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About *Calculus Volume 3*

*Calculus* is designed for the typical two- or three-semester general calculus course, incorporating innovative features to enhance student learning. The book guides students through the core concepts of calculus and helps them understand how those concepts apply to their lives and the world around them. Due to the comprehensive nature of the material, we are offering the book in three volumes for flexibility and efficiency. Volume 3 covers parametric equations and polar coordinates, vectors, functions of several variables, multiple integration, and second-order differential equations.

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Introduction

Parametric Equations

Parametric Equations and Their Graphs

Consider the orbit of Earth around the Sun. Our year lasts approximately 365.25 days, but for this discussion we will use 365 days. On January 1 of each year, the physical location of Earth with respect to the Sun is nearly the same, except for leap years, when the lag introduced by the extra $\frac{1}{4}$ day of orbiting time is built into the calendar. We call January 1 “day 1” of the year. Then, for example, day 31 is January 31, day 59 is February 28, and so on.

Eliminating the Parameter

To better understand the graph of a curve represented parametrically, it is useful to rewrite the two equations as a single equation relating the variables *x* and *y.* Then we can apply any previous knowledge of equations of curves in the plane to identify the curve. For example, the equations describing the plane curve in [link]b. are

Cycloids and Other Parametric Curves

Imagine going on a bicycle ride through the country. The tires stay in contact with the road and rotate in a predictable pattern. Now suppose a very determined ant is tired after a long day and wants to get home. So he hangs onto the side of the tire and gets a free ride. The path that this ant travels down a straight road is called a cycloid ([link]). A cycloid generated by a circle (or bicycle wheel) of radius *a* is given by the parametric equations

Calculus of Parametric Curves

Derivatives of Parametric Equations

We start by asking how to calculate the slope of a line tangent to a parametric curve at a point. Consider the plane curve defined by the parametric equations

Second-Order Derivatives

Our next goal is to see how to take the second derivative of a function defined parametrically. The second derivative of a function $y=f\left(x\right)$ is defined to be the derivative of the first derivative; that is,

Now that we have seen how to calculate the derivative of a plane curve, the next question is this: How do we find the area under a curve defined parametrically? Recall the cycloid defined by the equations $x\left(t\right)=t-\text{sin}{\rule{0.2em}{0ex}}t,{\rule{1em}{0ex}}y\left(t\right)=1-\text{cos}{\rule{0.2em}{0ex}}t.$ Suppose we want to find the area of the shaded region in the following graph.

Arc Length of a Parametric Curve

In addition to finding the area under a parametric curve, we sometimes need to find the arc length of a parametric curve. In the case of a line segment, arc length is the same as the distance between the endpoints. If a particle travels from point *A* to point *B* along a curve, then the distance that particle travels is the arc length. To develop a formula for arc length, we start with an approximation by line segments as shown in the following graph.

Recall the problem of finding the surface area of a volume of revolution. In Curve Length and Surface Area, we derived a formula for finding the surface area of a volume generated by a function $y=f\left(x\right)$ from $x=a$ to $x=b,$ revolved around the *x*-axis:

Polar Coordinates

Defining Polar Coordinates

To find the coordinates of a point in the polar coordinate system, consider [link]. The point $P$ has Cartesian coordinates $\left(x,y\right).$ The line segment connecting the origin to the point $P$ measures the distance from the origin to $P$ and has length $r.$ The angle between the positive $x$-axis and the line segment has measure $\theta .$ This observation suggests a natural correspondence between the coordinate pair $\left(x,y\right)$ and the values $r$ and $\theta .$ This correspondence is the basis of the polar coordinate system. Note that every point in the Cartesian plane has two values (hence the term *ordered pair*) associated with it. In the polar coordinate system, each point also has two values associated with it: $r$ and $\theta .$

Polar Curves

Now that we know how to plot points in the polar coordinate system, we can discuss how to plot curves. In the rectangular coordinate system, we can graph a function $y=f\left(x\right)$ and create a curve in the Cartesian plane. In a similar fashion, we can graph a curve that is generated by a function $r=f\left(\theta \right).$

Symmetry in Polar Coordinates

When studying symmetry of functions in rectangular coordinates (i.e., in the form $y=f\left(x\right)\right),$ we talk about symmetry with respect to the *y*-axis and symmetry with respect to the origin. In particular, if $f\left(\text{−}x\right)=f\left(x\right)$ for all $x$ in the domain of $f,$ then $f$ is an even function and its graph is symmetric with respect to the *y*-axis. If $f\left(\text{−}x\right)=\text{−}f\left(x\right)$ for all $x$ in the domain of $f,$ then $f$ is an odd function and its graph is symmetric with respect to the origin. By determining which types of symmetry a graph exhibits, we can learn more about the shape and appearance of the graph. Symmetry can also reveal other properties of the function that generates the graph. Symmetry in polar curves works in a similar fashion.

We have studied the formulas for area under a curve defined in rectangular coordinates and parametrically defined curves. Now we turn our attention to deriving a formula for the area of a region bounded by a polar curve. Recall that the proof of the Fundamental Theorem of Calculus used the concept of a Riemann sum to approximate the area under a curve by using rectangles. For polar curves we use the Riemann sum again, but the rectangles are replaced by sectors of a circle.

Arc Length in Polar Curves

Here we derive a formula for the arc length of a curve defined in polar coordinates.

Conic Sections

Parabolas

A parabola is generated when a plane intersects a cone parallel to the generating line. In this case, the plane intersects only one of the nappes. A parabola can also be defined in terms of distances.

Ellipses

An ellipse can also be defined in terms of distances. In the case of an ellipse, there are two foci (plural of focus), and two directrices (plural of directrix). We look at the directrices in more detail later in this section.

Hyperbolas

A hyperbola can also be defined in terms of distances. In the case of a hyperbola, there are two foci and two directrices. Hyperbolas also have two asymptotes.

Eccentricity and Directrix

An alternative way to describe a conic section involves the directrices, the foci, and a new property called eccentricity. We will see that the value of the eccentricity of a conic section can uniquely define that conic.

Polar Equations of Conic Sections

Sometimes it is useful to write or identify the equation of a conic section in polar form. To do this, we need the concept of the focal parameter. The focal parameter of a conic section *p* is defined as the distance from a focus to the nearest directrix. The following table gives the focal parameters for the different types of conics, where *a* is the length of the semi-major axis (i.e., half the length of the major axis), *c* is the distance from the origin to the focus, and *e* is the eccentricity. In the case of a parabola, *a* represents the distance from the vertex to the focus.

General Equations of Degree Two

A general equation of degree two can be written in the form

Vectors in Space

Introduction

Vectors in the Plane

Abstract:
* Describe a plane vector, using correct notation.
* Perform basic vector operations (scalar multiplication, addition, subtraction).
* Express a vector in component form.
* Explain the formula for the magnitude of a vector.
* Express a vector in terms of unit vectors.
* Give two examples of vector quantities.

Vector Representation

A vector in a plane is represented by a directed line segment (an arrow). The endpoints of the segment are called the initial point and the terminal point of the vector. An arrow from the initial point to the terminal point indicates the direction of the vector. The length of the line segment represents its magnitude. We use the notation $‖\text{v}‖$ to denote the magnitude of the vector $\text{v}.$ A vector with an initial point and terminal point that are the same is called the zero vector, denoted $0.$ The zero vector is the only vector without a direction, and by convention can be considered to have any direction convenient to the problem at hand.

Combining Vectors

Vectors have many real-life applications, including situations involving force or velocity. For example, consider the forces acting on a boat crossing a river. The boat’s motor generates a force in one direction, and the current of the river generates a force in another direction. Both forces are vectors. We must take both the magnitude and direction of each force into account if we want to know where the boat will go.

Vector Components

Working with vectors in a plane is easier when we are working in a coordinate system. When the initial points and terminal points of vectors are given in Cartesian coordinates, computations become straightforward.

Unit Vectors

A unit vector is a vector with magnitude $1.$ For any nonzero vector $\text{v},$ we can use scalar multiplication to find a unit vector $\text{u}$ that has the same direction as $\text{v}.$ To do this, we multiply the vector by the reciprocal of its magnitude:

Applications of Vectors

Because vectors have both direction and magnitude, they are valuable tools for solving problems involving such applications as motion and force. Recall the boat example and the quarterback example we described earlier. Here we look at two other examples in detail.

Vectors in Three Dimensions

Abstract:
* Describe three-dimensional space mathematically.
* Locate points in space using coordinates.
* Write the distance formula in three dimensions.
* Write the equations for simple planes and spheres.
* Perform vector operations in ${ℝ}^{3}.$

Three-Dimensional Coordinate Systems

As we have learned, the two-dimensional rectangular coordinate system contains two perpendicular axes: the horizontal *x*-axis and the vertical *y*-axis. We can add a third dimension, the *z*-axis, which is perpendicular to both the *x*-axis and the *y*-axis. We call this system the three-dimensional rectangular coordinate system. It represents the three dimensions we encounter in real life.

