CHV - n n n n R R T : n R A T = x x x , ) ( . Suppose that...

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COMMENTS ON CHANGE OF VARIABLES THEOREM The proof of this theorem in the textbook is long and will not be covered on the lecture in full. Its better to do practical examples instead. Anyway some informal explanations are necessary so that you understand the concept of the change of variables for multiple integral (its integration over the volume, and the volume elements change under a map). Below is a comment on linear maps and later a geometrical view in 2 R as usually is presented in less advanced texts. Volume Change Factor Suppose that the column vectors of the matrix B determine an n -box in n b b b ,..... , , 2 1 n R . It is proved in Linear Algebra that if the vectors are linearly independent then n b b b ,..... , , 2 1 the volume of the box is V = 0 ) det( B . This is very much clear in case n = 2, 3 see p.179 of your text (and section1.1 ). Associated with an matrix A is the linear transformation , where
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Unformatted text preview: n n n n R R T : n R A T = x x x , ) ( . Suppose that det( A ) , that is T is an invertible transformation. From the definition of matrix multiplication, the j th column vector of AB is so j A b . The linear transformation T thus carries the original n-box [ n A A A AB b b b . . . . . . 2 1 = ] determined by the column vectors of B into a new n-box determined by the column vectors of AB . The new n-box has volume ) det( ) det( ) det( B A AB = . That is, the volume of the new n-box, or image box, is equal to V A ) det( . Thus ) det( A is referred to as the volume-change factor for the linear transformation T . This is a basic property that explains the change of variables in multiple integral in case of linear maps. Above as well as in the next part the standard notation T is used for a transformation. In your text the notation is G ....
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CHV - n n n n R R T : n R A T = x x x , ) ( . Suppose that...

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