# Fundamental theorem in 2 dimensions.

1. Jun 9, 2013

### bobby2k

Hello

I have heard that Greens, Stokes and the Divergence theorem is the equivalent of the fundamental theorem in multiple dimensions. But is there some way to show the result under:

if
F(x,y) = $\int_{-\infty}^x\int_{-\infty}^yf(x^{*},y^{*})dx^{*}dy^{*}$
this implies that
f(x,y)=$\frac{\partial^{2} F(x,y)}{\partial x\partial y}$

Can tis be showed with Greens or Stokes, or derived on its own?

2. Jun 9, 2013

### lurflurf

Well Stoke's theorem is the fundamental theorem of every dimension.
$$\int_{\Omega} \mathop{d}\omega=\int_{\partial \Omega} \omega$$
Your example follows form Stoke's theorem or using the 1d fundamental theorem twice and equality of mixed partials.
$$\frac{\partial^2}{\partial y \partial x}=\frac{\partial^2}{\partial x \partial y}$$

3. Jun 10, 2013