Writing Equations in ℝ3

Now that we can represent points in space and find the distance between them, we can learn how to write equations of geometric objects such as lines, planes, and curved surfaces in ${ℝ}^{3}.$ First, we start with a simple equation. Compare the graphs of the equation $x=0$ in $ℝ,{ℝ}^{2},{\rule{0.2em}{0ex}}\text{and}{\rule{0.2em}{0ex}}{ℝ}^{3}$ ([link]). From these graphs, we can see the same equation can describe a point, a line, or a plane.

Working with Vectors in ℝ3

Just like two-dimensional vectors, three-dimensional vectors are quantities with both magnitude and direction, and they are represented by directed line segments (arrows). With a three-dimensional vector, we use a three-dimensional arrow.

The Dot Product

Abstract:
* Calculate the dot product of two given vectors.
* Determine whether two given vectors are perpendicular.
* Find the direction cosines of a given vector.
* Explain what is meant by the vector projection of one vector onto another vector, and describe how to compute it.
* Calculate the work done by a given force.

The Dot Product and Its Properties

We have already learned how to add and subtract vectors. In this chapter, we investigate two types of vector multiplication. The first type of vector multiplication is called the dot product, based on the notation we use for it, and it is defined as follows:

Using the Dot Product to Find the Angle between Two Vectors

When two nonzero vectors are placed in standard position, whether in two dimensions or three dimensions, they form an angle between them ([link]). The dot product provides a way to find the measure of this angle. This property is a result of the fact that we can express the dot product in terms of the cosine of the angle formed by two vectors.

Projections

As we have seen, addition combines two vectors to create a resultant vector. But what if we are given a vector and we need to find its component parts? We use vector projections to perform the opposite process; they can break down a vector into its components. The magnitude of a vector projection is a scalar projection. For example, if a child is pulling the handle of a wagon at a 55° angle, we can use projections to determine how much of the force on the handle is actually moving the wagon forward ([link]). We return to this example and learn how to solve it after we see how to calculate projections.

Work

Now that we understand dot products, we can see how to apply them to real-life situations. The most common application of the dot product of two vectors is in the calculation of work.

The Cross Product

Abstract:
* Calculate the cross product of two given vectors.
* Use determinants to calculate a cross product.
* Find a vector orthogonal to two given vectors.
* Determine areas and volumes by using the cross product.
* Calculate the torque of a given force and position vector.

The Cross Product and Its Properties

The dot product is a multiplication of two vectors that results in a scalar. In this section, we introduce a product of two vectors that generates a third vector orthogonal to the first two. Consider how we might find such a vector. Let $\text{u}=〈{u}_{1},{u}_{2},{u}_{3}〉$
and $\text{v}=〈{v}_{1},{v}_{2},{v}_{3}〉$
be nonzero vectors. We want to find a vector $\text{w}=〈{w}_{1},{w}_{2},{w}_{3}〉$ orthogonal to both $\text{u}$ and $\text{v}$—that is, we want to find $\text{w}$ such that $\text{u}·\text{w}=0$ and $\text{v}·\text{w}=0.$ Therefore, ${w}_{1},$ ${w}_{2},$ and ${w}_{3}$ must satisfy

Determinants and the Cross Product

Using [link] to find the cross product of two vectors is straightforward, and it presents the cross product in the useful component form. The formula, however, is complicated and difficult to remember. Fortunately, we have an alternative. We can calculate the cross product of two vectors using determinant notation.

Using the Cross Product

The cross product is very useful for several types of calculations, including finding a vector orthogonal to two given vectors, computing areas of triangles and parallelograms, and even determining the volume of the three-dimensional geometric shape made of parallelograms known as a *parallelepiped*. The following examples illustrate these calculations.

The Triple Scalar Product

Because the cross product of two vectors is a vector, it is possible to combine the dot product and the cross product. The dot product of a vector with the cross product of two other vectors is called the triple scalar product because the result is a scalar.

Applications of the Cross Product

The cross product appears in many practical applications in mathematics, physics, and engineering. Let’s examine some of these applications here, including the idea of torque, with which we began this section. Other applications show up in later chapters, particularly in our study of vector fields such as gravitational and electromagnetic fields (Introduction to Vector Calculus).

Equations of Lines and Planes in Space

Abstract:
* Write the vector, parametric, and symmetric equations of a line through a given point in a given direction, and a line through two given points.
* Find the distance from a point to a given line.
* Write the vector and scalar equations of a plane through a given point with a given normal.
* Find the distance from a point to a given plane.
* Find the angle between two planes.

Equations for a Line in Space

Let’s first explore what it means for two vectors to be parallel. Recall that parallel vectors must have the same or opposite directions. If two nonzero vectors, $\text{u}$ and $\text{v},$ are parallel, we claim there must be a scalar, $k,$ such that $\text{u}=k\text{v}.$ If $\text{u}$ and $\text{v}$ have the same direction, simply choose $k=\frac{‖\text{u}‖}{‖\text{v}‖}.$ If $\text{u}$ and $\text{v}$ have opposite directions, choose $k=-\frac{‖\text{u}‖}{‖\text{v}‖}.$ Note that the converse holds as well. If $\text{u}=k\text{v}$ for some scalar $\text{k},$ then either $\text{u}$ and $\text{v}$ have the same direction $\left(k>0\right)$ or opposite directions $\left(k<0\right),$ so $\text{u}$ and $\text{v}$ are parallel. Therefore, two nonzero vectors $\text{u}$ and $\text{v}$ are parallel if and only if $\text{u}=k\text{v}$ for some scalar $\text{k}.$ By convention, the zero vector $0$ is considered to be parallel to all vectors.

Distance between a Point and a Line

We already know how to calculate the distance between two points in space. We now expand this definition to describe the distance between a point and a line in space. Several real-world contexts exist when it is important to be able to calculate these distances. When building a home, for example, builders must consider “setback” requirements, when structures or fixtures have to be a certain distance from the property line. Air travel offers another example. Airlines are concerned about the distances between populated areas and proposed flight paths.

Relationships between Lines

Given two lines in the two-dimensional plane, the lines are equal, they are parallel but not equal, or they intersect in a single point. In three dimensions, a fourth case is possible. If two lines in space are not parallel, but do not intersect, then the lines are said to be skew lines ([link]).

Equations for a Plane

We know that a line is determined by two points. In other words, for any two distinct points, there is exactly one line that passes through those points, whether in two dimensions or three. Similarly, given any three points that do not all lie on the same line, there is a unique plane that passes through these points. Just as a line is determined by two points, a plane is determined by three.

Parallel and Intersecting Planes

We have discussed the various possible relationships between two lines in two dimensions and three dimensions. When we describe the relationship between two planes in space, we have only two possibilities: the two distinct planes are parallel or they intersect. When two planes are parallel, their normal vectors are parallel. When two planes intersect, the intersection is a line ([link]).

Quadric Surfaces

Abstract:
* Identify a cylinder as a type of three-dimensional surface.
* Recognize the main features of ellipsoids, paraboloids, and hyperboloids.
* Use traces to draw the intersections of quadric surfaces with the coordinate planes.

Identifying Cylinders

The first surface we’ll examine is the cylinder. Although most people immediately think of a hollow pipe or a soda straw when they hear the word *cylinder*, here we use the broad mathematical meaning of the term. As we have seen, cylindrical surfaces don’t have to be circular. A rectangular heating duct is a cylinder, as is a rolled-up yoga mat, the cross-section of which is a spiral shape.

Quadric Surfaces

We have learned about surfaces in three dimensions described by first-order equations; these are planes. Some other common types of surfaces can be described by second-order equations. We can view these surfaces as three-dimensional extensions of the conic sections we discussed earlier: the ellipse, the parabola, and the hyperbola. We call these graphs quadric surfaces.

Cylindrical and Spherical Coordinates

Abstract:
* Convert from cylindrical to rectangular coordinates.
* Convert from rectangular to cylindrical coordinates.
* Convert from spherical to rectangular coordinates.
* Convert from rectangular to spherical coordinates.

Cylindrical Coordinates

When we expanded the traditional Cartesian coordinate system from two dimensions to three, we simply added a new axis to model the third dimension. Starting with polar coordinates, we can follow this same process to create a new three-dimensional coordinate system, called the cylindrical coordinate system. In this way, cylindrical coordinates provide a natural extension of polar coordinates to three dimensions.

Spherical Coordinates

In the Cartesian coordinate system, the location of a point in space is described using an ordered triple in which each coordinate represents a distance. In the cylindrical coordinate system, location of a point in space is described using two distances $\left(r{\rule{0.2em}{0ex}}\text{and}{\rule{0.2em}{0ex}}z\right)$ and an angle measure $\left(\theta \right).$ In the spherical coordinate system, we again use an ordered triple to describe the location of a point in space. In this case, the triple describes one distance and two angles. Spherical coordinates make it simple to describe a sphere, just as cylindrical coordinates make it easy to describe a cylinder. Grid lines for spherical coordinates are based on angle measures, like those for polar coordinates.

Vector-Valued Functions

Introduction

Vector-Valued Functions and Space Curves

Abstract:
* Write the general equation of a vector-valued function in component form and unit-vector form.
* Recognize parametric equations for a space curve.
* Describe the shape of a helix and write its equation.
* Define the limit of a vector-valued function.

