form
Theorem 1. Stokes' Theorem
For blades , and volume element ,
Here the volume integral is over a dimensional surface (manifold), is the projection of the gradient onto the tangent space of the manifold, and indicates integration over the boundary of .
It takes some work to give this more concrete meaning. I will attempt to do so in a gradual fashion, and provide a number of examples that illustrate some of the relevant details.
Basic notation
A finite vector space, not necessarily Euclidean, with basis will be assumed to be the generator of the geometric algebra. A dual or reciprocal basis for this basis can be calculated, defined by the property
This is an Euclidean space when .
To select from a multivector the grade portion, say we write
The scalar portion of a multivector will be written as
The grade selection operators can be used to define the outer and inner products. For blades , and of grade and respectively, these are
Written out explicitly for odd grade blades (vector, trivector, …), and vector the dot and wedge products are respectively
Similarly for even grade blades these are
It will be useful to employ the cyclic scalar reordering identity for the scalar selection operator
For an dimensional vector space, a product of orthonormal (up to a sign) unit vectors is referred to as a pseudoscalar for the space, typically denoted by
The pseudoscalar may commute or anticommute with other blades in the space. We may also form a pseudoscalar for a subspace spanned by vectors by unit scaling the wedge products of those vectors .
Curvilinear coordinates
For our purposes a manifold can be loosely defined as a parameterized surface. For example, a 2D manifold can be considered a surface in an dimensional vector space, parameterized by two variables
Note that the indices here do not represent exponentiation. We can construct a basis for the manifold as
On the manifold we can calculate a reciprocal basis , defined by requiring, at each point on the surface
Associated implicitly with this basis is a curvilinear coordinate representation defined by the projection operation
(sums over mixed indices are implied). These coordinates can be calculated by taking dot products with the reciprocal frame vectors
In this document all coordinates are with respect to a specific curvilinear basis, and not with respect to the standard basis or its dual basis unless otherwise noted.
Similar to the usual notation for derivatives with respect to the standard basis coordinates we form a lower index partial derivative operator
so that when the complete vector space is spanned by the gradient has the curvilinear representation
This can be motivated by noting that the directional derivative is defined by
When the basis does not span the space, the projection of the gradient onto the tangent space at the point of evaluation
This is called the vector derivative.
See [6] for a more complete discussion of the gradient and vector derivatives in curvilinear coordinates.
Green's theorem
Given a two parameter () surface parameterization, the curvilinear coordinate representation of a vector has the form
We assume that the vector space is of dimension two or greater but otherwise unrestricted, and need not have an Euclidean basis. Here denotes the rejection of from the tangent space at the point of evaluation. Green's theorem relates the integral around a closed curve to an "area" integral on that surface
Theorem 2. Green's Theorem
Following the arguments used in [7] for Stokes theorem in three dimensions, we first evaluate the loop integral along the differential element of the surface at the point evaluated over the range , as shown in the infinitesimal loop of fig. 1.1.
Over the infinitesimal area, the loop integral decomposes into
where the differentials along the curve are
It is assumed that the parameterization change is small enough that this loop integral can be considered planar (regardless of the dimension of the vector space). Making use of the fact that for , the loop integral is
With the distances being infinitesimal, these differences can be rewritten as partial differentials
We can now sum over a larger area as in fig. 1.2
All the opposing oriented loop elements cancel, so the integral around the complete boundary of the surface is given by the area integral of the partials difference.
We will see that Green's theorem is a special case of the Curl (Stokes) theorem. This observation will also provide a geometric interpretation of the right hand side area integral of thm. 2, and allow for a coordinate free representation.
Special case:
An important special case of Green's theorem is for a Euclidean two dimensional space where the vector function is
Here Green's theorem takes the form
Curl theorem, two volume vector field
Having examined the right hand side of thm. 1 for the very simplest geometric object , let's look at the right hand side, the area integral in more detail. We restrict our attention for now to vectors still defined by eq. 1.19.
First we need to assign a meaning to . By this, we mean the wedge products of the two differential elements. With
that area element is
This is the oriented area element that lies in the tangent plane at the point of evaluation, and has the magnitude of the area of that segment of the surface, as depicted in fig. 1.3.
Observe that we have no requirement to introduce a normal to the surface to describe the direction of the plane. The wedge product provides the information about the orientation of the place in the space, even when the vector space that our vector lies in has dimension greater than three.
Proceeding with the expansion of the dot product of the area element with the curl, using eq. 1.0.6, eq. 1.0.7, and eq. 1.0.8, and a scalar selection operation, we have
Let's proceed to expand the inner dot product
The complete curl term is thus
This almost has the form of eq. 1.23, although that is not immediately obvious. Working backwards, using the shorthand , we can show that this coordinate representation can be eliminated
This relates the two parameter surface integral of the curl to the loop integral over its boundary
This is the very simplest special case of Stokes theorem. When written in the general form of Stokes thm. 1
we must remember (the is to remind us of this) that it is implied that both the vector and the differential elements are evaluated on the boundaries of the integration ranges respectively. A more exact statement is
Expanded out in full this is
which can be cross checked against fig. 1.4 to demonstrate that this specifies a clockwise orientation. For the surface with oriented area , the clockwise loop is designated with line elements (1)-(4), we see that the contributions around this loop (in boxes) match eq. 1.0.35.
Example: Green's theorem, a 2D Cartesian parameterization for a Euclidean space
For a Cartesian 2D Euclidean parameterization of a vector field and the integration space, Stokes theorem should be equivalent to Green's theorem eq. 1.0.25. Let's expand both sides of eq. 1.0.32 independently to verify equality. The parameterization is
Here the dual basis is the basis, and the projection onto the tangent space is just the gradient
The volume element is an area weighted pseudoscalar for the space
and the curl of a vector is
So, the LHS of Stokes theorem takes the coordinate form
For the RHS, following fig. 1.5, we have
As expected, we can also obtain this by integrating eq. 1.0.38.
Example: Cylindrical parameterization
Let's now consider a cylindrical parameterization of a 4D space with Euclidean metric or Minkowski metric . For such a space let's do a brute force expansion of both sides of Stokes theorem to gain some confidence that all is well.
With , such a space is conveniently parameterized as illustrated in fig. 1.6 as
Note that the Euclidean case where rejection of the non-axial components of expands to
whereas for the Minkowski case where we have a hyperbolic expansion
Within such a space consider the surface along , for which the vectors are parameterized by
The tangent space unit vectors are
Observe that both of these vectors have their origin at the point of evaluation, and aren't relative to the absolute origin used to parameterize the complete space.
We wish to compute the volume element for the tangent plane. Noting that and both anticommute with we have for
The tangent space volume element is thus
With the tangent plane vectors both perpendicular we don't need the general lemma 6 to compute the reciprocal basis, but can do so by inspection
Observe that the latter depends on the metric signature.
The vector derivative, the projection of the gradient on the tangent space, is
From this we see that acting with the vector derivative on a scalar radial only dependent function is a vector function that has a radial direction, whereas the action of the vector derivative on an azimuthal only dependent function is a vector function that has only an azimuthal direction. The interpretation of the geometric product action of the vector derivative on a vector function is not as simple since the product will be a multivector.
Expanding the curl in coordinates is messier, but yields in the end when tackled with sufficient care
After all this reduction, we can now state in coordinates the LHS of Stokes theorem explicitly
Now compare this to the direct evaluation of the loop integral portion of Stokes theorem. Expressing this using eq. 1.0.34, we have the same result
This example highlights some of the power of Stokes theorem, since the reduction of the volume element differential form was seen to be quite a chore (and easy to make mistakes doing.)
Example: Composition of boost and rotation
Working in a space with basis where and , an active composition of boost and rotation has the form
where is a bivector of a timelike unit vector and perpendicular spacelike unit vector, and is a bivector of two perpendicular spacelike unit vectors. For example, and . For such the respective Lorentz transformation matrices are
Let's calculate the tangent space vectors for this parameterization, assuming that the particle is at an initial spacetime position of . That is
To calculate the tangent space vectors for this subspace we note that
The tangent space vectors are therefore
Continuing a specific example where let's also pick , the spacetime position of a particle at the origin of a frame at that frame's . The tangent space vectors for the subspace parameterized by this transformation and this initial position is then reduced to
By inspection the dual basis for this parameterization is
So, Stokes theorem, applied to a spacetime vector , for this subspace is
Since the point is to avoid the curl integral, we did not actually have to state it explicitly, nor was there any actual need to calculate the dual basis.
Example: Dual representation in three dimensions
It's clear that there is a projective nature to the differential form . This projective nature allows us, in three dimensions, to re-express Stokes theorem using the gradient instead of the vector derivative, and to utilize the cross product and a normal direction to the plane.
When we parameterize a normal direction to the tangent space, so that for a 2D tangent space spanned by curvilinear coordinates and the vector is normal to both, we can write our vector as
and express the orientation of the tangent space area element in terms of a pseudoscalar that includes this normal direction
Inserting this into an expansion of the curl form we have
Observe that this last term, the contribution of the component of the gradient perpendicular to the tangent space, has no components
Now scale the normal vector and its dual to have unit norm as follows
so that for , the volume element can be
This scaling choice is illustrated in fig. 1.7, and represents the "outwards" normal. With such a scaling choice we have
and almost have the desired cross product representation
With the duality identity , we have the traditional 3D representation of Stokes theorem
Note that the orientation of the loop integral in the traditional statement of the 3D Stokes theorem is counterclockwise instead of clockwise, as written here.
Stokes theorem, three variable volume element parameterization
We can restate the identity of thm. 1 in an equivalent dot product form.
Here , with the implicit assumption that it and the blade that it is dotted with, are both evaluated at the end points of integration variable that has been integrated against.
We've seen one specific example of this above in the expansions of eq. 1.28, and eq. 1.29, however, the equivalent result of eq. 1.0.78, somewhat magically, applies to any degree blade and volume element provided the degree of the blade is less than that of the volume element (i.e. ). That magic follows directly from lemma 1.
