Question

Certainly, here's the statement with Unicode characters:

Prove the following extension of the Fundamental Theorem of Calculus to two variables: If ∂²F/∂x∂y = f(x, y), then
∬[R] f(x, y) dA = F(b, d) - F(a, d) - F(b, c) + F(a, c) where R = [a, b] × [c, d].

Answer 1

The extension of the Fundamental Theorem of Calculus to two variables states that the double integral of a function over a region is equal to the difference of the values of the function at the corners of the region.

Step-by-step explanation:

The extension of the Fundamental Theorem of Calculus to two variables states that if the second-order partial derivative of a function F with respect to x and y is equal to a function f(x, y), then the double integral of f(x, y) over a region R is equal to the difference of the values of F at the corners of R. In mathematical notation, this can be written as:

∫∫[R] f(x, y) dA = F(b, d) - F(a, d) - F(b, c) + F(a, c)

where R = [a, b] × [c, d] represents the rectangular region over which the double integral is taken.