# Define continuity for mapping

1. Aug 4, 2011

### squenshl

1. The problem statement, all variables and given/known data
a) Let f: RN to RM. Define continuity for mapping f. How does this relate to the notion of metric (norm)?
b) Define the Jacobian J of f. Write Taylor series expansion (for f) up to first degree at x = x0. Explain the terms.
c) Let y = f(x) $\in$ RM and yj = |f(x)|j = sum from k = 1 to N of ajkxk. What is the Jacobian of f? How are the rows of the Jacobian related to the gradients of yj with respect to x?

2. Relevant equations
Taylor series

3. The attempt at a solution
I think I can do a but I am completely stuck on b and c. Any help please.

2. Aug 5, 2011

### HallsofIvy

Staff Emeritus
Re: Mapping

Okay, what answer did you give for (a)? And (b) just asks for the definition of "Jacobian". Isn't that given in your book?

3. Aug 17, 2011

### squenshl

Re: Mapping

Any ideas on 5c?