As an expositional example, consider a three variable volume element parameterization, and a vector blade
It should not be surprising that this has the structure found in the theory of differential forms. Using the differentials for each of the parameterization "directions", we can write this dot product expansion as
Observe that the sign changes with each element of that is skipped. In differential forms, the wedge product composition of 1-forms is an abstract quantity. Here the differentials are just vectors, and their wedge product represents an oriented volume element. This interpretation is likely available in the theory of differential forms too, but is arguably less obvious.
As was the case with the loop integral, we expect that the coordinate representation has a representation that can be expressed as a number of antisymmetric terms. A bit of experimentation shows that such a sum, after dropping the parameter space volume element factor, is
To proceed with the integration, we must again consider an infinitesimal volume element, for which the partial can be evaluated as the difference of the endpoints, with all else held constant. For this three variable parameterization, say, , let's delimit such an infinitesimal volume element by the parameterization ranges , , . The integral is
Extending this over the ranges , , , we have proved Stokes thm. 1 for vectors and a three parameter volume element, provided we have a surface element of the form
where the evaluation of the dot products with are also evaluated at the same points.
Example: Euclidean spherical polar parameterization of 3D subspace
Consider an Euclidean space where a 3D subspace is parameterized using spherical coordinates, as in
The tangent space basis for the subspace situated at some fixed , is easy to calculate, and is found to be
While we can use the general relation of lemma 7 to compute the reciprocal basis. That is
However, a naive attempt at applying this without algebraic software is a route that requires a lot of care, and is easy to make mistakes doing. In this case it is really not necessary since the tangent space basis only requires scaling to orthonormalize, satisfying for
This allows us to read off the dual basis for the tangent volume by inspection
Should we wish to explicitly calculate the curl on the tangent space, we would need these. The area and volume elements are also messy to calculate manually. This expansion can be found in the Mathematica notebook \nbref{sphericalSurfaceAndVolumeElements.nb}, and is
Those area elements have a Geometric algebra factorization that are perhaps useful
One of the beauties of Stokes theorem is that we don't actually have to calculate the dual basis on the tangent space to proceed with the integration. For that calculation above, where we had a normal tangent basis, I still used software was used as an aid, so it is clear that this can generally get pretty messy.
To apply Stokes theorem to a vector field we can use eq. 1.0.82 to write down the integral directly
Observe that eq. 1.0.90 is a vector valued integral that expands to
This could easily be a difficult integral to evaluate since the vectors evaluated at the endpoints are still functions of two parameters. An easier integral would result from the application of Stokes theorem to a bivector valued field, say , for which we have
There is a geometric interpretation to these oriented area integrals, especially when written out explicitly in terms of the differentials along the parameterization directions. Pulling out a sign explicitly to match the geometry (as we had to also do for the line integrals in the two parameter volume element case), we can write this as
When written out in this differential form, each of the respective area elements is an oriented area along one of the faces of the parameterization volume, much like the line integral that results from a two parameter volume curl integral. This is visualized in fig. 1.8. In this figure, faces (1) and (3) are "top faces", those with signs matching the tops of the evaluation ranges eq. 1.0.94, whereas face (2) is a bottom face with a sign that is correspondingly reversed.
Example: Minkowski hyperbolic-spherical polar parameterization of 3D subspace
Working with a three parameter volume element in a Minkowski space does not change much. For example in a 4D space with , we can employ a hyperbolic-spherical parameterization similar to that used above for the 4D Euclidean space
This has tangent space basis elements
This is a normal basis, but again not orthonormal. Specifically, for we have
where we see that the radial vector is timelike. We can form the dual basis again by inspection
The area elements are
The volume element also reduces nicely, and is
The area and volume element reductions were once again messy, done in software using \nbref{sphericalSurfaceAndVolumeElementsMinkowski.nb}. However, we really only need eq. 1.0.96 to perform the Stokes integration.
Stokes theorem, four variable volume element parameterization
Volume elements for up to four parameters are likely of physical interest, with the four volume elements of interest for relativistic physics in spaces. For example, we may wish to use a parameterization , with a four volume
We follow the same procedure to calculate the corresponding boundary surface "area" element (with dimensions of volume in this case). This is
Our boundary value surface element is therefore
where it is implied that this (and the dot products with ) are evaluated on the boundaries of the integration ranges of the omitted index. This same boundary form can be used for vector, bivector and trivector variations of Stokes theorem.
Duality and its relation to the pseudoscalar.
Looking to eq. 1.0.181 of lemma 6, and scaling the wedge product by its absolute magnitude, we can express duality using that scaled bivector as a pseudoscalar for the plane that spans . Let's introduce a subscript notation for such scaled blades
This allows us to express the unit vector in the direction of as
Following the pattern of eq. 1.0.181, it is clear how to express the dual vectors for higher dimensional subspaces. For example
or for the unit vector in the direction of ,
Divergence theorem.
When the curl integral is a scalar result we are able to apply duality relationships to obtain the divergence theorem for the corresponding space. We will be able to show that a relationship of the following form holds
Here is a vector, is normal to the boundary surface, and is the area of this bounding surface element. We wish to quantify these more precisely, especially because the orientation of the normal vectors are metric dependent. Working a few specific examples will show the pattern nicely, but it is helpful to first consider some aspects of the general case.
First note that, for a scalar Stokes integral we are integrating the vector derivative curl of a blade over a k-parameter volume element. Because the dimension of the space matches the number of parameters, the projection of the gradient onto the tangent space is exactly that gradient
Multiplication of by the pseudoscalar will always produce a vector. With the introduction of such a dual vector, as in
Stokes theorem takes the form
where we will see that the vector can roughly be characterized as a normal to the boundary surface. Using primes to indicate the scope of the action of the gradient, cyclic permutation within the scalar selection operator can be used to factor out the pseudoscalar
The second last step uses lemma 8, and the last writes , where we have assumed (without loss of generality) that has the same orientation as the pseudoscalar for the space. We also assume that the parameterization is non-degenerate over the integration volume (i.e. no ), so the sign of this product cannot change.
Let's now return to the normal vector . With (the indexed differential omitted), and , we have
We've seen in eq. 1.0.106 and lemma 7 that the dual of vector with respect to the unit pseudoscalar in a subspace spanned by is
This allows us to write
where , and is the area of the boundary area element normal to . Note that the term will now cancel cleanly from both sides of the divergence equation, taking both the metric and the orientation specific dependencies with it.
This leaves us with
To spell out the details, we have to be very careful with the signs. However, that is a job best left for specific examples.
Example: 2D divergence theorem
Let's start back at
On the left our integral can be rewritten as
where and we pick the pseudoscalar with the same orientation as the volume (area in this case) element .
For the boundary form we have
The duality relations for the tangent space are
Back substitution into the line element gives
Writing (no sum) , we have
This provides us a divergence and normal relationship, with terms on each side that can be canceled. Restoring explicit range evaluation, that is
Let's consider this graphically for an Euclidean metric as illustrated in fig. 1.9.
We see that
- along the outwards normal is ,
- along the outwards normal is ,
- along the outwards normal is , and
- along the outwards normal is .
Writing that outwards normal as , we have
Note that we can use the same algebraic notion of outward normal for non-Euclidean spaces, although cannot expect the geometry to look anything like that of the figure.
Example: 3D divergence theorem
As with the 2D example, let's start back with
In a 3D space, the pseudoscalar commutes with all grades, so we have
where , and we have used a pseudoscalar with the same orientation as the volume element
In the boundary integral our dual two form is
where , and
Observe that we can do a cyclic permutation of a 3 blade without any change of sign, for example
Because of this we can write the dual two form as we expressed the normals in lemma 7
We can now state the 3D divergence theorem, canceling out the metric and orientation dependent term on both sides
where (sums implied)
The outwards normals at the upper integration ranges of a three parameter surface are depicted in fig. 1.10.
This sign alternation originates with the two form elements from the Stokes boundary integral, which were explicitly evaluated at the endpoints of the integral. That is, for ,
In the context of the divergence theorem, this means that we are implicitly requiring the dot products to be evaluated specifically at the end points of the integration where , accounting for the alternation of sign required to describe the normals as uniformly outwards.
Example: 4D divergence theorem
Applying Stokes theorem to a trivector in the 4D case we find
Here the pseudoscalar has been picked to have the same orientation as the hypervolume element . Writing the dual of the three form is
Here, we've written
Observe that the dual representation nicely removes the alternation of sign that we had in the Stokes theorem boundary integral, since each alternation of the wedged vectors in the pseudoscalar changes the sign once.
As before, we define the outwards normals as on the upper and lower integration ranges respectively. The scalar area elements on these faces can be written in a dual form
so that the 4D divergence theorem looks just like the 2D and 3D cases
Here we define the volume scaled normal as
As before, we have made use of the implicit fact that the three form (and it's dot product with ) was evaluated on the boundaries of the integration region, with a toggling of sign on the lower limit of that evaluation that is now reflected in what we have defined as the outwards normal.
We also obtain explicit instructions from this formalism how to compute the "outwards" normal for this surface in a 4D space (unit scaling of the dual basis elements), something that we cannot compute using any sort of geometrical intuition. For free we've obtained a result that applies to both Euclidean and Minkowski (or other non-Euclidean) spaces.
Volume integral coordinate representations
It may be useful to formulate the curl integrals in tensor form. For vectors , and bivectors , the coordinate representations of those differential forms (\cref{pr:stokesTheoremGeometricAlgebraII:1}) are
Here the bivector and trivector is expressed in terms of their curvilinear components on the tangent space
For the trivector components are also antisymmetric, changing sign with any interchange of indices.