Definition of a Vector-Valued Function

Our first step in studying the calculus of vector-valued functions is to define what exactly a vector-valued function is. We can then look at graphs of vector-valued functions and see how they define curves in both two and three dimensions.

Graphing Vector-Valued Functions

Recall that a plane vector consists of two quantities: direction and magnitude. Given any point in the plane (the *initial point*), if we move in a specific direction for a specific distance, we arrive at a second point. This represents the *terminal point* of the vector. We calculate the components of the vector by subtracting the coordinates of the initial point from the coordinates of the terminal point.

Limits and Continuity of a Vector-Valued Function

We now take a look at the limit of a vector-valued function. This is important to understand to study the calculus of vector-valued functions.

Calculus of Vector-Valued Functions

Abstract:
* Write an expression for the derivative of a vector-valued function.
* Find the tangent vector at a point for a given position vector.
* Find the unit tangent vector at a point for a given position vector and explain its significance.
* Calculate the definite integral of a vector-valued function.

Derivatives of Vector-Valued Functions

Now that we have seen what a vector-valued function is and how to take its limit, the next step is to learn how to differentiate a vector-valued function. The definition of the derivative of a vector-valued function is nearly identical to the definition of a real-valued function of one variable. However, because the range of a vector-valued function consists of vectors, the same is true for the range of the derivative of a vector-valued function.

Tangent Vectors and Unit Tangent Vectors

Recall from the Introduction to Derivatives that the derivative at a point can be interpreted as the slope of the tangent line to the graph at that point. In the case of a vector-valued function, the derivative provides a tangent vector to the curve represented by the function. Consider the vector-valued function $\text{r}\left(t\right)=\text{cos}{\rule{0.1em}{0ex}}t{\rule{0.1em}{0ex}}\text{i}+\text{sin}{\rule{0.1em}{0ex}}t{\rule{0.1em}{0ex}}\text{j}.$ The derivative of this function is ${r}^{\prime }\left(t\right)=-\text{sin}{\rule{0.1em}{0ex}}t{\rule{0.1em}{0ex}}\text{i}+\text{cos}{\rule{0.1em}{0ex}}t{\rule{0.1em}{0ex}}\text{j}.$ If we substitute the value $t=\pi \text{/}6$ into both functions we get

Integrals of Vector-Valued Functions

We introduced antiderivatives of real-valued functions in Antiderivatives and definite integrals of real-valued functions in The Definite Integral. Each of these concepts can be extended to vector-valued functions. Also, just as we can calculate the derivative of a vector-valued function by differentiating the component functions separately, we can calculate the antiderivative in the same manner. Furthermore, the Fundamental Theorem of Calculus applies to vector-valued functions as well.

Arc Length and Curvature

Abstract:
* Determine the length of a particle’s path in space by using the arc-length function.
* Explain the meaning of the curvature of a curve in space and state its formula.
* Describe the meaning of the normal and binormal vectors of a curve in space.

Arc Length for Vector Functions

We have seen how a vector-valued function describes a curve in either two or three dimensions. Recall [link], which states that the formula for the arc length of a curve defined by the parametric functions $x=x\left(t\right),y=y\left(t\right),{t}_{1}\le t\le {t}_{2}$ is given by

Arc-Length Parameterization

We now have a formula for the arc length of a curve defined by a vector-valued function. Let’s take this one step further and examine what an arc-length function is.

Curvature

An important topic related to arc length is curvature. The concept of curvature provides a way to measure how sharply a smooth curve turns. A circle has constant curvature. The smaller the radius of the circle, the greater the curvature.

The Normal and Binormal Vectors

We have seen that the derivative ${r}^{\prime }\left(t\right)$ of a vector-valued function is a tangent vector to the curve defined by $\text{r}\left(t\right),$ and the unit tangent vector $\text{T}\left(t\right)$ can be calculated by dividing ${r}^{\prime }\left(t\right)$ by its magnitude. When studying motion in three dimensions, two other vectors are useful in describing the motion of a particle along a path in space: the principal unit normal vector and the binormal vector.

Motion in Space

Abstract:
* Describe the velocity and acceleration vectors of a particle moving in space.
* Explain the tangential and normal components of acceleration.
* State Kepler’s laws of planetary motion.

Motion Vectors in the Plane and in Space

Our starting point is using vector-valued functions to represent the position of an object as a function of time. All of the following material can be applied either to curves in the plane or to space curves. For example, when we look at the orbit of the planets, the curves defining these orbits all lie in a plane because they are elliptical. However, a particle traveling along a helix moves on a curve in three dimensions.

Components of the Acceleration Vector

We can combine some of the concepts discussed in Arc Length and Curvature with the acceleration vector to gain a deeper understanding of how this vector relates to motion in the plane and in space. Recall that the unit tangent vector **T** and the unit normal vector **N** form an osculating plane at any point *P* on the curve defined by a vector-valued function $\text{r}\left(t\right).$ The following theorem shows that the acceleration vector $\text{a}\left(t\right)$ lies in the osculating plane and can be written as a linear combination of the unit tangent and the unit normal vectors.

Projectile Motion

Now let’s look at an application of vector functions. In particular, let’s consider the effect of gravity on the motion of an object as it travels through the air, and how it determines the resulting trajectory of that object. In the following, we ignore the effect of air resistance. This situation, with an object moving with an initial velocity but with no forces acting on it other than gravity, is known as projectile motion. It describes the motion of objects from golf balls to baseballs, and from arrows to cannonballs.

Kepler’s Laws

During the early 1600s, Johannes Kepler was able to use the amazingly accurate data from his mentor Tycho Brahe to formulate his three laws of planetary motion, now known as Kepler’s laws of planetary motion. These laws also apply to other objects in the solar system in orbit around the Sun, such as comets (e.g., Halley’s comet) and asteroids. Variations of these laws apply to satellites in orbit around Earth.

Differentiation of Functions of Several Variables

Introduction

Functions of Several Variables

Abstract:
* Recognize a function of two variables and identify its domain and range.
* Sketch a graph of a function of two variables.
* Sketch several traces or level curves of a function of two variables.
* Recognize a function of three or more variables and identify its level surfaces.

Functions of Two Variables

The definition of a function of two variables is very similar to the definition for a function of one variable. The main difference is that, instead of mapping values of one variable to values of another variable, we map ordered pairs of variables to another variable.

Graphing Functions of Two Variables

Suppose we wish to graph the function $z=\left(x,y\right).$ This function has two independent variables $\left(x{\rule{0.2em}{0ex}}\text{and}{\rule{0.2em}{0ex}}y\right)$ and one dependent variable $\left(z\right).$ When graphing a function $y=f\left(x\right)$ of one variable, we use the Cartesian plane. We are able to graph any ordered pair $\left(x,y\right)$ in the plane, and every point in the plane has an ordered pair $\left(x,y\right)$ associated with it. With a function of two variables, each ordered pair $\left(x,y\right)$ in the domain of the function is mapped to a real number $z.$ Therefore, the graph of the function $f$ consists of ordered triples $\left(x,y,z\right).$ The graph of a function $z=\left(x,y\right)$ of two variables is called a surface.

Level Curves

If hikers walk along rugged trails, they might use a topographical map that shows how steeply the trails change. A topographical map contains curved lines called *contour lines*. Each contour line corresponds to the points on the map that have equal elevation ([link]). A level curve of a function of two variables $f\left(x,y\right)$ is completely analogous to a contour line on a topographical map.

Functions of More Than Two Variables

So far, we have examined only functions of two variables. However, it is useful to take a brief look at functions of more than two variables. Two such examples are

Limits and Continuity

Abstract:
* Calculate the limit of a function of two variables.
* Learn how a function of two variables can approach different values at a boundary point, depending on the path of approach.
* State the conditions for continuity of a function of two variables.
* Verify the continuity of a function of two variables at a point.
* Calculate the limit of a function of three or more variables and verify the continuity of the function at a point.

Limit of a Function of Two Variables

Recall from The Limit of a Function the definition of a limit of a function of one variable:

Interior Points and Boundary Points

To study continuity and differentiability of a function of two or more variables, we first need to learn some new terminology.

Continuity of Functions of Two Variables

In Continuity, we defined the continuity of a function of one variable and saw how it relied on the limit of a function of one variable. In particular, three conditions are necessary for $f\left(x\right)$ to be continuous at point $x=a\text{:}$

Functions of Three or More Variables

The limit of a function of three or more variables occurs readily in applications. For example, suppose we have a function $f\left(x,y,z\right)$ that gives the temperature at a physical location $\left(x,y,z\right)$ in three dimensions. Or perhaps a function $g\left(x,y,z,t\right)$ can indicate air pressure at a location $\left(x,y,z\right)$ at time $t.$ How can we take a limit at a point in ${ℝ}^{3}?$ What does it mean to be continuous at a point in four dimensions?

Partial Derivatives

Abstract:
* Calculate the partial derivatives of a function of two variables.
* Calculate the partial derivatives of a function of more than two variables.
* Determine the higher-order derivatives of a function of two variables.
* Explain the meaning of a partial differential equation and give an example.

