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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Famous quotes containing the word circle:

    “That three times five is equal to the half of thirty, expresses a relation between these numbers. Propositions of this kind are discoverable by the mere operation of thought, without dependence on what is any where existent in the universe. Though there never were a circle or triangle in nature, the truths, demonstrated by Euclid, would for ever retain their certainty and evidence.”
    —David Hume (1711–1776)