Note that eq. 1.0.144d and eq. 1.0.144f appear much different on the surface, but both have the same structure. This can be seen by writing for former as
In both of these we have an alternation of sign, where the tensor index skips one of the volume element indices is sequence. We've seen in the 4D divergence theorem that this alternation of sign can be related to a duality transformation.
In integral form (no sum over indexes in terms), these are
Of these, I suspect that only eq. 1.0.148a and eq. 1.0.148d are of use.
Final remarks
Because we have used curvilinear coordinates from the get go, we have arrived naturally at a formulation that works for both Euclidean and non-Euclidean geometries, and have demonstrated that Stokes (and the divergence theorem) holds regardless of the geometry or the parameterization. We also know explicitly how to formulate both theorems for any parameterization that we choose, something much more valuable than knowledge that this is possible.
For the divergence theorem we have introduced the concept of outwards normal (for example in 3D, eq. 1.0.136), which still holds for non-Euclidean geometries. We may not be able to form intuitive geometrical interpretations for these normals, but do have an algebraic description of them.
Question: Expand volume elements in coordinates
Show that the coordinate representation for the volume element dotted with the curl can be represented as a sum of antisymmetric terms. That is
- (a)Prove eq. 1.0.144a
- (b)Prove eq. 1.0.144b
- (c)Prove eq. 1.0.144c
- (d)Prove eq. 1.0.144d
- (e)Prove eq. 1.0.144e
- (f)Prove eq. 1.0.144f
(a) Two parameter volume, curl of vector
(b) Three parameter volume, curl of vector
(c) Four parameter volume, curl of vector
(d) Three parameter volume, curl of bivector
(e) Four parameter volume, curl of bivector
To start, we require lemma 3. For convenience lets also write our wedge products as a single indexed quantity, as in for . The expansion is
This last step uses an intermediate result from the eq. 1.0.152 expansion above, since each of the four terms has the same structure we have previously observed.
(f) Four parameter volume, curl of trivector
Using the shorthand again, the initial expansion gives
Applying lemma 4 to expand the inner products within the braces we have
We can cancel those last terms using lemma 5. Using the same reverse chain rule expansion once more we have
The final result follows after permuting the indices slightly.
Some helpful identities
Lemma 1. Distribution of inner products
Given two blades with grades subject to , and a vector , the inner product distributes according to
This will allow us, for example, to expand a general inner product of the form .
The proof is straightforward, but also mechanical. Start by expanding the wedge and dot products within a grade selection operator
Solving for in
we have
The last term above is zero since we are selecting the grade element of a multivector with grades and , which has no terms for . Now we can expand the multivector product, for
The latter multivector (with the wedge product factor) above has grades and , so this selection operator finds nothing. This leaves
The first dot products term has grade and is selected, whereas the wedge term has grade (for ).
Lemma 2. Distribution of two bivectors
For vectors , , and bivector , we have
Proof follows by applying the scalar selection operator, expanding the wedge product within it, and eliminating any of the terms that cannot contribute grade zero values
Lemma 3. Inner product of trivector with bivector
Given a bivector , and trivector where and are vectors, the inner product is
This is also problem 1.1(c) from Exercises 2.1 in [3], and submits to a dumb expansion in successive dot products with a final regrouping. With
Lemma 4. Distribution of two trivectors
Given a trivector and three vectors , and , the entire inner product can be expanded in terms of any successive set inner products, subject to change of sign with interchange of any two adjacent vectors within the dot product sequence
To show this, we first expand within a scalar selection operator
Now consider any single term from the scalar selection, such as the first. This can be reordered using the vector dot product identity
The vector-trivector product in the latter grade selection operation above contributes only bivector and quadvector terms, thus contributing nothing. This can be repeated, showing that
Substituting this back into eq. 1.0.168 proves lemma 4.
Lemma 5. Permutation of two successive dot products with trivector
Given a trivector and two vectors and , alternating the order of the dot products changes the sign
This and lemma 4 are clearly examples of a more general identity, but I'll not try to prove that here. To show this one, we have
Cancellation of terms above was because they could not contribute to a grade one selection. We also employed the relation for bivector and vector .
Lemma 6. Duality in a plane
For a vector , and a plane containing and , the dual of this vector with respect to this plane is
To demonstrate, we start with the expansion of
Dotting with we have
but dotting with yields zero
To complete the proof, we note that the product in eq. 1.177 is just the wedge squared
This duality relation can be recast with a linear denominator
We can use this form after scaling it appropriately to express duality in terms of the pseudoscalar.
Lemma 7. Dual vector in a three vector subspace
In the subspace spanned by , the dual of is
Consider the dot product of with .
The canceled term is eliminated since it is the product of a vector and trivector producing no scalar term. Substituting , and noting that , we have
Lemma 8. Pseudoscalar selection
For grade blade (i.e. a pseudoscalar), and vectors , the grade selection of this blade sandwiched between the vectors is
To show this, we have to consider even and odd grades separately. First for even we have
Similarly for odd , we have
Adjusting for the signs completes the proof.
[1] John Denker. Magnetic field for a straight wire., 2014. URL http://www.av8n.com/physics/straight-wire.pdf. [Online; accessed 11-May-2014].
[2] H. Flanders. Differential Forms With Applications to the Physical Sciences. Courier Dover Publications, 1989.
[3] D. Hestenes. New Foundations for Classical Mechanics. Kluwer Academic Publishers, 1999.
[4] Peeter Joot. Collection of old notes on Stokes theorem in Geometric algebra, 2014. URL https://sites.google.com/site/peeterjoot3/math2014/bigCollectionOfPartiallyIncorrectStokesTheoremMusings.pdf.
[5] Peeter Joot. Synposis of old notes on Stokes theorem in Geometric algebra, 2014. URL https://sites.google.com/site/peeterjoot3/math2014/synopsisOfBigCollectionOfPartiallyIncorrectStokesTheoremMusings.pdf.
[6] A. Macdonald. Vector and Geometric Calculus. CreateSpace Independent Publishing Platform, 2012.
[7] M. Schwartz. Principles of Electrodynamics. Dover Publications, 1987.
[8] Michael Spivak. Calculus on manifolds, volume 1. Benjamin New York, 1965.


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Peanuts is a syndicated daily and Sunday American comic strip written and illustrated by Charles M. Schulz, which ran from October 2, 1950, to February 13, 2000, continuing in reruns afterward. The strip is the most popular and influential in the history of comic strips, with 17,897 strips published in all,[1] making it "arguably the longest story ever told by one human being".[2] At its peak, Peanuts ran in over 2,600 newspapers, with a readership of 355 million in 75 countries, and was translated into 21 languages.[3] It helped to cement the four-panel gag strip as the standard in the United States,[4] and together with its merchandise earned Schulz more than $1 billion.[1] Reprints of the strip are still syndicated and run in almost every U.S. newspaper.
Peanuts achieved considerable success with its television specials, several of which, including A Charlie Brown Christmas[5] and It's the Great Pumpkin, Charlie Brown,[6] won or were nominated for Emmy Awards. The holiday specials remain popular and are currently broadcast on ABC in the United States during the corresponding seasons. The Peanuts franchise met acclaim in theatre, with the stage musical You're a Good Man, Charlie Brown being a successful and often-performed production.
Peanuts has been described as "the most shining example of the American success story in the comic strip field"; this is ironic, given its theme is "the great American unsuccess story." The main character, Charlie Brown, is meek, nervous, and lacks self-confidence. He is unable to fly a kite, win a baseball game, or kick a football.[7] In 2013, TV Guide ranked Peanuts the fourth Greatest TV Cartoon of All Time.[8] A computer-animated feature film based on the strip, The Peanuts Movie, will be released on November 6, 2015.
Peanuts had its origin in Li'l Folks, a weekly panel comic that appeared in Schulz's hometown paper, the St. Paul Pioneer Press, from 1947 to 1950. He first used the name Charlie Brown for a character there, although he applied the name in four gags to three different boys and one buried in sand. The series also had a dog that looked much like the early 1950s version of Snoopy.[9] In 1948, Schulz sold a cartoon to The Saturday Evening Post which published 17 single-panel cartoons by Schulz. The first of these was of a boy sitting with his feet on an ottoman.
In 1948, Schulz tried to have Li'l Folks syndicated through the Newspaper Enterprise Association. Schulz would have been an independent contractor for the syndicate, unheard of in the 1940s, but the deal fell through. Li'l Folks was dropped in early 1950. Later that year, Schulz approached the United Feature Syndicate with his best work from Li'l Folks. When his work was picked up by United Feature Syndicate, they decided to run the new comic strip he had been working on. This strip was similar in spirit to the panel comic, but it had a set cast of characters, rather than different nameless little folk for each page. The name Li'l Folks was too close to the names of two other comics of the time: Al Capp's Li'l Abner and a strip titled Little Folks. To avoid confusion, the syndicate settled on the name Peanuts, after the peanut gallery featured in the Howdy Doody TV show.[10] Peanuts was a title Schulz always disliked. In a 1987 interview, Schulz said of the title Peanuts: "It's totally ridiculous, has no meaning, is simply confusing, and has no dignity—and I think my humor has dignity."[11] The periodic collections of the strips in paperback book form typically had either "Charlie Brown" or "Snoopy" in the title, not "Peanuts", because of Schulz's distaste for his strip's title. From November 20, 1966 to January 4, 1987, the opening Sunday panels typically read Peanuts, featuring Good Ol ' Charlie Brown.
