In Cartesian coordinates, a vector field takes the form F = 2rzi+2yzj + (x² + y²) k (b) Calculate the line integral of F along a straight-line path starting at the origin and ending at the point (a,b,c). This path has the parametric representation x = at, y=bt, z=ct (0 ≤t≤ 1). (c) Given that the point (a, b, c) could be anywhere, use your answer to part (b) to find the scalar potential function U(x, y, z) corresponding to F, such that F = -VU. (d) Hence, or otherwise, calculate the value of the line integral of F along a path defined by the parametric equations z = cost, y = sint, z = = t (0 ≤ t ≤ π). (Hint: You can use the potential function U calculated in part (c) to evaluate this line integral.)

Find the curl of the vector field at the given point. F(x, y, z) = x²zi - 2xzj + yzk; (3, −9, 9) curl F = 15i + 9j - 18k X

Consider the vector field v(x) j+6k. a) Calculate fv• dr, where I is the straight line from (4,0,4) to (0,4,8). b) Calculate fu• dr where C'is the curve from (4,0,4) to (0,4, 8) in the first quadrant given by x? + 4y? = 22 ,x + z = 8.

Find the curl of the vector field F = curl F = 7 + J+ k