Derivatives of a Function of Two Variables

When studying derivatives of functions of one variable, we found that one interpretation of the derivative is an instantaneous rate of change of $y$ as a function of $x.$ Leibniz notation for the derivative is $dy\text{/}dx,$ which implies that $y$ is the dependent variable and $x$ is the independent variable. For a function $z=f\left(x,y\right)$ of two variables, $x$ and $y$ are the independent variables and $z$ is the dependent variable. This raises two questions right away: How do we adapt Leibniz notation for functions of two variables? Also, what is an interpretation of the derivative? The answer lies in partial derivatives.

Functions of More Than Two Variables

Suppose we have a function of three variables, such as $w=f\left(x,y,z\right).$ We can calculate partial derivatives of $w$ with respect to any of the independent variables, simply as extensions of the definitions for partial derivatives of functions of two variables.

Higher-Order Partial Derivatives

Consider the function

Partial Differential Equations

In Introduction to Differential Equations, we studied differential equations in which the unknown function had one independent variable. A partial differential equation is an equation that involves an unknown function of more than one independent variable and one or more of its partial derivatives. Examples of partial differential equations are

Tangent Planes and Linear Approximations

Abstract:
* Determine the equation of a plane tangent to a given surface at a point.
* Use the tangent plane to approximate a function of two variables at a point.
* Explain when a function of two variables is differentiable.
* Use the total differential to approximate the change in a function of two variables.

Tangent Planes

Intuitively, it seems clear that, in a plane, only one line can be tangent to a curve at a point. However, in three-dimensional space, many lines can be tangent to a given point. If these lines lie in the same plane, they determine the tangent plane at that point. A tangent plane at a regular point contains all of the lines tangent to that point. A more intuitive way to think of a tangent plane is to assume the surface is smooth at that point (no corners). Then, a tangent line to the surface at that point in any direction does not have any abrupt changes in slope because the direction changes smoothly.

Linear Approximations

Recall from Linear Approximations and Differentials that the formula for the linear approximation of a function $f\left(x\right)$ at the point $x=a$ is given by

Differentiability

When working with a function $y=f\left(x\right)$ of one variable, the function is said to be differentiable at a point $x=a$ if ${f}^{\prime }\left(a\right)$ exists. Furthermore, if a function of one variable is differentiable at a point, the graph is “smooth” at that point (i.e., no corners exist) and a tangent line is well-defined at that point.

Differentials

In Linear Approximations and Differentials we first studied the concept of differentials. The differential of $y,$ written $dy,$ is defined as ${f}^{\prime }\left(x\right)dx.$ The differential is used to approximate $\text{Δ}y=f\left(x+\text{Δ}x\right)-f\left(x\right),$ where $\text{Δ}x=dx.$ Extending this idea to the linear approximation of a function of two variables at the point $\left({x}_{0},{y}_{0}\right)$ yields the formula for the total differential for a function of two variables.

Differentiability of a Function of Three Variables

All of the preceding results for differentiability of functions of two variables can be generalized to functions of three variables. First, the definition:

The Chain Rule

Abstract:
* State the chain rules for one or two independent variables.
* Use tree diagrams as an aid to understanding the chain rule for several independent and intermediate variables.
* Perform implicit differentiation of a function of two or more variables.

Chain Rules for One or Two Independent Variables

Recall that the chain rule for the derivative of a composite of two functions can be written in the form

The Generalized Chain Rule

Now that we’ve see how to extend the original chain rule to functions of two variables, it is natural to ask: Can we extend the rule to more than two variables? The answer is yes, as the generalized chain rule states.

Implicit Differentiation

Recall from Implicit Differentiation that implicit differentiation provides a method for finding $dy\text{/}dx$ when $y$ is defined implicitly as a function of $x.$ The method involves differentiating both sides of the equation defining the function with respect to $x,$ then solving for $dy\text{/}dx.$ Partial derivatives provide an alternative to this method.

Directional Derivatives and the Gradient

Abstract:
* Determine the directional derivative in a given direction for a function of two variables.
* Determine the gradient vector of a given real-valued function.
* Explain the significance of the gradient vector with regard to direction of change along a surface.
* Use the gradient to find the tangent to a level curve of a given function.
* Calculate directional derivatives and gradients in three dimensions.

Directional Derivatives

We start with the graph of a surface defined by the equation $z=f\left(x,y\right).$ Given a point $\left(a,b\right)$ in the domain of $f,$ we choose a direction to travel from that point. We measure the direction using an angle $\theta ,$ which is measured counterclockwise in the *x*, *y*-plane, starting at zero from the positive *x*-axis ([link]). The distance we travel is $h$ and the direction we travel is given by the unit vector $u=\left(\text{cos}{\rule{0.2em}{0ex}}\theta \right)i+\left(\text{sin}{\rule{0.2em}{0ex}}\theta \right)j.$ Therefore, the *z*-coordinate of the second point on the graph is given by $z=f\left(a+h{\rule{0.2em}{0ex}}\text{cos}{\rule{0.2em}{0ex}}\theta ,b+h{\rule{0.2em}{0ex}}\text{sin}{\rule{0.2em}{0ex}}\theta \right).$

Gradient

The right-hand side of [link] is equal to ${f}_{x}\left(x,y\right)\text{cos}{\rule{0.2em}{0ex}}\theta +{f}_{y}\left(x,y\right)\text{sin}{\rule{0.2em}{0ex}}\theta ,$ which can be written as the dot product of two vectors. Define the first vector as $\nabla f\left(x,y\right)={f}_{x}\left(x,y\right)\text{i}+{f}_{y}\left(x,y\right)\text{j}$ and the second vector as $u=\left(\text{cos}{\rule{0.2em}{0ex}}\theta \right)i+\left(\text{sin}{\rule{0.2em}{0ex}}\theta \right)j.$ Then the right-hand side of the equation can be written as the dot product of these two vectors:

Gradients and Level Curves

Recall that if a curve is defined parametrically by the function pair $\left(x\left(t\right),y\left(t\right)\right),$ then the vector ${x}^{\prime }\left(t\right)i+{y}^{\prime }\left(t\right)j$ is tangent to the curve for every value of $t$ in the domain. Now let’s assume $z=f\left(x,y\right)$ is a differentiable function of $x{\rule{0.2em}{0ex}}\text{and}{\rule{0.2em}{0ex}}y,$ and $\left({x}_{0},{y}_{0}\right)$ is in its domain. Let’s suppose further that ${x}_{0}=x\left({t}_{0}\right)$ and ${y}_{0}=y\left({t}_{0}\right)$ for some value of $t,$ and consider the level curve $f\left(x,y\right)=k.$ Define $g\left(t\right)=f\left(x\left(t\right),y\left(t\right)\right)$ and calculate ${g}^{\prime }\left(t\right)$ on the level curve. By the chain Rule,

Three-Dimensional Gradients and Directional Derivatives

The definition of a gradient can be extended to functions of more than two variables.

Maxima/Minima Problems

Abstract:
* Use partial derivatives to locate critical points for a function of two variables.
* Apply a second derivative test to identify a critical point as a local maximum, local minimum, or saddle point for a function of two variables.
* Examine critical points and boundary points to find absolute maximum and minimum values for a function of two variables.

Critical Points

For functions of a single variable, we defined critical points as the values of the function when the derivative equals zero or does not exist. For functions of two or more variables, the concept is essentially the same, except for the fact that we are now working with partial derivatives.

Second Derivative Test

Consider the function $f\left(x\right)={x}^{3}.$ This function has a critical point at $x=0,$ since $f\prime \left(0\right)=3{\left(0\right)}^{2}=0.$ However, $f$ does not have an extreme value at $x=0.$ Therefore, the existence of a critical value at $x={x}_{0}$ does not guarantee a local extremum at $x={x}_{0}.$ The same is true for a function of two or more variables. One way this can happen is at a saddle point. An example of a saddle point appears in the following figure.

Absolute Maxima and Minima

When finding global extrema of functions of one variable on a closed interval, we start by checking the critical values over that interval and then evaluate the function at the endpoints of the interval. When working with a function of two variables, the closed interval is replaced by a closed, bounded set. A set is *bounded* if all the points in that set can be contained within a ball (or disk) of finite radius. First, we need to find the critical points inside the set and calculate the corresponding critical values. Then, it is necessary to find the maximum and minimum value of the function on the boundary of the set. When we have all these values, the largest function value corresponds to the global maximum and the smallest function value corresponds to the absolute minimum. First, however, we need to be assured that such values exist. The following theorem does this.

Lagrange Multipliers

Abstract:
* Use the method of Lagrange multipliers to solve optimization problems with one constraint.
* Use the method of Lagrange multipliers to solve optimization problems with two constraints.

Lagrange Multipliers

[link] was an applied situation involving maximizing a profit function, subject to certain constraints. In that example, the constraints involved a maximum number of golf balls that could be produced and sold in $1$ month $\left(x\right),$ and a maximum number of advertising hours that could be purchased per month $\left(y\right).$ Suppose these were combined into a budgetary constraint, such as $20x+4y\le 216,$ that took into account the cost of producing the golf balls and the number of advertising hours purchased per month. The goal is, still, to maximize profit, but now there is a different type of constraint on the values of $x$ and $y.$ This constraint, when combined with the profit function $f\left(x,y\right)=48x+96y-{x}^{2}-2xy-9{y}^{2},$ is an example of an optimization problem, and the function $f\left(x,y\right)$ is called the objective function. A graph of various level curves of the function $f\left(x,y\right)$ follows.