Peanuts premiered on October 2, 1950, in nine newspapers: The Washington Post, The Chicago Tribune, The Minneapolis Tribune, The Allentown Call-Chronicle, The Bethlehem Globe-Times, The Denver Post, The Seattle Times, The New York World-Telegram & Sun, and The Boston Globe. It began as a daily strip. The first strip was four panels long and showed Charlie Brown walking by two other young children, Shermy and Patty. Shermy lauds Charlie Brown as he walks by, but then tells Patty how he hates him in the final panel. This was groundbreaking. Until then, rarely had children expressed hatred for others in comic strips. Snoopy was also an early character in the strip, first appearing in the third strip, which ran on October 4. Its first Sunday strip appeared January 6, 1952, in the half-page format, which was the only complete format for the entire life of the Sunday strip. Most of the other characters that eventually became the main characters of Peanuts did not appear until later: Violet (February 1951), Schroeder (May 1951), Lucy (March 1952), Linus (September 1952), Pig-Pen (July 1954), Sally (August 1959), Frieda (March 1961), "Peppermint" Patty (August 1966), Woodstock (introduced April 1967; given a name in June 1970), Franklin (July 1968), Marcie (July 1971), and Rerun (March 1973).
Schulz decided to produce all aspects of the strip himself from the script to the finished art and lettering. (Schulz did, however, hire help to produce the comic book adaptations of Peanuts.[12]) Thus, the strip was able to be presented with a unified tone, and Schulz was able to employ a minimalistic style. Backgrounds were generally not used, and when they were, Schulz's frazzled lines imbued them with a fraught, psychological appearance. This style has been described by art critic John Carlin as forcing "its readers to focus on subtle nuances rather than broad actions or sharp transitions."[13] Schulz held this belief all his life, reaffirming in 1994 the importance of crafting the strip himself: "This is not a crazy business about slinging ink. This is a deadly serious business."[14]
While the strip in its early years resembles its later form, there are significant differences. The art was cleaner, sleeker, and simpler, with thicker lines and short, squat characters. For example, in these early strips, Charlie Brown's famous round head is closer to the shape of an American football or rugby football. Most of the kids were initially fairly round-headed. As another example, all the characters (except Charlie Brown) had their mouths longer and had smaller eyes when they looked sideways.
The 1960s is known as the "golden age" for the comic strip. During this period some of the most well known themes and characters appeared, including: Peppermint Patty, Snoopy as the "World War One Flying Ace", Frieda and her "naturally curly hair", and Franklin. Peanuts is remarkable for its deft social commentary, especially compared with other strips appearing in the 1950s and early 1960s. Schulz did not explicitly address racial and gender equality issues so much as he assumed them to be self-evident in the first place. Peppermint Patty's athletic skill and self-confidence is simply taken for granted, for example, as is Franklin's presence in a racially integrated school and neighborhood. (Franklin came about at least in part as a result of Schulz's correspondence in 1968 with a socially progressive fan.[15]) The fact that Charlie Brown's baseball team had three girls was also at least ten years ahead of its time (and in fact, the TV special Charlie Brown's All-Stars dealt with Charlie Brown refusing sponsorship of the team because the sponsor said the league does not allow girls or dogs to play).
Schulz would throw satirical barbs at any number of topics when he chose. Over the years he tackled everything from the snowman-building leagues and criticize Charlie Brown when he insists on building his own snowmen without leagues or coaches.
Peanuts did not shy away from cartoon violence. The most obvious example might be Charlie Brown's annual, futile effort to kick the football while Lucy holds it. At the last moment, she would pull the ball away just as he was kicking. The off-balance Charlie Brown would sail into the air and land on his back with a loud thud. There was also the ever-present (and often executed) threat by Lucy to "slug" someone, especially her brother Linus. Though violence would happen from time to time, only once or twice was a boy ever depicted hitting a girl (Charlie Brown, who accidentally hit Lucy; when Lucy complained about it, Charlie Brown went down to her psychiatric booth where she returned the slug much harder) August 8, 1965.
Peanuts touched on religious themes on many occasions, most notably the classic television special A Charlie Brown Christmas in 1965, which features the character Linus van Pelt quoting the King James Version of the Bible (Luke 2:8–14) to explain to Charlie Brown what Christmas is all about (in personal interviews, Schulz mentioned that Linus represented his spiritual side). Because of the explicit religious material in A Charlie Brown Christmas, many have interpreted Schulz's work as having a distinct Christian theme, though the popular perspective has been to view the franchise through a secular lens.[16]
In the late 1970s, during Schulz 's negotiations with United Feature Syndicate over a new contract, syndicate president William C. Payette hired comics artist Al Plastino to draw a backlog of Peanuts strips to hold in reserve in case Schulz left the strip. When Schulz and the syndicate reached a successful agreement, United Media stored these unpublished strips, the existence of which eventually became public.[17]
Though Garfield and Calvin and Hobbes rivaled Peanuts in popularity in the 1980s and the 1990s respectively, the strip still remained the most popular comic in history.[18]
The daily Peanuts strips were formatted in a four-panel "space saving" format beginning in the 1950s, with a few very rare eight-panel strips, that still fit into the four-panel mold. In 1975, the panel format was shortened slightly horizontally, and shortly after the lettering became larger to accommodate the shrinking format. Beginning on Leap Day in 1988, Schulz abandoned the four-panel format in favor of three-panel dailies and occasionally used the entire length of the strip as one panel, partly for experimentation, but also to combat the dwindling size of the comics page.
Schulz continued the strip until he had to retire because of health reasons; he died the day before the final Sunday strip was published.
The final daily original Peanuts comic strip was published on January 3, 2000. At that point, five more original Sunday Peanuts strips had yet to be published.
On February 13, 2000, the day following Schulz's death, the last ever Peanuts strip ran in papers. The strip began with Charlie Brown answering the phone with someone on the end presumably asking for Snoopy. Charlie Brown responded with "No, I think he's writing." The bottom panel consisted of the final daily strip in its entirety, reprinted in color, and included various Peanuts characters surrounding it. The very last panel consisted simply of Snoopy sitting at his typewriter in thought with a note from Schulz that read as follows:
Dear Friends,
I have been fortunate to draw Charlie Brown and his friends for almost fifty years. It has been the fulfillment of my childhood ambition.
Unfortunately, I am no longer able to maintain the schedule demanded by a daily comic strip. My family does not wish "Peanuts" to be continued by anyone else, therefore I am announcing my retirement.
I have been grateful over the years for the loyalty of our editors and the wonderful support and love expressed to me by fans of the comic strip.
Charlie Brown, Snoopy, Linus, Lucy... how can I ever forget them...
— Charles M. Schulz
Fittingly, Charlie Brown was the only character to appear in both the first strip in 1950 and the last in 2000.
Many other cartoonists paid tribute to Peanuts and Schulz by homages in their own strips, appearing on February 13, 2000 or in the week beforehand. The comic was reprinted the day after that, but only had the farewell letter.
After Peanuts came to an end, United Feature Syndicate began offering the newspapers that ran it a package of reprinted strips under the title Classic Peanuts. The syndicate limited the choices to either strips from the 1960s or from the 1990s, although a newspaper was also given the option to carry both reprint packages if it desired. All Sunday strips in the package, however, come from the 1960s.
Despite the end of the strip, Peanuts continues to be prevalent in multiple media, through widespread syndication, the publication of The Complete Peanuts, the release of several new television specials (all of which Schulz had worked on, but had not finished, before his death), and Peanuts Motion Comics. Additionally, BOOM! Studios has published a series of comic books, which feature new material by new writers and artists, although some of the new material is based on classic Schulz stories from decades past, as well as including some classic strips by Schulz, mostly Sunday color strips.
On June 3, 2010, United Media sold all its Peanuts-related assets, including its strips and branding, to a new company, Peanuts Worldwide, LLC, a joint venture of the Iconix Brand Group (which owns 80%) and Charles M. Schulz Creative Associates (20%). In addition, United Media sold its United Media Licensing arm, which represents licensing for its other properties, to Peanuts Worldwide.[19][20] United Feature Syndicate continued to syndicate the strip, until February 27, 2011, when Universal Uclick took over syndication, ending United Media's 60-plus-years stewardship of Peanuts.[21] (Universal Uclick would take over all of United Media's syndicated offerings later that year.)
The initial cast of Peanuts was small, featuring only Charlie Brown, Shermy, Patty (not to be confused with Peppermint Patty) and (two days after the release of the first strip) a beagle, Snoopy. The first addition, Violet, was made on February 7, 1951. Other character introductions that soon followed were Schroeder, on May 30, 1951, as a baby; Lucy, on March 3, 1952; Lucy's baby brother Linus, on September 19, 1952 (after his existence was first mentioned back on July 14); and Pig-Pen, on July 13, 1954.
Though the strip did not have a lead character at the onset, it soon began to focus on Charlie Brown, a character developed from some of the painful experiences of Schulz's formative years. In early strips Charlie Brown was depicted as distinctly younger than his cohorts Patty and Shermy. Charlie Brown's main characteristic is either self-defeating stubbornness or admirable determined persistence to try his best against all odds: he can never win a ballgame but continues playing baseball; he can never fly a kite successfully but continues trying to do so. Though his inferiority complex was evident from the start, in the earlier strips he also got in his own jabs when verbally sparring with Patty and Shermy. Some early strips also involved romantic attractions between Charlie Brown and Patty or Violet. On September 1, 1958 Charlie Brown's father was formally revealed to be a barber (after earlier instances in the strip that linked Charlie Brown to barbers by implication). In 1960, the now popular line of Charlie Brown greeting cards was introduced by Hallmark Cards. Charlie Brown and Snoopy reached new heights on May 18, 1969 as they became the names of the command module and lunar module, respectively, for Apollo 10.
As the years went by, Shermy in particular, but Patty and Violet as well, appeared less often and were demoted to supporting roles (eventually disappearing from the strip in 1969, 1976, and 1984 respectively, although Patty and Violet were still seen as late as April 9, 1995), while new major characters were introduced. Schroeder, Lucy van Pelt, and her brother Linus debuted as very young children—with Schroeder and Linus both in diapers and pre-verbal. Snoopy also started to verbalize his thoughts via thought bubbles.
