find the domain of the function

a) f(x,y) = e^(-xy)

b) h,(u,v)= √(4-u^2 - v^2)

find the second-order partial derivatives of f(x,y)=x^3 + x^2y^2 + y^3 + x+y

and show that the mixed partial derivatives fxy and fyx are equal

f'(x)=3x^2+2xy^2+1

f'(Y)=2x^2+3y^2+1

f'(xy)=4xy
f'(yx)=4xy

For part B, your figure there is an ellipsoid, x^2+y^2+z^2=4, so the domain should be (x,y,z):X<4,y<4,z<4

To find the domain of a function, we need to determine the values of the variables for which the function is defined.

a) For the function f(x, y) = e^(-xy), exponential functions are defined for all values of x and y. Therefore, there are no restrictions on the domain. The domain is (-∞, +∞) for both x and y.

b) For the function h(u, v) = √(4 - u^2 - v^2), we need to consider the values that are within the square root. For the inside of the square root to be a real number, the value (4 - u^2 - v^2) must be greater than or equal to zero.

Setting 4 - u^2 - v^2 ≥ 0, we can solve for u and v:

u^2 + v^2 ≤ 4

This represents a circular region in the xy-plane centered at (0, 0) with a radius of 2. Therefore, the domain of h(u, v) is the set of all points (u, v) inside or on the circle of radius 2 centered at (0, 0). In interval notation, the domain can be expressed as D = { (u, v) | u^2 + v^2 ≤ 4 } or in set-builder notation as D = { (u, v) : u^2 + v^2 ≤ 4 }.