The expression
is confusing. What is the limit equal to?
Suppose . Then is differentiable at a point if there is an n by m matrix such that .
My question is, does it matter HOW goes to 0? For example, consider the function . Say I wanted to calculate its derivative.
When taking the limit of the derivative difference quotient, is it permissible to let the entries of the matrix approach 0 in any fashion I want?
I am asking this because I am just learning multi-variable calculus, and when calculating the derivatives I've always let = (h h) and was wondering if I am allowed to do this.
If you look here, you will see that all of the partial derivatives can exist, and the function still not be differentiable at the point. So, it must be that must be allowed to approach zero in any direction whatever. That is the condition of differentiability. Now, if you're asking how to compute the derivative, then I would think you could let it go to zero in any particular way want; you should be able to get the same answer regardless.
Suppose that f is a function that isn't differentiable, but the derivative difference quotient goes to 0 if h goes to 0 along a straight line through the origin.
If I am computing the derivative, and I let h go to 0 along a straight line through the origin, then the derivative difference quotient would go to 0, and I would think that the derivative of f is some matrix B, even though f isn't really differentiable, since if I let h go to 0 in another way, the limit wouldn't exist.
How do I get past this problem?
Well, I would think you'd need to convince yourself the derivative existed before you went looking for it. That, at least, is the typical thing a mathematician would do. According to the wiki, "It is known that if the partial derivatives of a function all exist and are continuous in a neighborhood of a point, then the function must be differentiable at that point,..."