Fundamental theorem in 2 dimensions.

[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?

 

[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.
$$\frac{\partial^2}{\partial y \partial x}=\frac{\partial^2}{\partial x \partial y}$$

 

[bobby2k]

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.
$$\frac{\partial^2}{\partial y \partial x}=\frac{\partial^2}{\partial x \partial y}$$

Hi, thank you for your answer.
Can you please show how it follos from these 2? I think I get how to use the 1 fundamental theorem twice to see this, but how can you use stokes?