Mathematical Methods for Physics and Engineering - Matematica.NET

6.4 CHANGE OF VARIABLES IN MULTIPLE INTEGRALSis constant along the line element KL, the latter has components (∂x/∂u) du and(∂y/∂u) du in the directions of the x- andy-axes respectively. Similarly, since uis constant along the line element KN, the latter has corresponding components(∂x/∂v) dv and (∂y/∂v) dv. Using the result for the area of a parallelogram givenin chapter 7, we find that the area of the parallelogram KLMN is given by∣ dA uv =∂x∣ ∂u du∂y ∂x ∂y ∣∣∣dv − dv∂v ∂v ∂u du =∂x ∂y∣ ∂u ∂v − ∂x∂y∂v ∂u∣ du dv.Defining the Jacobian of x, y with respect to u, v as∂(x, y)J =∂(u, v) ≡ ∂x ∂y∂u ∂v − ∂x ∂y∂v ∂u ,we havedA uv =∂(x, y)∣ ∂(u, v) ∣ du dv.The reader acquainted with determinants will notice that the Jacobian can alsobe written as the 2 × 2 determinant∣ ∣∣∣∣∣∣∣ ∂x ∂y∂(x, y)J =∂(u, v) = ∂u ∂u∂x ∂y.∣∂v ∂vSuch determinants can be evaluated using the methods of chapter 8.So, in summary, the relationship between the size of the area element generatedby dx, dy and the size of the corresponding area element generated by du, dv isdx dy =∂(x, y)∣ ∂(u, v) ∣ du dv.This equality should be taken as meaning that when transforming from coordinatesx, y to coordinates u, v, the area element dx dy should be replaced by theexpression on the RHS of the above equality. Of course, the Jacobian can, andin general will, vary over the region of integration. We may express the doubleintegral in either coordinate system as∫∫∫∫I = f(x, y) dx dy = g(u, v)∂(x, y)∣RR ∂(u, v) ∣ du dv. (6.12)′When evaluating the integral in the new coordinate system, it is usually advisableto sketch the region of integration R ′ in the uv-plane.201