Implicit Function Theorem - Application: Change of Coordinates

Application: Change of Coordinates

Suppose we have an m-dimensional space, parametrised by a set of coordinates . We can introduce a new coordinate system by supplying m functions . These functions allow to calculate the new coordinates of a point, given the point's old coordinates using . One might want to verify if the opposite is possible: given coordinates, can we 'go back' and calculate the same point's original coordinates ? The implicit function theorem will provide an answer to this question. The (new and old) coordinates are related by, with

f(x'_1,\ldots,x'_m,x_1,\ldots x_m)=(h_1(x_1,\ldots x_m)-x'_1,\ldots, h_m(x_1,\ldots, x_m)-x'_m).

Now the Jacobian matrix of f at a certain point is given by

\begin{matrix} (Df)(a,b) & = & \begin{bmatrix} -1 & \cdots & 0 & \frac{\partial h_1}{\partial x_1}(b) & \cdots & \frac{\partial h_1}{\partial x_m}(b)\\ \vdots & \ddots & \vdots & \vdots & \ddots & \vdots\\ 0 & \cdots & -1 & \frac{\partial h_m}{\partial x_1}(b) & \cdots & \frac{\partial h_m}{\partial x_m}(b)\\ \end{bmatrix}\\ & = & \begin{bmatrix} -1_m & | & J \end{bmatrix}.\\ \end{matrix}

where denotes the identity matrix, and J is the matrix of partial derivatives, evaluated at . (In the above, these blocks were denoted by X and Y. As it happens, in this particular application of the theorem, neither matrix depends on .) The implicit function theorem now states that we can locally express as a function of if J is invertible. Demanding J is invertible is equivalent to, thus we see that we can go back from the primed to the unprimed coordinates if the determinant of the Jacobian J is non-zero. This statement is also known as the inverse function theorem.

Read more about this topic:  Implicit Function Theorem

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