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.

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    “A beauty is not suddenly in a circle. It comes with rapture. A great deal of beauty is rapture. A circle is a necessity. Otherwise you would see no one. We each have our circle.”
    —Gertrude Stein (1874–1946)