Please solve this math

Question

If f(x,y) is continuous over R:a≤x≤b,c≤y≤d and F(x,y)=∫ x a ∫ y c f(u,v) dudv on the interior of R, find the second partial deri.

Answer 1

The mixed partial second derivative of F(x,y) = f(x,y).

The other second partial derivatives are more complicated.

Answer 2

DF / Dy = ∫ x a f (u,y) du, where DF / Dy denotes the partial derivative with respect to y, and ∫ x a stands for the integral from a to x.

To see this, first switch the order of the double integral:

F(x,y)=∫ x a ∫ y c f(u,v) dudv = ∫ y c ( ∫ x a f(u,v) du) dv.

Now, apply the Fundamental Theorem of Calculus, and you get the result right away.