Question

Does there exist a function f(x, y, z) such that fx = x2yz − e2xyz and fy = 2xyz − ye2xyz?

Answer 1

No, there does not exist a function f(x, y, z) that satisfies the given partial derivatives.

Step-by-step explanation:

To determine if a function f(x, y, z) exists given the partial derivatives fx = x²yz - e²xyz and fy = 2xyz - ye²xyz, we need to check if the mixed partial derivatives fyx and fxy are equal.

Calculating fyx = (fx)y = (2xyz - ye²xyz)y = 2xz - e²xyz - 2xye²xyz = 2xz - e²xyz(1 + 2ye)

Calculating fxy = (fy)x = (x²yz - e²xyz)x = 2xyz - x²ye²xyz = 2xyz - e²xyz(x²y)

Since fyx and fxy are not equal, there does not exist a function f(x, y, z) that satisfies the given partial derivatives.