Suppose that F: Rn -> Rn is continuously differentiable everywhere

[matthewturner]

Please help me with this. I don't know even how to start

Definition: Suppose that F: Rn Rn is continuously differentiable everywhere. A point P∈R^n is called an isolated singularity of F if DF_p is not invertible but DF_y is invertible for all Y≠P in some neighborhood of P.

a. Let f: R -> R by f(x) = 3 x^4 - 20 x^3. Prove that f has exactly two points which are isolated singularities. Describe what happens at each point.

b. Let f denote the transformation that takes rectangular coordinates into polar coordinates. Does f have any isolated singularities? If so, identify them and explain what happens at each point?

c. Write down explicitly a continuously differentiable function F : R2 -> R2 that has an isolated singularity at the origin and no other singularity.

 

[pat_connell]

a. The function f(x) = 3 x^4 - 20 x^3 has two points which are isolated singularities. At the point x = 0, the derivative is 0 and therefore DF_0 is not invertible. However, for any other point Y ≠ 0, the derivative is non-zero and therefore DF_y is invertible. The second point of isolated singularity is at x = 5/2, where the derivative is zero and DF_5/2 is not invertible. b. The transformation f that takes rectangular coordinates into polar coordinates does not have any isolated singularities. This is because at every point (x, y) in the rectangular coordinates, the Jacobian matrix of f is always invertible. c. An example of a continuously differentiable function F : R2 -> R2 that has an isolated singularity at the origin is given by F(x, y) = (x^2 + y^2, x y). At the origin (0, 0), the derivative is (0, 0) and therefore DF_0 is not invertible. However, at any other point (x, y) ≠ (0, 0), the derivative is non-zero and therefore DF_x is invertible.