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# Vector Calculus watch

1. Can anyone give me starting to point to prove that a linear map f is differentiable everywhere when it goes from Real N to Real M and f(x) is an mxn matrix acting on a vector x?

I can't see how to show something is differentiable?

What I've looked up online talks about smooth functions - what's a smooth function?

Also does anyone have a simple guide/definition and/or example on how to determine the derivative of a vector function from one dimension to another and also what a jacobian matix is?

Sorry for all the questions... I'm trying to catch up with everything so I don't get dragged down for the exams.
2. I've found a similar question in a past exam where they state the function is differentiable as all the partial derivatives are continuous. Is there a theorem that backs up this reasoning?
3. The gradient of a function , denoted by , is the mxn matrix , where . This matrix is called the Jacobian matrix.

Smooth function: http://mathworld.wolfram.com/SmoothFunction.html

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