Finding potential Functions of Vector Field

Question

F‎ind all p‎otential fu‎nctions of the con‎servative ve‎ctor field F~ (x, y, z) = < 2xz + y ²,2xy, x ² +3z²>. Th‎en use the Funda‎mental Theor‎em for Line In‎tegrals to ev‎aluate ∫c F~ · d R~ , where C is the cu‎rve defin‎ed by R~ ( t) = <t ² , t + 1 , 2 t − 1 > , 0 ≤ t‎ ≤ 1

Answer 1

Let find curlF = det{(i j k) (∂/∂x ∂/∂y ∂/∂z) (2xz + y² 2xy x² + 3z²)] = (0)i - (2x - 2x)j + (2y - 2y) =k = 0.

So, field F is potential field.

Let potential of field is G(x,y,z); ∂G/∂x = 2xz + y²; G(x,y,z) = ∫(2xz + y²)dx = x²z + y²x + h(y,z);

∂G/∂y = 2yx + ∂h(y,z)/∂y = 2xy; ∂h(y,z)/∂y = 0; h(y,z) = k(z) + C;

G(x,y,z) = x²z + y²x + k(z) + C; ∂G/∂z = x² + k'(z) = x² + 3z²; k'(z) = 3z²; k(z) = ∫3z²dz = z³ + D

G(x,y,z) = x²z + xy² + z³ + D, D is arbitrary constant.

∫_CFdR = ∫_C∇GdR = G(B) - G(A);

A = R(0) = (0,1, -1); B = R(1) = (1, 2, 1). ∫_CFdR = (1 + 4 + 1 + D) - (-1 + D) = 7