the cylindrical axis. Generalization It is also possible to deal with ellipsoids in Cartesian coordinates by using a modified version of the spherical coordinates. Let P be an ellipsoid specified by the level set The modified spherical coordinates of a point in P in the ISO convention (i.e. for physics: radius , inclination , azimuth ) can be obtained from its Cartesian coordinates by the formulae An infinitesimal volume element is given by The square-root factor comes from the property of the determinant that allows a constant to be pulled out from a column: Integration and differentiation in spherical coordinates The following equations (Iyanaga 1977) assume that the colatitude is the inclination from the (polar) axis (ambiguous since , , and are mutually normal), as in the physics convention discussed. The line element for an infinitesimal displacement from to is where are the local orthogonal unit vectors in the directions of increasing , , and , respectively, and , , and are the unit vectors in Cartesian coordinates. The linear transformation to this right-handed coordinate triplet is a rotation matrix, This gives the transformation from the spherical to the cartesian, the other way around is given by its inverse. Note: the matrix is an orthogonal matrix, that is, its inverse is simply its transpose. The Cartesian unit vectors are thus related to the spherical unit vectors by: The general form of the formula to prove the differential line element, is that is, the change in is decomposed into individual changes corresponding to changes in the individual coordinates. To apply this to the present case, one needs to calculate how changes with each of the coordinates. In the conventions used, Thus, The desired coefficients are the magnitudes of these vectors: The surface element spanning from to and to on a spherical surface at (constant) radius is then Thus the differential solid angle is The surface element in a surface of polar angle constant (a cone with vertex the origin) is The surface element in a surface of azimuth constant (a vertical half-plane) is The volume element spanning from to , to , and to is specified by the determinant of the Jacobian matrix of partial derivatives, namely Thus, for example, a function can be integrated over every point in by the triple integral The del operator in this system leads to the following expressions for the gradient, divergence, curl and (scalar)