Problems with Two Constraints

The method of Lagrange multipliers can be applied to problems with more than one constraint. In this case the optimization function, $w$ is a function of three variables:

Multiple Integration

Introduction

Double Integrals over Rectangular Regions

Abstract:
* Recognize when a function of two variables is integrable over a rectangular region.
* Recognize and use some of the properties of double integrals.
* Evaluate a double integral over a rectangular region by writing it as an iterated integral.
* Use a double integral to calculate the area of a region, volume under a surface, or average value of a function over a plane region.

Volumes and Double Integrals

We begin by considering the space above a rectangular region *R*. Consider a continuous function $f\left(x,y\right)\ge 0$ of two variables defined on the closed rectangle *R*:

Properties of Double Integrals

The properties of double integrals are very helpful when computing them or otherwise working with them. We list here six properties of double integrals. Properties 1 and 2 are referred to as the linearity of the integral, property 3 is the additivity of the integral, property 4 is the monotonicity of the integral, and property 5 is used to find the bounds of the integral. Property 6 is used if $f\left(x,y\right)$ is a product of two functions $g\left(x\right)$ and $h\left(y\right).$

Iterated Integrals

So far, we have seen how to set up a double integral and how to obtain an approximate value for it. We can also imagine that evaluating double integrals by using the definition can be a very lengthy process if we choose larger values for $m$ and $n.$ Therefore, we need a practical and convenient technique for computing double integrals. In other words, we need to learn how to compute double integrals without employing the definition that uses limits and double sums.

Applications of Double Integrals

Double integrals are very useful for finding the area of a region bounded by curves of functions. We describe this situation in more detail in the next section. However, if the region is a rectangular shape, we can find its area by integrating the constant function $f\left(x,y\right)=1$ over the region $R.$

Double Integrals over General Regions

Abstract:
* Recognize when a function of two variables is integrable over a general region.
* Evaluate a double integral by computing an iterated integral over a region bounded by two vertical lines and two functions of $x,$ or two horizontal lines and two functions of $y.$
* Simplify the calculation of an iterated integral by changing the order of integration.
* Use double integrals to calculate the volume of a region between two surfaces or the area of a plane region.
* Solve problems involving double improper integrals.

General Regions of Integration

An example of a general bounded region $D$ on a plane is shown in [link]. Since $D$ is bounded on the plane, there must exist a rectangular region $R$ on the same plane that encloses the region $D,$ that is, a rectangular region $R$ exists such that $D$ is a subset of $R\left(D\subseteq R\right).$

Double Integrals over Nonrectangular Regions

To develop the concept and tools for evaluation of a double integral over a general, nonrectangular region, we need to first understand the region and be able to express it as Type I or Type II or a combination of both. Without understanding the regions, we will not be able to decide the limits of integrations in double integrals. As a first step, let us look at the following theorem.

Changing the Order of Integration

As we have already seen when we evaluate an iterated integral, sometimes one order of integration leads to a computation that is significantly simpler than the other order of integration. Sometimes the order of integration does not matter, but it is important to learn to recognize when a change in order will simplify our work.

Calculating Volumes, Areas, and Average Values

We can use double integrals over general regions to compute volumes, areas, and average values. The methods are the same as those in Double Integrals over Rectangular Regions, but without the restriction to a rectangular region, we can now solve a wider variety of problems.

Improper Double Integrals

An improper double integral is an integral $\underset{D}{\iint }f{\rule{0.2em}{0ex}}dA$ where either $D$ is an unbounded region or $f$ is an unbounded function. For example, $D=\left\{\left(x,y\right)||x-y|\ge 2\right\}$ is an unbounded region, and the function $f\left(x,y\right)=1\text{/}\left(1-{x}^{2}-2{y}^{2}\right)$ over the ellipse ${x}^{2}+2{y}^{2}\le 1$ is an unbounded function. Hence, both of the following integrals are improper integrals:

Double Integrals in Polar Coordinates

Abstract:
* Recognize the format of a double integral over a polar rectangular region.
* Evaluate a double integral in polar coordinates by using an iterated integral.
* Recognize the format of a double integral over a general polar region.
* Use double integrals in polar coordinates to calculate areas and volumes.

Polar Rectangular Regions of Integration

When we defined the double integral for a continuous function in rectangular coordinates—say, $g$ over a region $R$ in the $xy$-plane—we divided $R$ into subrectangles with sides parallel to the coordinate axes. These sides have either constant $x$-values and/or constant $y$-values. In polar coordinates, the shape we work with is a polar rectangle, whose sides have constant $r$-values and/or constant $\theta$-values. This means we can describe a polar rectangle as in [link](a), with $R=\left\{\left(r,\theta \right)|a\le r\le b,\alpha \le \theta \le \beta \right\}.$

General Polar Regions of Integration

To evaluate the double integral of a continuous function by iterated integrals over general polar regions, we consider two types of regions, analogous to Type I and Type II as discussed for rectangular coordinates in Double Integrals over General Regions. It is more common to write polar equations as $r=f\left(\theta \right)$ than $\theta =f\left(r\right),$ so we describe a general polar region as $D=\left\{\left(r,\theta \right)|\alpha \le \theta \le \beta ,{h}_{1}\left(\theta \right)\le r\le {h}_{2}\left(\theta \right)\right\}$ (see the following figure).

Polar Areas and Volumes

As in rectangular coordinates, if a solid $S$ is bounded by the surface $z=f\left(r,\theta \right),$ as well as by the surfaces $r=a,r=b,\theta =\alpha ,$ and $\theta =\beta ,$ we can find the volume $V$ of $S$ by double integration, as

Triple Integrals

Abstract:
* Recognize when a function of three variables is integrable over a rectangular box.
* Evaluate a triple integral by expressing it as an iterated integral.
* Recognize when a function of three variables is integrable over a closed and bounded region.
* Simplify a calculation by changing the order of integration of a triple integral.
* Calculate the average value of a function of three variables.

Integrable Functions of Three Variables

We can define a rectangular box $B$ in ${ℝ}^{3}$ as $B=\left\{\left(x,y,z\right)|a\le x\le b,c\le y\le d,e\le z\le f\right\}.$ We follow a similar procedure to what we did in Double Integrals over Rectangular Regions. We divide the interval $\left[a,b\right]$ into $l$ subintervals $\left[{x}_{i-1},{x}_{i}\right]$ of equal length $\text{Δ}x=\frac{b-a}{l},$ divide the interval $\left[c,d\right]$ into $m$ subintervals $\left[{y}_{j-1},{y}_{j}\right]$ of equal length $\text{Δ}y=\frac{d-c}{m},$ and divide the interval $\left[e,f\right]$ into $n$ subintervals $\left[{z}_{k-1},{z}_{k}\right]$ of equal length $\text{Δ}z=\frac{f-e}{n}.$ Then the rectangular box $B$ is subdivided into $lmn$ subboxes ${B}_{ijk}=\left[{x}_{i-1},{x}_{i}\right]{\rule{0.2em}{0ex}}×{\rule{0.2em}{0ex}}\left[{y}_{j-1},{y}_{j}\right]{\rule{0.2em}{0ex}}×{\rule{0.2em}{0ex}}\left[{z}_{k-1},{z}_{k}\right],$ as shown in [link].

Triple Integrals over a General Bounded Region

We now expand the definition of the triple integral to compute a triple integral over a more general bounded region $E$ in ${ℝ}^{3}.$ The general bounded regions we will consider are of three types. First, let $D$ be the bounded region that is a projection of $E$ onto the $xy$-plane. Suppose the region $E$ in ${ℝ}^{3}$ has the form

Changing the Order of Integration

As we have already seen in double integrals over general bounded regions, changing the order of the integration is done quite often to simplify the computation. With a triple integral over a rectangular box, the order of integration does not change the level of difficulty of the calculation. However, with a triple integral over a general bounded region, choosing an appropriate order of integration can simplify the computation quite a bit. Sometimes making the change to polar coordinates can also be very helpful. We demonstrate two examples here.

Average Value of a Function of Three Variables

Recall that we found the average value of a function of two variables by evaluating the double integral over a region on the plane and then dividing by the area of the region. Similarly, we can find the average value of a function in three variables by evaluating the triple integral over a solid region and then dividing by the volume of the solid.

Triple Integrals in Cylindrical and Spherical Coordinates

Abstract:
* Evaluate a triple integral by changing to cylindrical coordinates.
* Evaluate a triple integral by changing to spherical coordinates.

Review of Cylindrical Coordinates

As we have seen earlier, in two-dimensional space ${ℝ}^{2},$ a point with rectangular coordinates $\left(x,y\right)$ can be identified with $\left(r,\theta \right)$ in polar coordinates and vice versa, where $x=r{\rule{0.2em}{0ex}}\text{cos}{\rule{0.2em}{0ex}}\theta ,$ $y=r{\rule{0.2em}{0ex}}\text{sin}{\rule{0.2em}{0ex}}\theta ,$ ${r}^{2}={x}^{2}+{y}^{2}$ and $\text{tan}{\rule{0.2em}{0ex}}\theta =\left(\frac{y}{x}\right)$ are the relationships between the variables.

