Suppose S is a rectangle in the uv plane with vertices O(0,0), P(delta u, 0), (delta u, delta v), Q(0, delta v). The image of S under the transformation x=g(u,v), y=h(u,v) is a region R in the xy plane. Let O', P' and Q' be the images of O,P,Q, respectively, in the xy plane where O',P',Q' do not all lie on the same line. The coordinates of O' , P', and Q' are (g(0,0),h(0,0)), (g(delta u,0),h(delta u,0)), and (g(0,delta v),h(0,delta v)), respectively. Consider the parallelogram determined by the vectors O'P' and O'Q'. Use the cross product to show that the area of the parallelogram is approximately |J(u,v)|delta(u)delta(v). |J(u,v)| is the jacobian determinant.

Suppose S is a rectangle in the uv plane with vertices O(0,0), P(delta u, 0), (delta u, delta v), Q(0, delta v). The image of S under the transformation x=g(u,v), y=h(u,v) is a region R in the xy plane. Let O', P' and Q' be the images of O,P,Q, respectively, in the xy plane where O',P',Q' do not all lie on the same line. The coordinates of O' , P', and Q' are (g(0,0),h(0,0)), (g(delta u,0),h(delta u,0)), and (g(0,delta v),h(0,delta v)), respectively. Consider the parallelogram determined by the vectors O'P' and O'Q'. Use the cross product to show that the area of the parallelogram is approximately |J(u,v)|delta(u)delta(v). |J(u,v)| is the jacobian determinant.

Elementary Linear Algebra (MindTap Course List)

8th Edition

ISBN:9781305658004

Author:Ron Larson

Publisher:Ron Larson

Chapter7: Eigenvalues And Eigenvectors

Section7.CM: Cumulative Review

Problem 3CM: Let T:RnRm be the linear transformation defined by T(v)=Av, where A=[30100302]. Find the dimensions...

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