Let \(\displaystyle f\) be a real valued function on an open subset \(\displaystyle E^2.\) Prove that if \(\displaystyle \frac{\partial}{\partial x}\left(\frac{\partial f}{\partial y}\right)\) and \(\displaystyle \frac{\partial}{\partial y}\left(\frac{\partial f}{\partial x}\right)\) exist and are continuous then they are equal.
So I need to show that if the set \(\displaystyle \{(x,y)\in E^2 : x\in[a,b], \ y\in[c,d]\}\) is entirely contained in the set in which \(\displaystyle f\) is defined, then \(\displaystyle \int^b_a\left(\int^d_c\frac{\partial}{\partial x}\left(\frac{\partial f}{\partial y}\right)dy\right)dx=\int^b_a\left(\int^d_c\frac{\partial}{\partial y}\left(\frac{\partial f}{\partial x}\right)dy\right)dx.\) But I am not sure how I can continue the proof.
So I need to show that if the set \(\displaystyle \{(x,y)\in E^2 : x\in[a,b], \ y\in[c,d]\}\) is entirely contained in the set in which \(\displaystyle f\) is defined, then \(\displaystyle \int^b_a\left(\int^d_c\frac{\partial}{\partial x}\left(\frac{\partial f}{\partial y}\right)dy\right)dx=\int^b_a\left(\int^d_c\frac{\partial}{\partial y}\left(\frac{\partial f}{\partial x}\right)dy\right)dx.\) But I am not sure how I can continue the proof.