Integration in Cylindrical Coordinates

Triple integrals can often be more readily evaluated by using cylindrical coordinates instead of rectangular coordinates. Some common equations of surfaces in rectangular coordinates along with corresponding equations in cylindrical coordinates are listed in [link]. These equations will become handy as we proceed with solving problems using triple integrals.

Review of Spherical Coordinates

In three-dimensional space ${ℝ}^{3}$ in the spherical coordinate system, we specify a point $P$ by its distance $\rho$ from the origin, the polar angle $\theta$ from the positive $x\text{-axis}$ (same as in the cylindrical coordinate system), and the angle $\phi$ from the positive $z\text{-axis}$ and the line $OP$ ([link]). Note that $\rho \ge 0$ and $0\le \phi \le \pi .$ (Refer to Cylindrical and Spherical Coordinates for a review.) Spherical coordinates are useful for triple integrals over regions that are symmetric with respect to the origin.

Integration in Spherical Coordinates

We now establish a triple integral in the spherical coordinate system, as we did before in the cylindrical coordinate system. Let the function $f\left(\rho ,\theta ,\phi \right)$ be continuous in a bounded spherical box, $B=\left\{\left(\rho ,\theta ,\phi \right)|a\le \rho \le b,\alpha \le \theta \le \beta ,\gamma \le \phi \le \psi \right\}.$ We then divide each interval into $l,m{\rule{0.2em}{0ex}}\text{and}{\rule{0.2em}{0ex}}n$ subdivisions such that $\text{Δ}\rho =\frac{b-a}{l},\text{Δ}\theta =\frac{\beta -\alpha }{m},\text{Δ}\phi =\frac{\psi -\gamma }{n}.$

Calculating Centers of Mass and Moments of Inertia

Abstract:
* Use double integrals to locate the center of mass of a two-dimensional object.
* Use double integrals to find the moment of inertia of a two-dimensional object.
* Use triple integrals to locate the center of mass of a three-dimensional object.

Center of Mass in Two Dimensions

The center of mass is also known as the center of gravity if the object is in a uniform gravitational field. If the object has uniform density, the center of mass is the geometric center of the object, which is called the centroid. [link] shows a point $P$ as the center of mass of a lamina. The lamina is perfectly balanced about its center of mass.

Moments of Inertia

For a clear understanding of how to calculate moments of inertia using double integrals, we need to go back to the general definition of moments and centers of mass in Section 6.6 of Volume 1. The moment of inertia of a particle of mass $m$ about an axis is $m{r}^{2},$ where $r$ is the distance of the particle from the axis. We can see from [link] that the moment of inertia of the subrectangle ${R}_{ij}$ about the $x\text{-axis}$ is ${\left({y}_{ij}^{\*}\right)}^{2}\rho \left({x}_{ij}^{\*},{y}_{ij}^{\*}\right)\text{Δ}A.$ Similarly, the moment of inertia of the subrectangle ${R}_{ij}$ about the $y\text{-axis}$ is ${\left({x}_{ij}^{\*}\right)}^{2}\rho \left({x}_{ij}^{\*},{y}_{ij}^{\*}\right)\text{Δ}A.$ The moment of inertia is related to the rotation of the mass; specifically, it measures the tendency of the mass to resist a change in rotational motion about an axis.

Center of Mass and Moments of Inertia in Three Dimensions

All the expressions of double integrals discussed so far can be modified to become triple integrals.

Change of Variables in Multiple Integrals

Abstract:
* Determine the image of a region under a given transformation of variables.
* Compute the Jacobian of a given transformation.
* Evaluate a double integral using a change of variables.
* Evaluate a triple integral using a change of variables.

Planar Transformations

A planar transformation $T$ is a function that transforms a region $G$ in one plane into a region $R$ in another plane by a change of variables. Both $G$ and $R$ are subsets of ${ℝ}^{2}.$ For example, [link] shows a region $G$ in the $uv\text{-plane}$ transformed into a region $R$ in the $xy\text{-plane}$ by the change of variables $x=g\left(u,v\right)$ and $y=h\left(u,v\right),$ or sometimes we write $x=x\left(u,v\right)$ and $y=y\left(u,v\right).$ We shall typically assume that each of these functions has continuous first partial derivatives, which means ${g}_{u},{g}_{v},{h}_{u},$ and ${h}_{v}$ exist and are also continuous. The need for this requirement will become clear soon.

Jacobians

Recall that we mentioned near the beginning of this section that each of the component functions must have continuous first partial derivatives, which means that ${g}_{u},{g}_{v},{h}_{u},$ and ${h}_{v}$ exist and are also continuous. A transformation that has this property is called a ${C}^{1}$ transformation (here $C$ denotes continuous). Let $T\left(u,v\right)=\left(g\left(u,v\right),h\left(u,v\right)\right),$ where $x=g\left(u,v\right)$ and $y=h\left(u,v\right),$ be a one-to-one ${C}^{1}$ transformation. We want to see how it transforms a small rectangular region $S,$ $\text{Δ}u$ units by $\text{Δ}v$ units, in the $uv\text{-plane}$ (see the following figure).

Change of Variables for Double Integrals

We have already seen that, under the change of variables $T\left(u,v\right)=\left(x,y\right)$ where $x=g\left(u,v\right)$ and $y=h\left(u,v\right),$ a small region $\text{Δ}A$ in the $xy\text{-plane}$ is related to the area formed by the product $\text{Δ}u\text{Δ}v$ in the $uv\text{-plane}$ by the approximation

Change of Variables for Triple Integrals

Changing variables in triple integrals works in exactly the same way. Cylindrical and spherical coordinate substitutions are special cases of this method, which we demonstrate here.

Vector Calculus

Introduction

Vector Fields

Abstract:
* Recognize a vector field in a plane or in space.
* Sketch a vector field from a given equation.
* Identify a conservative field and its associated potential function.

Examples of Vector Fields

How can we model the gravitational force exerted by multiple astronomical objects? How can we model the velocity of water particles on the surface of a river? [link] gives visual representations of such phenomena.

Drawing a Vector Field

We can now represent a vector field in terms of its components of functions or unit vectors, but representing it visually by sketching it is more complex because the domain of a vector field is in ${ℝ}^{2},$ as is the range. Therefore the “graph” of a vector field in ${ℝ}^{2}$ lives in four-dimensional space. Since we cannot represent four-dimensional space visually, we instead draw vector fields in ${ℝ}^{2}$ in a plane itself. To do this, draw the vector associated with a given point at the point in a plane. For example, suppose the vector associated with point $\left(4,-1\right)$ is $〈3,1〉.$ Then, we would draw vector $〈3,1〉$ at point $\left(4,-1\right).$

Gradient Fields

In this section, we study a special kind of vector field called a gradient field or a conservative field. These vector fields are extremely important in physics because they can be used to model physical systems in which energy is conserved. Gravitational fields and electric fields associated with a static charge are examples of gradient fields.

Line Integrals

Abstract:
* Calculate a scalar line integral along a curve.
* Calculate a vector line integral along an oriented curve in space.
* Use a line integral to compute the work done in moving an object along a curve in a vector field.
* Describe the flux and circulation of a vector field.

Scalar Line Integrals

A line integral gives us the ability to integrate multivariable functions and vector fields over arbitrary curves in a plane or in space. There are two types of line integrals: scalar line integrals and vector line integrals. Scalar line integrals are integrals of a scalar function over a curve in a plane or in space. Vector line integrals are integrals of a vector field over a curve in a plane or in space. Let’s look at scalar line integrals first.

Vector Line Integrals

The second type of line integrals are vector line integrals, in which we integrate along a curve through a vector field. For example, let

Applications of Line Integrals

Scalar line integrals have many applications. They can be used to calculate the length or mass of a wire, the surface area of a sheet of a given height, or the electric potential of a charged wire given a linear charge density. Vector line integrals are extremely useful in physics. They can be used to calculate the work done on a particle as it moves through a force field, or the flow rate of a fluid across a curve. Here, we calculate the mass of a wire using a scalar line integral and the work done by a force using a vector line integral.

Flux and Circulation

We close this section by discussing two key concepts related to line integrals: flux across a plane curve and circulation along a plane curve. Flux is used in applications to calculate fluid flow across a curve, and the concept of circulation is important for characterizing conservative gradient fields in terms of line integrals. Both these concepts are used heavily throughout the rest of this chapter. The idea of flux is especially important for Green’s theorem, and in higher dimensions for Stokes’ theorem and the divergence theorem.

Conservative Vector Fields

Abstract:
* Describe simple and closed curves; define connected and simply connected regions.
* Explain how to find a potential function for a conservative vector field.
* Use the Fundamental Theorem for Line Integrals to evaluate a line integral in a vector field.
* Explain how to test a vector field to determine whether it is conservative.

