Implicit Function Theorem - The Circle Example

The Circle Example

Let us go back to the example of the unit circle. In this case and . The matrix of partial derivatives is just a 1×2 matrix, given by

\begin{matrix} (Df)(a,b) & = & \begin{bmatrix} \frac{\partial f}{\partial x}(a,b) & \frac{\partial f}{\partial y}(a,b)\\ \end{bmatrix}\\ & = & \begin{bmatrix} 2a & 2b \end{bmatrix}.\\ \end{matrix}

Thus, here, is just a number; the linear map defined by it is invertible iff . By the implicit function theorem we see that we can locally write the circle in the form for all points where . For and we run into trouble, as noted before.

Read more about this topic:  Implicit Function Theorem

Famous quotes containing the word circle:

    “Men go out to admire the heights of mountains, the huge waves of the sea, the broadest spans of rivers, the circle of ocean, the revolutions of stars, and leave themselves behind.”
    —St. Augustine (354–430)