Consider the vector field Fx y z 2xzi z2j x2 2yzk Calcul

Consider the vector field F(x, y, z) = 2xzi - z^2j + (x^2 - 2yz)k. Calculate the curl and divergence of F. Show that this is a conservative field. Find a function G(x, y, z) such that F = nabla G. Evaluate integral_C F.dr where C is the curve r(t) = 2ti +t^2j + t^3k and 0 lessthanorequalto t lessthanorequalto 1.

Solution

] Curl F =

           = i [-2z+2z] –j [2x-2x] + k[0]

            = 0

Div F =[ i] . [2xzi-z² j + (x²-2yz)k]

          = 2z – 0 -2y

           =2z-2y

b] Also, it is clearly a conservative field.

c]F =

2xzi-z² j + (x²-2yz)k = i

Equating, =2xz, = -z², =x²-2yz

Integrating w.r.t x,y,z resply.,

G =x²z, G=-z²y, G=x²z-z²y

Combining, G = x²z-z²y

d] r=2t i + t²j+ t³k

    dr = [2i + 2t j + 3t² k]dt

F.dr =[4xz-2z²t +3(x²-2yz)t²]dt

=[ 4xzt-2z²t²/2+3(x²-2yz)t³/3]¹₀

          =4xz-z²+x²-2yz