Differentiation/integration of two variables

[Hobold]

Though this problem came to me when studying thermodynamics, it's purely mathematical and seems to be easy, but I'm not getting anywhere.

Suppose I have a function H = H(x,y). Differentiating this function I get:

$$dH = \left (\frac{\partial H}{\partial x} \right )_y dx + \left (\frac{\partial H}{\partial y} \right )_x dy$$

in which the subscript means that x or y is taken constant during the operation.

It is also given that $$\left (\frac{\partial H}{\partial y} \right )_x = \left (\frac{\partial F}{\partial x} \right )_y$$

so

$$dH = \left (\frac{\partial H}{\partial x} \right )_y dx + \left (\frac{\partial F}{\partial x} \right )_y dy$$

When I integrate from (x0,y0) to (x1,y1) I get

$$\int_{ (x_0,y_0)}^{ (x_1,y_1)} dH = \int_{x_0}^{x_1} dH |_{y_0} + \int_{y_0}^{y_1} dH |_{x_1} = \int_{x_0}^{x_1} \left (\frac{\partial H}{\partial x} \right )_y dx |_{y_0} + \int_{y_0}^{y_1} \left (\frac{\partial F}{\partial x} \right )_y dy |_{x_1}$$

What is killing me is if the integral

$$\int_{y_0}^{y_1} \left (\frac{\partial F}{\partial x} \right )_y dy |_{x_1}$$

is null, because in the derivative, you take y constant, but when integrating, x is constant. Therefore, as F = F(x,y) and both variables are taken constant, any partial derivative must be zero, so the integral is also zero. Is that right?

 

[Hobold]

Nevermind, I just noticed that my question makes no sense at all, as you are differentiating first then integrating.

Thanks, nonetheless.