Chapter 17.1, Problem 46E

Gradient fields on curves For the potential function φ and points A, B, C, and D on the level curve φ(x, y) = 0, complete the following steps.

a. Find the gradient field F = ∇φ.

b. Evaluate F at the points A, B, C, and D.

c. Plot the level curve φ(x, y) = 0 and the vectors F at the points A, B, C, and D.

46. $\phi (x,y)=\frac{32-x^4-y^4}{32}$; A(2, 2), B(−2, 2), C(−2, −2), and D(2, −2)