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
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:
“Change begets change. Nothing propagates so fast. If a man habituated to a narrow circle of cares and pleasures, out of which he seldom travels, step beyond it, though for never so brief a space, his departure from the monotonous scene on which he has been an actor of importance would seem to be the signal for instant confusion.... The mine which Time has slowly dug beneath familiar objects is sprung in an instant; and what was rock before, becomes but sand and dust.”
—Charles Dickens (1812–1870)