Comments on Colley, Section 5.5 In single variable calculus, change of variables is an extremely useful technique for rewriting integrals in more computable forms, and not surprisingly there are correspondingly important results for multiple integrals. Before discussing these, it might be helpful to review some aspects of the single variable theory in a form that will generalize to several variables. The Chain Rule for derivatives and the Fundamental Theorem of Calculus combine to yield the following integral identity; for the sake of simplicity, we assume that f and g are continuous on the relevant intervals and that g has a continuous derivative. Strictly speaking, when we think of change of variables we think of something that is reversible: Not only do we want x to be uniquely expressible in terms of t, but conversely we also want t to be uniquely expressible in terms of x. In order for this to happen, the change of variables function x = g( t ) must be either strictly increasing or strictly decreasing. Suppose first that g is strictly increasing. In this case g(a) < g(b) and the right hand side is just fine. However, if g is strictly decreasing, so that g(a) > g(b), then the lower limit in the right hand integral is greater than the upper limit. By convention, this expression is equal to the negative of the integral with the limits written in the usual order. For our purposes it is useful to note that both cases can be combined into a single formula as follows: Adopt the notation of the preceding formula, and assume that the function g is either strictly increasing or decreasing with image given by the closed interval [c, d]. Then we have the following change of variables formula: Notice the absolute value signs which surround g′( t ) on the right hand side. These play an important role in the generalizations to several variables. In one dimension, change of variables takes one interval to another; the two intervals often have different lengths, but their shapes are the same. However, in two or three dimensions, both the size and shape can change, so the first step instudying change of variables in multiple dimensions is to become familiar with some of the things that can happen and to adjust the setting so that it reflects these complications. Transformations of regions It is useful to view change of variables phenomena in two and three dimensions as geometrical transformations which send the points in one region S into the points in another region R. For the time being we shall restrict attention to the case of two variables. (Source: http://math.etsu.edu/Multicalc/Chap4/Chap4-4/index.htm ) As indicated in the illustration, the coordinates in the source are denoted by (u, v) and viewed as points in the uv – plane, while the coordinates in the target are denoted by (x, y) and viewed as points in the xy – plane. Analytically, the change of variables transformation is given by a formula like T(u, v) = ( f (u, v), g (u, v) ) Where (1) f and g are functions with continuous partial derivatives, (2) the system of equations x = f (u, v) and y = g(u, v) can be solved uniquely for u and v either everywhere or “almost everywhere” on R, and (3) these solutions are expressible as functions of x and y with continuous derivatives almost everywhere. Here are some simple but important examples: Translations. In this case f(u, v) = u + a and g(u, v) = v + b for some fixed vector (a, b). Physically, this corresponds to moving everything a units in the u – direction and b units in the v – direction. The functions f and g obviously have continuous partial derivatives, and clearly we also have unique solutions to the system of equations x = f(u, v), y = g(u, v) given by u = x – a and v = y – b. The latter formulas clearly show that u and v have continuous partial derivatives with respect to x and y.Invertible linear transformations. In this case x = f(u, v) = au + bv and y = g(u, v) = cu + dv for some fixed constants a, b, c, d. As usual, we need to assume that the determinant ad – bc is nonzero in order to solve for u and v. (Source: www.ies.co.jp/math/java/misc/don_trans/pict.gif ) The solutions for u and v are linear expressions in x and y, so once again these solutions clearly have continuous first partial derivatives. Polar coordinates. In this case it is customary to view the source as the rθθθθ – plane, and then the polar coordinate transformation takes the standard form; namely, x = r cos θθθθ and y = r sin θθθθ. We can solve this uniquely for r and θθθθ in terms of x and y provided r is nonzero and the values of θθθθ lie within some interval whose length is less than 2ππππ; the “almost everywhere” condition means that we can extend everything to the case where r is nonnegative (and possibly zero).Typical problems Frequently we are given a change of variables transformation, and the question is to find R given S or vice versa. Sometime the objective is to solve for u and v in terms of x and y or vice versa. Problems of the second type are generally done using algebra. In problems of the first sort, one is usually given the boundary of a region in terms as the set of solutions to some equations. The usual procedure is to solve these equations to find the boundary of the other region in the problem. Three – dimensional transformations Everything said thus far carries over to three variables, but in this case one gets a system of three equations in three unknowns. There are similar examples of transformations given by translations and invertible linear transformations, and the polar coordinate transformation has two natural counterparts given by the transformations for cylindrical coordinates (x and y as before in terms of r and θθθθ, together with z) and spherical coordinates given by the usual formulas x = ρρρρ cos θθθθ sin φφφφ , y = ρρρρ sin θθθθ sin φφφφ ,,,, z = ρρρρ cos φφφφ . (Source: http://mathworld.wolfram.com/SphericalCoordinates.html ) In this drawing, the variable r corresponds to the variable ρρρρ in the displayed formulas; the three vectors indicate directions in which the coordinate values increase. The videos at the following site may be helpful for studying the material presented thus