Let's define a function
$$ I = \int dy\, dx\, \frac{d f_1 (y,x)}{d y dx}\, \tan x $$
What I want to do is to write the function as summation.
I'm sure there was no $d^2 f_1$ in the numerator I saw. Now I'm confused if this is a convention. Can someone explain this integration? Any reference would be much appreciated.
For example $f_1(x) = (x^2+2y^3)$. I can take the first differentiation for sure, but didn't understand how to work on with the $y$?
EDIT
Can it be done this way following the suggestion of @Allawonder:
$$I = \int dy dx \frac{f_x dx + f_y dy}{dy dx} tanx $$
$$I = \int df(x,y) \ tanx $$ If the above transformation is true can't we transform into summation like this? $$I = \int df(x,y) \ tanx = \sum_n f(x_i,y) \ tanx_i$$
I'll write $f=f_1$ for brevity. Then we have that $\mathrm df=f_x\mathrm dx+f_y\mathrm dy.$ Thus the integral becomes $$\int \tan x(f_x\mathrm dx+f_y\mathrm dy),$$ which is a line integral.