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:
“Sense is a line, the mind is a circle. Sense is like a line which is the flux of a point running out from itself, but intellect like a circle that keeps within itself.”
—Ralph J. Cudworth (16171688)