Changes

An interesting theorem is that these matrices are inverses of each other.  (It follows that their determinants are reciprocals of each other.)

An interesting theorem is that these matrices are inverses of each other.  (It follows that their determinants are reciprocals of each other.)

Exercise:  Prove that the matrix product of the J matrices of two sequential coordinate changes is the J matrix of the combined change.  That is, if <math>f_1,f_2,...,f_n \to g_1,g_2,...,g_n</math> and <math>g_1,g_2,...,g_n \to h_1,h_2,...,h_n</math> are two coordinate changes, so that <math>f_1,f_2,...,f_n \to h_1,h_2,...,h_n</math> is the result of doing both, then

Exercise:  Prove that the matrix product of the J matrices of two sequential coordinate changes is the J matrix of the combined change.  That is, if

::<math>\frac{\partial(h_1,h_2,...,h_n)}{\partial(f_1,f_2,...,f_n)} = \frac{\partial(h_1,h_2,...,h_n)}{\partial(g_1,g_2,...,g_n)} \frac{\partial(g_1,g_2,...,g_n)}{\partial(f_1,f_2,...,f_n)}</math>

::<math>f_1,f_2,...,f_n \to g_1,g_2,...,g_n</math>

and

::<math>g_1,g_2,...,g_n \to h_1,h_2,...,h_n</math>

are two coordinate changes, so that

::<math>f_1,f_2,...,f_n \to h_1,h_2,...,h_n</math>

is the result of doing both, then

::<math>\frac{\partial(h_1,h_2,...,h_n)}{\partial(f_1,f_2,...,f_n)} = \frac{\partial(h_1,h_2,...,h_n)}{\partial(g_1,g_2,...,g_n)}\ \frac{\partial(g_1,g_2,...,g_n)}{\partial(f_1,f_2,...,f_n)}</math>

Hint:  This is really just the chain rule for partial derivatives.  The adding up of terms in the chain rule is the same as the adding up of products in a matrix multiplication.

Hint:  This is really just the chain rule for partial derivatives.  The adding up of terms in the chain rule is the same as the adding up of products in a matrix multiplication.

It follows that, if the second change is just the inverse of the first, bringing us back to the original coordinate system, the two matrices are inverses.

 

It follows from this that, if the second change is just the inverse of the first, bringing us back to the original coordinate system, the two matrices are inverses.

==Vectors in Arbitrary Coordinate Systems==

==Vectors in Arbitrary Coordinate Systems==