One recurring theme in the strip is Charlie Brown's neighborhood baseball team. Charlie Brown is the player-manager of the team and, usually, its pitcher, and Schroeder is the catcher. The other characters of the strip comprise the rest of the team, including Linus playing as second baseman and Lucy as right fielder. Charlie Brown is a terrible pitcher, often giving up tremendous hits which either knock him off the mound or leave him with only his shorts on. The team itself is also poor, with only Snoopy, at shortstop, being particularly competent. Because of this, the team consistently loses. However, while the team is often referred to as "win-less," it does win at least ten games over the course of the strip's run, most of these when Charlie Brown is not playing, a fact that Charlie Brown finds highly dispiriting.[22]
In the 1960s, Snoopy, a character previously minor, began to get attention. Many of the strips from this point revolve around Snoopy's active, Walter Mitty-like fantasy life, in which he would imagine himself in various roles, including a World War One Flying Ace, a World Famous author, a college student named Joe Cool. Although the names give his imaginary roles self-praise, he fails in his fantasies.
Schulz continued to introduce new characters into the strip, particularly including a tomboyish, freckle-faced, shorts-and-sandals-wearing girl named Patricia Reichardt, better known as Peppermint Patty. Peppermint Patty is an assertive, athletic but rather obtuse girl who shakes up Charlie Brown's world by calling him "Chuck", flirting with him and giving him compliments he is not so sure he deserves. She also brings in a new group of friends (and heads a rival baseball team), including the strip's first black character, Franklin; a Mexican-Swedish kid named José Peterson, and Peppermint Patty's bookish sidekick Marcie, who calls Peppermint Patty "Sir" and Charlie Brown "Charles" and sometimes "Chuck" (most characters only referred to him as "Charlie Brown", though he was known as "Charles" to Eudora, "big brother" to his sister Sally Brown, "that round-headed kid" to Snoopy, and as "Brownie Charles" to Peggy Jean after misspeaking his name out of nervousness).
Several additional family members of the characters were also introduced: Charlie Brown's younger sister Sally, who became fixated on Linus; Linus and Lucy van Pelt's younger brother Rerun, who almost always found himself on the back of his mother's bike for a time; and Spike, Snoopy's desert-dwelling brother from Needles, California, who was apparently named for Schulz's own childhood dog.[23] Snoopy also had six other siblings, and a total of five of his siblings made some appearances in the strip (his brothers Andy, Olaf, and Marbles (in addition to Spike) and his sister Belle).
Other notable characters include Snoopy's friend Woodstock, a bird whose chirping is represented in print as hash marks but is nevertheless clearly understood by Snoopy; three of Woodstock's buddies who usually appeared when on a scouting trip with Snoopy as their scout leader; Pig-Pen, the perpetually dirty boy who could raise a cloud of dust on a clean sidewalk, in a snowstorm, or inside a building; and Frieda, a girl proud of her "naturally curly hair," and who owned a cat named Faron, much to Snoopy's chagrin (the way Faron hung over Frieda's arms prompted Snoopy to comment that they had "finally developed a boneless cat"). Frieda eventually disappeared from the strip.
Peanuts had several recurring characters that were actually absent from view. Some, such as the Great Pumpkin or Manfred von Richthofen (the Red Baron), were merely figments of the cast's imaginations. Others were not imaginary, such as the Little Red-Haired Girl (Charlie Brown's perennial dream girl who finally appeared in 1998, but only in silhouette), Joe Shlabotnik (Charlie Brown's baseball hero), World War II (the vicious cat who lives next door to Snoopy—not to be confused with Frieda's cat, Faron), and Charlie Brown's unnamed pen pal, referred to as his "pencil-pal" after Charlie Brown's failed mastery of the fountain pen. Adult figures only appeared in the strip during a four-week Sunday-comic sequence in 1954 in which Lucy plays in an amateur golf tournament, with Charlie Brown "coaching" her. At no time, however, were any adult faces seen (it was also in this story that Lucy's family name, "van Pelt", was first revealed.) There are adult voices in a few of the strips in its early years.
Schulz also added some fantastical characters, sometimes imbuing inanimate objects with sparks of life. Charlie Brown's nemesis, the Kite-Eating Tree, is one example. Sally Brown's school building, that expressed thoughts and feelings about the students (and the general business of being a brick building), is another. Linus' famous "security blanket" also displayed occasional signs of anthropomorphism. Another example is Charlie Brown's pitching mound, which at times would express thoughts and opinions ("Why don't you learn how to pitch, you stupid kid?").
Schulz was born in Minneapolis, and his family moved to St. Paul when he was four years old, first living at 1680 James Avenue and then moving to 473 Macalester Street, Saint Paul, Minnesota. There are hints in the early strips that the setting for Peanuts is Schulz's boyhood home, near North James Avenue, Minneapolis; in one early strip, Schroeder mentions his address as "1770 James St.", and in another Lucy shows Charlie Brown a trophy she has just won, with the words "Outstanding Fussbudget of Hennepin County" on it (Hennepin County consists primarily of Minneapolis and its suburbs). In You're Not Elected, Charlie Brown, Lucy receives a phone call from someone "looking for Harold in St. Paul". In the strip for March 23, 1984, Peppermint Patty loses her homework to a gust of wind and mentions that her English theme was last seen on Selby Avenue and her history paper is flying over Highland Park; both are locations in Saint Paul, Minnesota.
In a 1961 strip, Lucy illustrates flying from Texas to Singapore via the polar route by using Charlie Brown's head as a globe, implying that their location is in Texas.[24]
Some later strips tend to point to California, where Schulz lived the later part of his life, although regular snowfall can still be seen during the winter months. It's sometimes pointed out that "5" was a character introduced in 1963, whose full name was "555 95472." His family name and ZIP code, 95472, is the actual ZIP code for Sebastopol, California, in Sonoma County in Northern California where Schulz was living at the time that the character was introduced. However, within the context of the strip, it's clear that Sebastopol is where 5's family moved from, not where they are living now. Another clue that the location is California appears in "A Charlie Brown Celebration, where Charlie Brown is shown to be a fan of "Poly State" (California Polytechnic State University)
In the Schulz-authored collection Security Is a Thumb and a Blanket, Linus is shown hugging a sign that says "Pinetree Corners Population 3,260", with a caption that says "Security is having a home town." However, the name Pinetree Corners was never referenced in the actual comic strip; Schulz later admitted he used Pinetree Corners as an "inside joke", and that it was not really the kids' hometown.
Peanuts is regarded as one of the most influential and well-written comic strips of all time. Schulz received the National Cartoonist Society Humor Comic Strip Award for Peanuts in 1962, the Reuben Award in 1955 and 1964 (the first cartoonist to receive the honor twice), the Elzie Segar Award in 1980, and the Milton Caniff Lifetime Achievement Award in 1999. A Charlie Brown Christmas won a Peabody Award and an Emmy; Peanuts cartoon specials have received a total of two Peabody Awards and four Emmys. For his work on the strip, Charles Schulz is credited with a star on the Hollywood Walk of Fame and a place in the William Randolph Hearst Cartoon Hall of Fame. Peanuts was featured on the cover of Time Magazine on April 9, 1965, with the accompanying article praising the strip as being "the leader of a refreshing new breed that takes an unprecedented interest in the basics of life."[25]
Considered among the greatest comic strips of all time, Peanuts was declared second in a list of the greatest comics of the 20th century commissioned by Krazy Kat, a strip Schulz admired (and in fact was among his biggest inspirations), and he accepted the positioning in good grace, to the point of agreeing with the result.[27] In 2002 TV Guide declared Snoopy and Charlie Brown equal 8th[28] in their list of "Top 50 Greatest Cartoon Characters of All Time",[29] published to commemorate their 50th anniversary.
Cartoon tributes have appeared in other comic strips since Schulz's death in 2000, and are now displayed at the Charles Schulz Museum.[30] In May 2000, many cartoonists included a reference to Peanuts in their own strips. Originally planned as a tribute to Schulz's retirement, after his death that February it became a tribute to his life and career. Similarly, on October 30, 2005, several comic strips again included references to Peanuts, and specifically the It's the Great Pumpkin, Charlie Brown television special.
The December 1997 issue of The Comics Journal featured an extensive collection of testimonials to Peanuts. Over forty cartoonists, from mainstream newspaper cartoonists to underground, independent comic artists, shared reflections on the power and influence of Schulz's art. Gilbert Hernandez wrote "Peanuts was and still is for me a revelation. It's mostly from Peanuts where I was inspired to create the village of Palomar in Love and Rockets. Schulz's characters, the humor, the insight... gush, gush, gush, bow, bow, bow, grovel, grovel, grovel..." Tom Batiuk wrote "The influence of Charles Schulz on the craft of cartooning is so pervasive it is almost taken for granted." Batiuk also described the depth of emotion in Peanuts: "Just beneath the cheerful surface were vulnerabilities and anxieties that we all experienced, but were reluctant to acknowledge. By sharing those feelings with us, Schulz showed us a vital aspect of our common humanity, which is, it seems to me, the ultimate goal of great art."[31]
In 2001, the Sonoma County Board of Supervisors renamed the Sonoma County Airport, located a few miles northwest of Santa Rosa, California, the Charles M. Schulz Airport in his honor. The airport's logo features Snoopy as the World War I Flying Ace (goggles/scarf), taking to the skies on top of his red doghouse. A bronze statue of Charlie Brown and Snoopy stands in Depot Park in downtown Santa Rosa.[32]
Schulz was included in the touring exhibition "Masters of American Comics" based on his achievements in the art form while producing the strip. His gag work is hailed as being "psychologically complex," and his style on the strip is noted as being "perfectly in keeping with the style of its times."[13]
Despite the widespread acclaim generated by Peanuts as a whole, some critics have alleged a decline in the strip's quality in the later years of its run, as Schulz frequently digressed from the more cerebral socio-psychological themes that characterized his earlier work in favor of lighter, more whimsical fare. For example, in an essay published in the New York Press at the time of the final daily strip in January 2000, "Against Snoopy," Christopher Caldwell argued that the character of Snoopy, and the strip's increased focus on him in the 1970s, "went from being the strip's besetting artistic weakness to ruining it altogether".[33]
Video rights to all the films and TV specials were licensed by Media Home Entertainment and Kartes Video Communications in the 1980s, and by Paramount Home Entertainment from 1994 to 2007. The distribution rights to the TV specials are now with Warner Bros. Television and Warner Home Video, and managed by its classic animation division, while the theatrical films are still at Paramount, who produced the last two and acquired the first two through the merger of CBS, who in turn produced them via Cinema Center Films; the first two films were originally released to video by CBS/Fox Video and are now available through Paramount, with TV rights managed for the first two via CBS Television Distribution, and the latter two via Trifecta Entertainment and Media under Paramount's license.
