Understanding Continuity and the Jacobian Matrix in Multivariable Functions

[squenshl]

Homework Statement

a) Let f: R^N to R^M. 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) $\in$ R^M and y_j = |f(x)|_j = sum from k = 1 to N of a_jkx_k. What is the Jacobian of f? How are the rows of the Jacobian related to the gradients of y_j with respect to x?

Homework Equations

Taylor series

The Attempt at a Solution

I think I can do a but I am completely stuck on b and c. Any help please.

 

[HallsofIvy]

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

 

[squenshl]

Any ideas on 5c?