Question

2. (1 Point) Let r-2u and y-3u. (a) Let R be the rectangle in the uv-plane defined by the points (0,0), (2,0), (2,1), (0 , 1). Find the area of the image of R in the ry plane? (b) Find the area of R by computing the Jacobian of the transformation from uv-space to xy-space Change of Variables When working integrals, it is wise to choose a coordinate system that fits the problem; e.g. polar coordinates are a good choice for integrating over disks. Once we choose a coordinate system we must figure out the area form (dA) for that system. For example, when switching from rectangular to polar coordinates we must change the form of the area element from drdy to rdrd0. To determine that rdrde is the correct formula how the edges of the element varied with r and 0. With more spherical coordinates, this procedure becomes more difficult and the diagrams more intricate. Fortunately, there is a method to find the area form by calculation. we drew a careful picture of a typical area element and noticed complex coordinate systems, such as Suppose we are integrating a function over a to another set of variables u, v. Then region in the xy-plane f(x, y) and we want to change f (x, y)dxdy dudv f(r(u, v), y(u, v)) Det JR D where D is a region in the ry-plane, R is the corresponding region in the uv-plane. The determinant of the matrix Bu r(u,v) and y = y(u, v) sometimes is called the Jacobian of the transformation defined by a = denoted , y) It is important that there be a one-to-one correspondence between points inD 0u, υ) and points in R under the change of variables and that both r and y are differentiable functions of u and v. region of n-dimensional space would be integrating a function In complete generality, with the function and the region of integration given in terms of one set of variables 1, .,n that we want to change to another set of variables u1,..., un Then over a we Cboll = "p ap("α' ... Π ),( f a (u, un), . . . , πη(u]; s ν sy Un) | 9(ui un) | 0(α1. n duy- dun JR S3515

Answer 1

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