In addition to the strip and numerous books, the Peanuts characters have appeared in animated form on television numerous times. This started when the Ford Motor Company licensed the characters in early 1959 for a series of color television commercials for its automobiles and intros for The Tennessee Ernie Ford Show which they sponsored. While the show ended in 1961, the deal lasted another three years. The ads were animated by Bill Meléndez for Playhouse Pictures, a cartoon studio that had Ford as a client. Schulz and Meléndez became friends, and when producer Lee Mendelson decided to make a two-minute animated sequence for a TV documentary called A Boy Named Charlie Brown in 1963, he brought on Meléndez for the project. Before the documentary was completed, the three of them (with help from their sponsor, the Coca-Cola Company) produced their first half-hour animated special, the Emmy- and Peabody Award-winning A Charlie Brown Christmas, which was first aired on the CBS network on December 9, 1965. This episode is undoubtedly the most widely recognized of all Peanuts TV specials. This came after Coca-Cola asked Mendelson if he had a Christmas special. He said "yes." The next day he called Schulz up and said they were making A Charlie Brown Christmas.
The animated version of Peanuts differs in some aspects from the strip. In the strip, adult voices are heard, though conversations are usually only depicted from the children's end. To translate this aspect to the animated medium, the sound of a trombone with a solotone mute (created by Vince Guaraldi and performed by Dean Hubbard[34]) was used to simulate adult "voices." A more significant deviation from the strip was the treatment of Snoopy. In the strip, the dog's thoughts are verbalized in thought balloons; in animation, he is typically mute, his thoughts communicated through growls or laughs (voiced by Bill Meléndez), and pantomime, or by having human characters verbalizing his thoughts for him. These treatments have both been abandoned temporarily in the past. For example, they experimented with teacher dialogue in She's a Good Skate, Charlie Brown. The elimination of Snoopy's "voice" is probably the most controversial aspect of the adaptations, but Schulz apparently approved of the treatment. (Snoopy's thoughts were conveyed in voice over for the first time in the animated adaptations of the Broadway musicals You're a Good Man, Charlie Brown and Snoopy!!! The Musical, and later on occasion in the animated series The Charlie Brown and Snoopy Show.) The success of A Charlie Brown Christmas was the impetus for CBS to air many more prime-time Peanuts specials over the years, beginning with Charlie Brown's All-Stars and It's the Great Pumpkin, Charlie Brown in 1966. In total, more than thirty animated specials were produced. Until his death in 1976, jazz pianist Vince Guaraldi composed musical scores for the specials; in particular, the piece "Linus and Lucy" which has become popularly known as the signature theme song of the Peanuts franchise.
In addition to Coca-Cola, other companies that sponsored Peanuts specials over the years included Dolly Madison cakes, Kellogg's, McDonald's, Peter Paul-Cadbury candy bars, General Mills and Nabisco.
Schulz, Mendelson and Meléndez also collaborated on four theatrical feature films starring the characters, the first of which was A Boy Named Charlie Brown (1969). Most of these made use of material from Schulz's strips, which were then adapted, although in other cases plots were developed around areas where there were minimal strips to reference. Such was also the case with The Charlie Brown and Snoopy Show, a Saturday-morning TV series which debuted on CBS in 1983 and aired for two seasons.
By the mid-1990s, the specials' popularity had begun to wane, and CBS showed disinterest in new specials, even rejecting It's Spring Training, Charlie Brown completely. An eight-episode TV miniseries called This is America, Charlie Brown, for instance, was released during a writer's strike. Also, NBC aired You're in the Super Bowl, Charlie Brown in 1994 (the first special not to air on CBS) ten days before Super Bowl XXVIII. Eventually, the last Peanuts specials made during Schulz's lifetime were released direct-to-video, and no new ones were created until after the year 2000 when ABC obtained the rights to the three fall holiday specials. The Nickelodeon cable network re-aired a package of most of the specials produced before 1992, as well as The Charlie Brown and Snoopy Show and This Is America, Charlie Brown, under the umbrella title You're on Nickelodeon, Charlie Brown between 1997 and 2001. Eight Peanuts-based specials have been made posthumously. Of these, three are tributes to Peanuts or other Peanuts specials, and five are completely new specials based on dialogue from the strips and ideas given to ABC by Schulz before his death. He's a Bully, Charlie Brown, was telecast on ABC on November 20, 2006, following a repeat broadcast of A Charlie Brown Thanksgiving. Airing 43 years after the first special, the premiere of He's a Bully, Charlie Brown was watched by nearly 10 million viewers, winning its time slot and beating a Madonna concert special.[35] In the 2010 Macy's Thanksgiving Day Parade, it was announced that a new Peanuts animated special, Happiness Is a Warm Blanket, Charlie Brown, would debut in 2011. The special was released on DVD first, on March 29, 2011, and later premiered on Fox, on November 24, 2011.
Many of the specials and feature films have also been released on various home video formats over the years. To date, 20 of the specials, the two films A Boy Named Charlie Brown and Snoopy, Come Home, and the miniseries This Is America, Charlie Brown have all been released to DVD.
In October 2007, Warner Bros. acquired the Peanuts catalog from Paramount for an undisclosed amount of money. As aforementioned, they now hold the worldwide distribution rights for all Peanuts properties including over 50 television specials—these are originally managed by Warner Bros. Television and Warner Bros. Television Animation. Warner has made plans to develop new specials for television as well as the direct to video market, as well as short subjects for digital distribution, and some of these have in fact already been released via Warner Premiere.[36] Paramount retains the rights to the theatrical releases, as the first two movies (A Boy Named Charlie Brown and Snoopy, Come Home) are owned by CBS and distributed for home video through Paramount while CTD distributes for television, and the other two (Race for Your Life, Charlie Brown and Bon Voyage, Charlie Brown (and Don't Come Back!!)) were handled in-house by Paramount, with Trifecta holding TV distribution rights.
In October 2012, it was announced that a 3D computer-animated[37] feature film titled The Peanuts Movie will be released on November 6, 2015, coinciding with the 65th anniversary of the debut of the comic strip and the 50th anniversary of the television special, A Charlie Brown Christmas. Written by Charles Shulz's son, Craig, his grandson, Bryan, and Cornelius Uliano—who are also producing the film alongside Paul Feig[38]—it will be animated by 20th Century Fox's Blue Sky Studios, and directed by Steve Martino, the director of Horton Hears a Who! and Ice Age: Continental Drift.[39][40] A trailer for the film was released on March 18, 2014. The same day, it was also announced that archive recordings of Bill Melendez will be used to make up Snoopy and Woodstock's dialogue.
Peanuts Worldwide has partnered with Normaal Animation and France Télévisions to produce 500 90-second animated short films, adapted from the strip Peanuts, which will air in fall 2014, including on French television station France 3.[41]
The Peanuts characters have been featured in many books over the years.[42] Some represented chronological reprints of the newspaper strip, while others were thematic collections such as Snoopy's Tennis Book, or collections of inspirational adages such as Happiness Is a Warm Puppy. Some single-story books were produced, such as Snoopy and the Red Baron. In addition, many of the animated television specials and feature films were adapted into book form.
The first set of reprints was produced by Rinehart & Company beginning in 1952 (the company later merged into Holt, Rinehart and Winston in 1960, which continued the series thereafter), with the release of a collection simply titled Peanuts. This reprint series, which presented the strips in rough chronological order (albeit with many strips omitted each year) through the 1980s, after which reprint rights were handed off to various other publishers. Ballantine Books published the last original series of Peanuts reprints, including Peanuts 2000, which collected the final year of the strip's run.
Coinciding with these reprints were smaller paperback collections published by Fawcett Publications. Drawing material from the main reprints, the paperback series began with The Wonderful World of Peanuts in 1962 and continued through Get Back to Nature, Snoopy in 1990.
Charles Schulz had always resisted republication of the earliest Peanuts strips, as they did not reflect the characters as he eventually developed them. However, in 1997 he began talks with Fantagraphics Books to have the entire run of the strip, which would end up with 17,897 strips total, published chronologically in book form.[43] In addition to the post-millennium Peanuts publications are BOOM! Studios restyling of the comics and activity books, and "First Appearances" series. Of which its content produced by Peanuts Studio, subsequently an arm of Peanuts Worldwide LLC.
The entire run of Peanuts, covering nearly 50 years of comic strips, is being reprinted in Fantagraphics' The Complete Peanuts, a 25-volume set to be released over a 12-year period, two volumes per year, published every May and October. The first volume (collecting strips from 1950 to 1952) was published in May 2004; the final volume (which will include all the strips from 1999 and seven strips from 2000; plans for the remaining space are unknown) is expected to be published in May 2016.[45] A companion series, titled Peanuts Every Sunday and presenting the complete Sunday strips in color (as the main Complete Peanuts books reproduce them in black and white only), was launched in December 2013; this series will run ten volumes, with the last expected to be published in 2022.
In addition, almost all Peanuts strips are now also legally available online at GoComics.com[46] (there are some strips missing from the digital archive). Peanuts strips were previously featured on Comics.com.
