What is the Theorem for Differentiability in Advanced Calculus?

[nateHI]

Homework Statement

This isn't a standard homework problem. We were asked to do research and to find a theorem of the form:
If something about the partial derivatives of u and v is true then the implication is $D(u,v)$ at $(x_0,y_0)$ exists from $R^2$ to $R^2$

Homework Equations

The Attempt at a Solution

I've done a lot of reading on the the difference in differentiability between $R^2$ and $\mathbb{C}$ but haven't been lucky enough to stumble upon an exact theorem. Anyway, a push in the right direction would save me a lot of time.

 

[Svein]

A little hint: In ℝ² , you may calculate the partial derivatives $\frac{\partial u}{\partial x}, \frac{\partial u}{\partial y}, \frac{\partial v}{\partial x}, \frac{\partial v}{\partial y}$ and you do not expect any obvious relation between them. On the other hand, in ℂ, you expect the derivative of f(z) to be just a function of z and nothing else.
So, if you put z = x + iy and f(z) = u(z) + iv(z), this implies a relation between u and v...

 

[nateHI]

I found it thanks! It's kind of a long theorem but if you're interested to know what it is let me know and I'll type. It doesn't have a distinct name that I can just reference for you.

 

[BvU]

Wouldn't they be called the Cauchy-Riemann equations ?

 

[nateHI]

Wouldn't they be called the Cauchy-Riemann equations ?

That's what I thought at first and I suppose the class will get to Cauchy-Riemann eventually. But the instructor stressed the fact that we were not working in the complex numbers for this problem. The Theorem he was looking for is from Advanced calculus. He probably wants to demonstrate the advantages of $\mathbb{C}$.