Curves and Regions

Before continuing our study of conservative vector fields, we need some geometric definitions. The theorems in the subsequent sections all rely on integrating over certain kinds of curves and regions, so we develop the definitions of those curves and regions here.

Fundamental Theorem for Line Integrals

Now that we understand some basic curves and regions, let’s generalize the Fundamental Theorem of Calculus to line integrals. Recall that the Fundamental Theorem of Calculus says that if a function $f$ has an antiderivative *F*, then the integral of $f$ from *a* to *b* depends only on the values of *F* at *a* and at *b*—that is,

Conservative Vector Fields and Potential Functions

As we have learned, the Fundamental Theorem for Line Integrals says that if **F** is conservative, then calculating ${\int }_{C}\text{F}·dr$ has two steps: first, find a potential function $f$ for **F** and, second, calculate $f\left({P}_{1}\right)-f\left({P}_{0}\right),$ where ${P}_{1}$ is the endpoint of *C* and ${P}_{0}$ is the starting point. To use this theorem for a conservative field **F**, we must be able to find a potential function $f$ for **F**. Therefore, we must answer the following question: Given a conservative vector field **F**, how do we find a function $f$ such that $\text{∇}f=\text{F}?$ Before giving a general method for finding a potential function, let’s motivate the method with an example.

Testing a Vector Field

Until now, we have worked with vector fields that we know are conservative, but if we are not told that a vector field is conservative, we need to be able to test whether it is conservative. Recall that, if **F** is conservative, then **F** has the cross-partial property (see [link]). That is, if $\text{F}=〈P,Q,R〉$ is conservative, then ${P}_{y}={Q}_{x},{P}_{z}={R}_{x},$ and ${Q}_{z}={R}_{y}.$ So, if **F** has the cross-partial property, then is **F** conservative? If the domain of **F** is open and simply connected, then the answer is yes.

Green’s Theorem

Abstract:
* Apply the circulation form of Green’s theorem.
* Apply the flux form of Green’s theorem.
* Calculate circulation and flux on more general regions.

Extending the Fundamental Theorem of Calculus

Recall that the Fundamental Theorem of Calculus says that

Circulation Form of Green’s Theorem

The first form of Green’s theorem that we examine is the circulation form. This form of the theorem relates the vector line integral over a simple, closed plane curve *C* to a double integral over the region enclosed by *C*. Therefore, the circulation of a vector field along a simple closed curve can be transformed into a double integral and vice versa.

Flux Form of Green’s Theorem

The circulation form of Green’s theorem relates a double integral over region *D* to line integral ${\oint }_{C}\text{F}·\text{T}ds,$ where *C* is the boundary of *D*. The flux form of Green’s theorem relates a double integral over region *D* to the flux across boundary *C*. The flux of a fluid across a curve can be difficult to calculate using the flux line integral. This form of Green’s theorem allows us to translate a difficult flux integral into a double integral that is often easier to calculate.

Green’s Theorem on General Regions

Green’s theorem, as stated, applies only to regions that are simply connected—that is, Green’s theorem as stated so far cannot handle regions with holes. Here, we extend Green’s theorem so that it does work on regions with finitely many holes ([link]).

Divergence and Curl

Abstract:
* Determine divergence from the formula for a given vector field.
* Determine curl from the formula for a given vector field.
* Use the properties of curl and divergence to determine whether a vector field is conservative.

Divergence

Divergence is an operation on a vector field that tells us how the field behaves toward or away from a point. Locally, the divergence of a vector field **F** in ${ℝ}^{2}$ or ${ℝ}^{3}$ at a particular point *P* is a measure of the “outflowing-ness” of the vector field at *P*. If **F** represents the velocity of a fluid, then the divergence of **F** at *P* measures the net rate of change with respect to time of the amount of fluid flowing away from *P* (the tendency of the fluid to flow “out of” *P*). In particular, if the amount of fluid flowing into *P* is the same as the amount flowing out, then the divergence at *P* is zero.

Curl

The second operation on a vector field that we examine is the curl, which measures the extent of rotation of the field about a point. Suppose that **F** represents the velocity field of a fluid. Then, the curl of **F** at point *P* is a vector that measures the tendency of particles near *P* to rotate about the axis that points in the direction of this vector. The magnitude of the curl vector at *P* measures how quickly the particles rotate around this axis. In other words, the curl at a point is a measure of the vector field’s “spin” at that point. Visually, imagine placing a paddlewheel into a fluid at *P*, with the axis of the paddlewheel aligned with the curl vector ([link]). The curl measures the tendency of the paddlewheel to rotate.

Using Divergence and Curl

Now that we understand the basic concepts of divergence and curl, we can discuss their properties and establish relationships between them and conservative vector fields.

Surface Integrals

Abstract:
* Find the parametric representations of a cylinder, a cone, and a sphere.
* Describe the surface integral of a scalar-valued function over a parametric surface.
* Use a surface integral to calculate the area of a given surface.
* Explain the meaning of an oriented surface, giving an example.
* Describe the surface integral of a vector field.
* Use surface integrals to solve applied problems.

Parametric Surfaces

A surface integral is similar to a line integral, except the integration is done over a surface rather than a path. In this sense, surface integrals expand on our study of line integrals. Just as with line integrals, there are two kinds of surface integrals: a surface integral of a scalar-valued function and a surface integral of a vector field.

Surface Area of a Parametric Surface

Our goal is to define a surface integral, and as a first step we have examined how to parameterize a surface. The second step is to define the surface area of a parametric surface. The notation needed to develop this definition is used throughout the rest of this chapter.

Surface Integral of a Scalar-Valued Function

Now that we can parameterize surfaces and we can calculate their surface areas, we are able to define surface integrals. First, let’s look at the surface integral of a scalar-valued function. Informally, the surface integral of a scalar-valued function is an analog of a scalar line integral in one higher dimension. The domain of integration of a scalar line integral is a parameterized curve (a one-dimensional object); the domain of integration of a scalar surface integral is a parameterized surface (a two-dimensional object). Therefore, the definition of a surface integral follows the definition of a line integral quite closely. For scalar line integrals, we chopped the domain curve into tiny pieces, chose a point in each piece, computed the function at that point, and took a limit of the corresponding Riemann sum. For scalar surface integrals, we chop the domain *region* (no longer a curve) into tiny pieces and proceed in the same fashion.

Orientation of a Surface

Recall that when we defined a scalar line integral, we did not need to worry about an orientation of the curve of integration. The same was true for scalar surface integrals: we did not need to worry about an “orientation” of the surface of integration.

Surface Integral of a Vector Field

With the idea of orientable surfaces in place, we are now ready to define a surface integral of a vector field. The definition is analogous to the definition of the flux of a vector field along a plane curve. Recall that if **F** is a two-dimensional vector field and *C* is a plane curve, then the definition of the flux of **F** along *C* involved chopping *C* into small pieces, choosing a point inside each piece, and calculating $\text{F}·\text{N}$ at the point (where **N** is the unit normal vector at the point). The definition of a surface integral of a vector field proceeds in the same fashion, except now we chop surface *S* into small pieces, choose a point in the small (two-dimensional) piece, and calculate $\text{F}·\text{N}$ at the point.

Stokes’ Theorem

Abstract:
* Explain the meaning of Stokes’ theorem.
* Use Stokes’ theorem to evaluate a line integral.
* Use Stokes’ theorem to calculate a surface integral.
* Use Stokes’ theorem to calculate a curl.

Stokes’ Theorem

Stokes’ theorem says we can calculate the flux of curl **F** across surface *S* by knowing information only about the values of **F** along the boundary of *S*. Conversely, we can calculate the line integral of vector field **F** along the boundary of surface *S* by translating to a double integral of the curl of **F** over *S*.

Applying Stokes’ Theorem

Stokes’ theorem translates between the flux integral of surface *S* to a line integral around the boundary of *S*. Therefore, the theorem allows us to compute surface integrals or line integrals that would ordinarily be quite difficult by translating the line integral into a surface integral or vice versa. We now study some examples of each kind of translation.

Interpretation of Curl

In addition to translating between line integrals and flux integrals, Stokes’ theorem can be used to justify the physical interpretation of curl that we have learned. Here we investigate the relationship between curl and circulation, and we use Stokes’ theorem to state Faraday’s law—an important law in electricity and magnetism that relates the curl of an electric field to the rate of change of a magnetic field.

The Divergence Theorem

Abstract:
* Explain the meaning of the divergence theorem.
* Use the divergence theorem to calculate the flux of a vector field.
* Apply the divergence theorem to an electrostatic field.

Overview of Theorems

Before examining the divergence theorem, it is helpful to begin with an overview of the versions of the Fundamental Theorem of Calculus we have discussed:

Stating the Divergence Theorem

The divergence theorem follows the general pattern of these other theorems. If we think of divergence as a derivative of sorts, then the divergence theorem relates a triple integral of derivative div**F** over a solid to a flux integral of **F** over the boundary of the solid. More specifically, the divergence theorem relates a flux integral of vector field **F** over a closed surface *S* to a triple integral of the divergence of **F** over the solid enclosed by *S*.