Several books have been released to commemorate key anniversaries of Peanuts:
Peanuts characters even found their way to the live stage, appearing in the musicals You're a Good Man, Charlie Brown and Snoopy!!! — The Musical, and in "Snoopy on Ice", a live Ice Capades-style show aimed primarily at young children, all of which have had several touring productions over the years.[48] You're a Good Man, Charlie Brown was originally a successful Off-Broadway musical that ran for four years (1967–1971) in New York City and on tour, with Gary Burghoff as the original Charlie Brown. An updated revival opened on Broadway in 1999, and by 2002 it had become the most frequently produced musical in American theatre history.[4] It was also adapted for television twice, as a live-action NBC special and an animated CBS special. Snoopy!!! The Musical was a musical comedy based on the Peanuts comic strip, originally performed at Lamb's Theatre Off-Broadway in 1982. In its 1983 run in London's West End, it won an Olivier Award. In 1988, it was adapted into an animated TV special. The New Players Theatre in London staged a revival in 2004 to honor its 21st anniversary, but some reviewers noted that its "feel good" sentiments had not aged well. The Off-Broadway drama Dog Sees God: Confessions of a Teenage Blockhead centers on the Peanuts characters becoming teenagers, though it is an unauthorized parody.[49]
In 1962 Columbia Records issued an album titled Peanuts, with Kaye Ballard and Arthur Siegel performing (as Lucy and Charlie Brown, respectively) to music composed by Fred Karlin.
Fantasy Records issued several albums featuring Vince Guaraldi's jazz scores from the animated specials, including Jazz Impressions of a Boy Named Charlie Brown (1964), A Charlie Brown Christmas (1965), Oh, Good Grief! (1968), Charlie Brown's Holiday Hits (1998), and Peanuts Portraits (2010). All were later reissued on CD.
Columbia Records released soundtrack albums for the films A Boy Named Charlie Brown (1969) and Snoopy, Come Home (1972), although neither has been reissued on CD.
Other jazz artists have recorded Peanuts-themed albums, often featuring cover versions of Guaraldi's compositions. These include David Benoit (Here's to You, Charlie Brown!, 2000, and Jazz for Peanuts, 2008); and Cyrus Chestnut (A Charlie Brown Christmas, 2000). The 1989 GRP Records release Happy Anniversary, Charlie Brown! and the 2005 Peak Records release 40 Years: A Charlie Brown Christmas also include interpretations of Guaraldi's themes by various smooth jazz and blues artists.
The 1960s American rock band The Royal Guardsmen recorded several songs about Snoopy's fantasies of flying against the Red Baron in World War I, including the hit singles "Snoopy Vs. The Red Baron" and "Snoopy's Christmas." The first song was released without Schulz's consent, and he and UFS sued successfully for royalties, but allowed the group to make future songs and even contributed album artwork for such releases as Snoopy and His Friends.
Cast recordings (in both original and revival productions) of the stage musicals You're a Good Man, Charlie Brown and Snoopy!!! The Musical have been released over the years.
Numerous animated Peanuts specials were adapted into book-and-record sets, issued on the "Charlie Brown Records" label by Disney Read-Along in the 1970s and '80s. Also issued on Charlie Brown Records, via Disneyland Records, was the soundtrack to Flashbeagle in 1984, which featured Desiree Goyette and Joey Scarbury (of "Theme from the Greatest American Hero" fame) on the title track and other songs written by Ed Bogas and Goyette.
In 1992 RCA Victor released an album of classical piano music ostensibly performed by Schroeder himself. Titled Schroeder's Greatest Hits, the album contains solo piano works by Beethoven, Brahms, Chopin, and others, performed by John Miller, Ronnie Zito, Ken Bichel, and Nelly Kokinos.
In 1983, Knott's Berry Farm, in Southern California, was the first theme park to license the Peanuts characters, creating the first Camp Snoopy area and making Snoopy the park's mascot. Knott's expanded its operation in 1992 by building an indoor amusement park in the Mall of America, called Knott's Camp Snoopy. The Knott's theme parks were acquired by the national amusement park chain Cedar Fair Entertainment Company in 1997, which continued to operate Knott's Camp Snoopy park until the mall took over its operation in March 2005.[50] Cedar Fair had already licensed the Peanuts characters for use in 1992 as atmosphere,[51] so its acquisition of Knott's Berry Farm did not alter the use of those characters.
Snoopy is currently the official mascot of all the Cedar Fair parks. It was previously used in all of the park logos but it has since been removed. Cedar Fair also operated a Camp Snoopy area at Dorney Park & Wildwater Kingdom, Worlds of Fun, and Valleyfair featuring various Peanuts-themed attractions until 2011. There is still a Camp Snoopy area at Cedar Point and Knott's Berry Farm.
In 2008, Cedar Point introduced Planet Snoopy, a children's area where Peanuts Playground used to be. This area consists of family and children rides that were relocated from Cedar Point's sister park Geauga Lake after its closing. The rides are themed around the Peanuts characters. The area also consists of a "Kids Only" restaurant called Joe Cool Cafe (there is a small menu for adults). In 2010, the Nickelodeon Central and Nickelodeon Universe areas in the former Paramount Parks (California's Great America, Canada's Wonderland, Carowinds, Kings Dominion, and Kings Island) were replaced by Planet Snoopy . In 2011, Cedar Fair announced it would also add Planet Snoopy to Valleyfair, Dorney Park & Wildwater Kingdom, and Worlds of Fun, replacing the Camp Snoopy areas. Planet Snoopy is now at every Cedar Fair park besides Knott's Berry Farm and Michigan's Adventure.
Over the years, the Peanuts characters have appeared in ads for Dolly Madison snack cakes, Chex Mix, Bounty, Cheerios, A&W Root Beer, Kraft Foods, and Ford automobiles.[52][53][54] Pig-Pen appeared in a memorable spot for Regina vacuum cleaners.[55]
They are currently spokespeople in print and television advertisements for the MetLife insurance company.[56] MetLife usually uses Snoopy in its advertisements as opposed to other characters: for instance, the MetLife blimps are named "Snoopy One" and "Snoopy Two" and feature him in his World War I flying ace persona.[57] Starting in 2011, the annual New York Giants and New York Jets preseason football game is called the MetLife Bowl. The winning team receives a trophy which depicts Snoopy carrying a football, wearing an old-style football helmet and holding his arm up as if to ward off a tackler.[58]
The characters have been featured on Hallmark Cards since 1960,[59] and can be found adorning clothing, figurines, plush dolls, flags, balloons, posters, Christmas ornaments, and countless other bits of licensed merchandise.[60][61][62][63][64]
The Apollo 10 lunar module was nicknamed "Snoopy" and the command module "Charlie Brown".[65] While not included in the official mission logo, Charlie Brown and Snoopy became semi-official mascots for the mission.[66][67] Charles Schulz drew an original picture of Charlie Brown in a spacesuit that was hidden aboard the craft to be found by the astronauts once they were in orbit. This drawing is now on display at the Kennedy Space Center. Snoopy is the personal safety mascot for NASA astronauts,[68] and NASA issues a Silver Snoopy award to employees that promote flight safety. The black "caps" that the Apollo astronauts wore on their heads were referred to as "Snoopy hats."
In East Asia, RM Licensing has licensed the Peanuts characters for fashion stores, children's apparel, and restaurants[69] including Charlie Brown Cafe, a Hong Kong-based fast causal restaurant chain.[70] There is also a Charlie Brown Cafe, themed with Peanuts characters, in Hongdae, Mapo-gu, Seoul, South Korea.[71]
The 1960s pop band The Royal Guardsmen drew inspiration from Peanuts, and their single "Snoopy vs. the Red Baron" reached number two on the charts.[72]
In the 1960s, Robert L. Short interpreted certain themes and conversations in Peanuts as being consistent with parts of Christian theology, and used them as illustrations during his lectures about the gospel, and as source material for several books, as he explained in his bestselling paperback book, The Gospel According to Peanuts.
In 1980, Charles Schulz was introduced to artist Tom Everhart during a collaborative art project.[73] Everhart became fascinated with Schulz's art style and worked Peanuts themed art into his own work. Schulz encouraged Everhart to continue with his work. Everhart continues to be the only artist authorized to paint Peanuts characters.[74]
Giant helium balloons of Charlie Brown and Snoopy have long been a feature in the annual Macy's Thanksgiving Day Parade in New York City. This was referenced in a Super Bowl XLII commercial for Coca-Cola, in which the Charlie Brown balloon snags a Coca-Cola bottle from two battling balloons (Underdog and Stewie Griffin).
Peanuts on Parade is St. Paul, Minnesota's tribute to Peanuts.[75] It began in 2000, with the placing of 101 5-foot-tall (1.5 m) statues of Snoopy throughout the city of Saint Paul. The statues were later auctioned at the Mall of America in Bloomington, Minnesota. In 2001, there was "Charlie Brown Around Town", 2002 brought "Looking for Lucy", and in 2003, "Linus Blankets Saint Paul".[76] Permanent bronze statues of the Peanuts characters are found in Landmark Plaza in downtown Saint Paul.[77]
The Peanuts characters have been licensed to Universal Studios Japan (while Peanuts merchandise in Japan has been licensed by Sanrio, best known for Hello Kitty).[78]
The Peanuts characters have appeared in various video games, such as Snoopy in a 1984 by Radarsoft, Snoopy and the Red Baron for the Atari 2600, Snoopy's Silly Sports Spectacular (1989, Nintendo Entertainment System), Snoopy's Magic Show (1990, Game Boy), Snoopy Tennis (2001, Game Boy Color), Snoopy Concert which was released in 1995 and sold to the Japanese market for the Super NES, and in October 2006, a second game titled Snoopy vs. The Red Baron by Namco Bandai for the PlayStation 2. In July 2007, the Peanuts characters appeared in the Snoopy the Flying Ace mobile-phone game by Namco Networks.
