Problem 3. Let V be a vector space over a field F. Prove that -(-v) = v for all v € V.

Let F(x, y, z) = (yz², xz², 2xyz) and f be a scalar field such that Vf = F. Evaluate f-(1, –1, 1). Select one: O a. 0 O b. None of these O c. -2 O d. 2 Ое. 1

For each of the following vector fields, find its curl and determine if it is a gradient field. (a) ♬ = 5yz i + (5xz + z²) j + (5xy+2yz) k curl F F? = (b) Ğ = (10xy + x³) i+ (5x² + z²) j + (2yz – 4z) k curl G G ? = (c) H = 5yzi+ (z² – 5xz) j + (5xy + 2yz) k: curl H H? =

Consider the following expressions for second derivatives of scalar field ø(x, y, z) and vector field A (x, y, z). State whether or not each expression makes sense, and if not, state why. Also point out any that are always equal to zero. grad(grad(ø)) div(grad(ø)) curl(grad(ø)) grad(div(A)) div(div(A)) curl(div(A)) grad(curl(A)) div(curl(A)) curl(curl(A))