Using the Divergence Theorem

The divergence theorem translates between the flux integral of closed surface *S* and a triple integral over the solid enclosed by *S*. Therefore, the theorem allows us to compute flux integrals or triple integrals that would ordinarily be difficult to compute by translating the flux integral into a triple integral and vice versa.

Application to Electrostatic Fields

The divergence theorem has many applications in physics and engineering. It allows us to write many physical laws in both an integral form and a differential form (in much the same way that Stokes’ theorem allowed us to translate between an integral and differential form of Faraday’s law). Areas of study such as fluid dynamics, electromagnetism, and quantum mechanics have equations that describe the conservation of mass, momentum, or energy, and the divergence theorem allows us to give these equations in both integral and differential forms.

Second-Order Differential Equations

Introduction

Second-Order Linear Equations

Abstract:
* Recognize homogeneous and nonhomogeneous linear differential equations.
* Determine the characteristic equation of a homogeneous linear equation.
* Use the roots of the characteristic equation to find the solution to a homogeneous linear equation.
* Solve initial-value and boundary-value problems involving linear differential equations.

Homogeneous Linear Equations

Consider the second-order differential equation

Second-Order Equations with Constant Coefficients

Now that we have a better feel for linear differential equations, we are going to concentrate on solving second-order equations of the form

Initial-Value Problems and Boundary-Value Problems

So far, we have been finding general solutions to differential equations. However, differential equations are often used to describe physical systems, and the person studying that physical system usually knows something about the state of that system at one or more points in time. For example, if a constant-coefficient differential equation is representing how far a motorcycle shock absorber is compressed, we might know that the rider is sitting still on his motorcycle at the start of a race, time $t={t}_{0}.$ This means the system is at equilibrium, so $y\left({t}_{0}\right)=0,$ and the compression of the shock absorber is not changing, so ${y}^{\prime }\left({t}_{0}\right)=0.$ With these two initial conditions and the general solution to the differential equation, we can find the *specific* solution to the differential equation that satisfies both initial conditions. This process is known as *solving an initial-value problem*. (Recall that we discussed initial-value problems in Introduction to Differential Equations.) Note that second-order equations have two arbitrary constants in the general solution, and therefore we require two initial conditions to find the solution to the initial-value problem.

Nonhomogeneous Linear Equations

Abstract:
* Write the general solution to a nonhomogeneous differential equation.
* Solve a nonhomogeneous differential equation by the method of undetermined coefficients.
* Solve a nonhomogeneous differential equation by the method of variation of parameters.

General Solution to a Nonhomogeneous Linear Equation

Consider the nonhomogeneous linear differential equation

Undetermined Coefficients

The method of undetermined coefficients involves making educated guesses about the form of the particular solution based on the form of $r\left(x\text{).}$ When we take derivatives of polynomials, exponential functions, sines, and cosines, we get polynomials, exponential functions, sines, and cosines. So when $r\left(x\right)$ has one of these forms, it is possible that the solution to the nonhomogeneous differential equation might take that same form. Let’s look at some examples to see how this works.

Variation of Parameters

Sometimes, $r\left(x\right)$ is not a combination of polynomials, exponentials, or sines and cosines. When this is the case, the method of undetermined coefficients does not work, and we have to use another approach to find a particular solution to the differential equation. We use an approach called the method of variation of parameters.

Applications

Abstract:
* Solve a second-order differential equation representing simple harmonic motion.
* Solve a second-order differential equation representing damped simple harmonic motion.
* Solve a second-order differential equation representing forced simple harmonic motion.
* Solve a second-order differential equation representing charge and current in an RLC series circuit.

Simple Harmonic Motion

Consider a mass suspended from a spring attached to a rigid support. (This is commonly called a spring-mass system.) Gravity is pulling the mass downward and the restoring force of the spring is pulling the mass upward. As shown in [link], when these two forces are equal, the mass is said to be at the equilibrium position. If the mass is displaced from equilibrium, it oscillates up and down. This behavior can be modeled by a second-order constant-coefficient differential equation.

Damped Vibrations

With the model just described, the motion of the mass continues indefinitely. Clearly, this doesn’t happen in the real world. In the real world, there is almost always some friction in the system, which causes the oscillations to die off slowly—an effect called *damping*. So now let’s look at how to incorporate that damping force into our differential equation.

Forced Vibrations

The last case we consider is when an external force acts on the system. In the case of the motorcycle suspension system, for example, the bumps in the road act as an external force acting on the system. Another example is a spring hanging from a support; if the support is set in motion, that motion would be considered an external force on the system. We model these forced systems with the nonhomogeneous differential equation

The *RLC* Series Circuit

Consider an electrical circuit containing a resistor, an inductor, and a capacitor, as shown in [link]. Such a circuit is called an *RLC* series circuit. *RLC* circuits are used in many electronic systems, most notably as tuners in AM/FM radios. The tuning knob varies the capacitance of the capacitor, which in turn tunes the radio. Such circuits can be modeled by second-order, constant-coefficient differential equations.

Series Solutions of Differential Equations

Abstract: * Use power series to solve first-order and second-order differential equations.

Table of Integrals

Basic Integrals

1. $\int {u}^{n}{\rule{0.2em}{0ex}}du=\frac{{u}^{n+1}}{n+1}+C,n\ne \text{−}1$

Trigonometric Integrals

18. $\int {\text{sin}}^{2}u{\rule{0.2em}{0ex}}du=\frac{1}{2}u-\frac{1}{4}\text{sin}{\rule{0.2em}{0ex}}2u+C$

Exponential and Logarithmic Integrals

42. $\int u{e}^{au}{\rule{0.2em}{0ex}}du=\frac{1}{{a}^{2}}\left(au-1\right){e}^{au}+C$

Hyperbolic Integrals

49. $\int \text{sinh}{\rule{0.2em}{0ex}}u{\rule{0.2em}{0ex}}du=\text{cosh}{\rule{0.2em}{0ex}}u+C$

Inverse Trigonometric Integrals

59. $\int {\text{sin}}^{-1}u{\rule{0.2em}{0ex}}du=u{\rule{0.2em}{0ex}}{\text{sin}}^{-1}u+\sqrt{1-{u}^{2}}+C$

Integrals Involving *a*2 + *u*2, *a* > 0

68. $\int \sqrt{{a}^{2}+{u}^{2}}{\rule{0.2em}{0ex}}du=\frac{u}{2}\sqrt{{a}^{2}+{u}^{2}}+\frac{{a}^{2}}{2}\text{ln}{\rule{0.1em}{0ex}}\left(u+\sqrt{{a}^{2}+{u}^{2}}\right)+C$

Integrals Involving *u*2 − *a*2, *a* > 0

77. $\int \sqrt{{u}^{2}-{a}^{2}}{\rule{0.2em}{0ex}}du=\frac{u}{2}\sqrt{{u}^{2}-{a}^{2}}-\frac{{a}^{2}}{2}\text{ln}{\rule{0.1em}{0ex}}|u+\sqrt{{u}^{2}-{a}^{2}}|+C$

Integrals Involving *a*2 − *u*2, *a* > 0

85. $\int \sqrt{{a}^{2}-{u}^{2}}{\rule{0.2em}{0ex}}du=\frac{u}{2}\sqrt{{a}^{2}-{u}^{2}}+\frac{{a}^{2}}{2}{\text{sin}}^{-1}\frac{u}{a}+C$

Integrals Involving 2*au* − *u*2, *a* > 0

94. $\int \sqrt{2au-{u}^{2}}{\rule{0.2em}{0ex}}du=\frac{u-a}{2}\sqrt{2au-{u}^{2}}+\frac{{a}^{2}}{2}{\text{cos}}^{-1}\left(\frac{a-u}{a}\right)+C$

Integrals Involving *a* + *bu*, *a* ≠ 0

98. $\int \frac{u{\rule{0.2em}{0ex}}du}{a+bu}=\frac{1}{{b}^{2}}\left(a+bu-a\text{ln}{\rule{0.1em}{0ex}}|a+bu|\right)+C$

Table of Derivatives

General Formulas

1. $\frac{d}{dx}\left(c\right)=0$

Trigonometric Functions

9. $\frac{d}{dx}\left(\text{sin}{\rule{0.1em}{0ex}}x\right)=\text{cos}{\rule{0.1em}{0ex}}x$

Inverse Trigonometric Functions

15. $\frac{d}{dx}\left({\text{sin}}^{-1}x\right)=\frac{1}{\sqrt{1-{x}^{2}}}$

Exponential and Logarithmic Functions

21. $\frac{d}{dx}\left({e}^{x}\right)={e}^{x}$

Hyperbolic Functions

25. $\frac{d}{dx}\left(\text{sinh}{\rule{0.1em}{0ex}}x\right)=\text{cosh}{\rule{0.1em}{0ex}}x$

Inverse Hyperbolic Functions

31. $\frac{d}{dx}\left({\text{sinh}}^{-1}x\right)=\frac{1}{\sqrt{{x}^{2}+1}}$

Review of Pre-Calculus

Formulas from Geometry

$A=\text{area},$ $V=\text{Volume},{\rule{0.2em}{0ex}}\text{and}$ $S=\text{lateral surface area}$

Formulas from Algebra

Formulas from Trigonometry