In 1980 (with a new edition published in 1990), the Funk & Wagnalls publishing house also produced a children's encyclopedia called the Charlie Brown's 'Cyclopedia. The 15-volume set features many of the Peanuts characters.
In April 2002, The Peanuts Collectors Edition Monopoly board game was released by USAopoly. The game was created by Justin Gage, a prolific collector and friend of Charles and Jeannie Schulz. The game was dedicated to Schulz in memory of his passing.
Cbs, Peanuts, Vince Guaraldi, Christmas traditions, Animation
Nickelodeon, Knott's Berry Farm, Nick Jr., Cedar Point, Valleyfair
Zamperla, Kansas City, Missouri, Knott's Berry Farm, Intamin, HUSS Park Attractions
Intamin, HUSS Park Attractions, HalloWeekends, Bolliger & Mabillard, National Register of Historic Places
Peanuts, Garfield, List of Peanuts media, The Critic, Family Guy
The Simpsons, The Flintstones, Dvd, Peanuts, United Kingdom
Peanuts, Cbs, The Flintstones, The Critic, Animation
Peanuts, Upn, The Critic, Animation, Family Guy
The Simpsons, Futurama, Family Guy, South Park, Loren Bouchard


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BATON ROUGE, La. (AP) — Baton Rouge-based Turner Industries is the target of a federal lawsuit alleging widespread discrimination against black employees and a hostile working environment in which the use of racial slurs is commonplace.
The lawsuit, filed Wednesday in Middle District Court in Baton Rouge on behalf of former employee Enrico Williams, alleges that Turner Industries fired Williams after learning that he had signed an affidavit corroborating charges by other co-workers of discriminatory practices. William's statement was part of a civil lawsuit filed last year against Tuner Industries by workers in Texas and Louisiana.
Williams, who is black, was a former supervisor and had worked for the company for 14 years, said Jay Ellwanger, an attorney representing Williams.
Ellwanger said Thursday that his client saw "first-hand an environment of discrimination and retaliation that shouldn't be allowed to continue."
The unfair treatment alleged includes discriminatory lay-offs and capricious disciplinary action against black workers, according to the suit.
Phyllis Cancienne, an attorney for Turner Industries, said Thursday that the company would not comment on the pending litigation. The company, with about 15,000 employees, provides skilled laborers to industrial plants.
The lawsuit, which mirrors legal action pending in federal court in Texas and Baton Rouge involving more than 200 former employees, alleges that Turner Industries has a history of ignoring complaints of discriminatory practices and the hostile treatment of minority workers.
The lawsuit alleges black workers were routine targets of racial slurs by co-workers and managers and subjected to racially offensive drawings and graffiti and nooses "throughout Turner facilities and job sites."
In 2010 the Equal Employment Opportunity Commission investigated worker complaints at a Paris, Texas, facility and found that blacks were subjected to racial slurs and intimidation. According to the EEOC investigation, managers at the plant were aware of the hostile environment, but targeted workers who complained and disciplined white employees who opposed the harassment.
In October, a Texas jury rendered a $4 million verdict against the company in an employment discrimination suit filed by workers at the Paris plant, said Ellwanger, who represented those employees.
"This lawsuit is another volley is this war that we hope Turner Industries will listen to," Ellwanger said Thursday.
To bring attention to the suit, the Louisiana NAACP held a rally and news conference in front of the federal courthouse on Thursday. Among the group were former employees of Turner Industries who carried pictures of hanging nooses and signs with scrawled racial slurs, images they said were placed in work sites in Louisiana and Texas.
"Nothing was ever done about it," said Leamos Byrd, a Baton Rouge carpenter who said he was fired in 2005 after reporting racial slurs and a work bus incident in which co-workers brought a noose on board.
Ernest Johnson, state chairman of the NAACP, called the allegations shameful and said the group will continue to oppose discrimination.
"We want Turner Industries to know that we will continue to fight until they get it right!"
In addition to back pay for Williams, the Baton Rouge suit also seeks to require Turner Industries to provide diversity training for managers, employment discrimination training for all employees and to have human resources representatives located at all Turner facilities.


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Late last year, the FIA announced a new race series called the FIA GT3 European Championship. Team VETTE attended the GT3 press conference at Monte Carlo, where a total of nine manufacturers expressed interest in fielding entries for the series. The teams would be required to enter three cars at each of the '06 season's five races.
According to GT3 Director Stephane Ratel, GT3 would be "a gentlemen's race series." This meant that no professional race drivers would be allowed to compete unless they were over the age of 55. All five events were to include two one-hour races, and teams would be required to perform a driver change during each race.
Callaway Competition, in Leingarten, Germany, was contracted by the Netherlands' Team GLPK Carsport to build six GT3-spec Z06 Corvettes. Connecticut-based Callaway Cars provided engineering and technical support. Work began in late November, and all six cars-dubbed Z06.Rs-were on the grid May 7 for the first race event at England's historic Silverstone circuit. (Three were purchased by France's Team Riverside, while the remaining three are campaigned by a joint effort known as Team Carsport/Callaway.)
Unfortunately, the Silverstone races did not go well for the Vettes, as rain and mechanical gremlins kept the cars out of the top finishing positions. With the season opener behind them, the teams could only hope for a better showing at the series' next round in Oschersleben, Germany.
Oschersleben is a small city located in the farmlands of the former East Germany. The 2.8-mile, nine-turn Oschersleben Motorsports Arena, which opened in 1997, is a tight, twisty course that requires drivers to run over high curbs in order to log quick lap times. Forty-one GT3 entries, including the six Z06.Rs, arrived at the track hoping to qualify for a top spot.
Prior to the race, the FIA had ruled the Z06.Rs were too fast and ordered that the cars run both 260 pounds of ballast and a 10mm inlet restrictor. Oschersleben would be the Corvettes' first race under these new restrictions.
When Carsport/Callaway driver Uli Martini's regular copilot, Klaus Ludwig, was unable to make the race, Berberich and the team offered the open seat to five-time SCCA T-1 National Champion John Heinricy. John, who is the director of GM's Performance Division, was in Germany working on a project and quickly accepted.
The FIA scheduled two 20-minute qualifying runs for the Saturday session-one for that afternoon's race and another for the Sunday morning event. For Race One, Team Riverside qualified 22nd, 23rd, and 24th, while Carsport/Callaway took the 28th and 34th spots. In Race Two qualifying, Riverside secured the first, third, and fifth starting spots, while Carsport/Callaway grabbed seventh and twenty-second.
Forty-one starters took the green flag for Saturday's race under clear skies. The first 20 minutes saw plenty of banging and spinning among the competitors, and the safety car was brought out to clear up the mess. Most cars, including the Corvettes, pitted during the caution and changed drivers. When the race resumed, the No. 19 Riverside car of Rabineau/Ruffier and the No. 16 Berberich/Heinricy Carsport/Callaway were running third and fourth, respectively, overall.
And that is how they finished, with only 2.24 seconds separating First and Fourth place. Martini and Heinricy were all smiles, and spectators were quick to congratulate the American driver on his impressive European debut.
Race Two was held Sunday morning before a record crowd. When the green flag fell, the pole-sitting No. 19 Corvette streaked out to an impressive lead. Heinricy moved from seventh into third place behind an Aston Martin and stayed in that position until he pitted for a driver change. During the driver change it was learned that the Z06.R had lost its clutch at the start. Incredibly, Heinricy had driven the first half of the race without a clutch!
Berberich managed to pull No. 16 to within eighth place before spinning on the last lap while avoiding another car. By the time he got restarted, he had fallen to 13th. The No. 19 Corvette, meanwhile, had pulled out to a 30-second lead, which it held through the end of the race. Unfortunately, the team's crew had to help start the car during the driver change, incurring a 30-second penalty. This dropped No. 19 from First to Third overall, 27 seconds behind the winning Aston Martin DBR9. Corvettes finished in the Third, Ninth, Thirteenth, Twenty-seventh, and Thirty-second places.
In spite of FIA's "performance adjustment" penalty, the Z06.Rs are showing themselves as formidable competitors against the world's best exotic cars. According to Heinricy, "The car has a lot of grip in mid- and high-speed corners. It tends to understeer in slow corners, but that can be easily controlled with the throttle. The weight penalty and the engine restrictor make the engine feel like a stock LS7 with a good aero package and slick tires. It was a blast to drive, and I am ready to do it again."
Racer Profile: John Heinricy
John Heinricy believes he has a dream job, and we tend to agree. John holds the position of Director of High Performance Vehicle Operations for GM's Performance Division. He and his team are instrumental in developing, building, and testing every high-performance vehicle that GM sells. It is in the latter capacity that Heinricy and his staff make frequent trips to Germany's Eiffel region, home of the famous Nordschleife Nrburgring. John has personally made over 600 laps around this 13-mile, 170-turn race track.
Heinricy joined GM in 1970, after graduating from The South Dakota School of Mines and Technology. He has held many desirable positions during his long career, including that of vehicle-development manager for the Corvette platform.
As if managing the team responsible for GM's performance image wasn't enough, John also plays with cars in his down-time. Since getting started in racing in 1984, he has won eight SCCA National Championships, including five consec-utive Touring 1 titles in a Phoenix Motorsports/Goodyear-sponsored C5 Z06.
Heinricy has also won numerous 24-hour races with Morrison Motorsports and competed at the Daytona 24 Hours and the 12 Hours of Sebring in Morrison Corvettes. He was even part of the team that drove a stock, Morrison-prepped ZR-1 to an FIA speed record at Fort Stockton, Texas, in 1990. The car averaged 174.5 mph for 24 hours, breaking a 50-year-old record held by Mercedes Benz.
Heinricy was also a factory team driver for the '99 C5-R racing effort, and he has notched several Second Place finishes at the wheel of a SPEED GT Cadillac CTS-V. Recently, he drove a Carsport/Callaway Z06.R to three impressive finishes in European GT3 racing.
John lives in Michigan with his wife, Rita.