Final answer:
The function f such that ∇f = 3x²ᵃ - 3y²ᵇ is f(x, y) = x³ - y³ + K. The line integral of the gradient of f over a curve C is equivalent to evaluating f at the endpoints of C.
Step-by-step explanation:
The student is asking for a function f(x, y) whose gradient is ∇f = 3x²ᵃ - 3y²ᵇ. To find this function, we integrate the partial derivatives. By integrating 3x² with respect to x, we obtain x³, and by integrating -3y² with respect to y, we get -y³. Therefore, a potential function for f is f(x, y) = x³ - y³ + K, where K is the constant of integration.
Using the potential function, we can evaluate the line integral ∫_C (3x²ᵃ - 3y²ᵇ) over a curve C, which by the gradient theorem (also known as the fundamental theorem for line integrals), reduces to evaluating the function f at the endpoints of C.