Determine if the vector field F(x, y,z)=(xy2z2+x)i+(x2yz2+y)j+(x2y2z)k is conservative. curl(F)= 0i+0j+0k functionsequation editor Therefore F A. Is conservative B. Is not conservative If F is conservative find a function f such that F=∇f. If not write "DNE". f= x^2y^2z^2+x^2 functionsequation editor + K

1) Option A is correct.

F is conservative because the curl of F = (0î + 0j + 0k)

2) f = (1/2) [x²y²z² + x² + y²] + K

Step-by-step explanation:

F(x,y,z) = (xy²z² + x)i + (x²yz² + y)j + (x²y²z)k

1) For a vector field to be conservative, the curl of that vector field has to be a zero vector. That is,

∇ × F = 0î + 0j + 0k

∇ = (∂/∂x) î + (∂/∂y) j + (∂/∂z) k

F = (xy²z² + x)i + (x²yz² + y)j + (x²y²z)k

Vector cross product is usually evaluated in form of a determinant.

The curl is evaluated on the attached image to this question.

As shown in the attached image,

(∇ × F) is indeed equal to (0î + 0j + 0k)

2) F = ∇f

F = (xy²z² + x)i + (x²yz² + y)j + (x²y²z)k

∇ = (∂/∂x) î + (∂/∂y) j + (∂/∂z) k

∇f = (∂f/∂x) î + (∂f/∂y) j + (∂f/∂z) k

(xy²z² + x)i + (x²yz² + y)j + (x²y²z)k = (∂f/∂x) î + (∂f/∂y) j + (∂f/∂z) k

Therefore,

(∂f/∂x) = (xy²z² + x) (eqn 1)

(∂f/∂y) = (x²yz² + y) (eqn 2)

(∂f/∂z) = (x²y²z) (eqn 3)

We can start the integration from any of the above, but the constant of integration will be what is a bit technical to evaluate.

Starting with the first one; (eqn 1)

(∂f/∂x) = (xy²z² + x)

f = ∫ (xy²z² + x) dx

f = (x²y²z²/2) + (x²/2) + h(y,z)

note that h(y,z) is the constant of integration, since, (∂f/∂x) was done keeping y and z constant.

f = (x²y²z²/2) + (x²/2) + h(y,z)

We then differentiate with respect to y

(∂f/∂y) = (x²yz²) + h'(y,z)

Comparing this with the expression for (∂f/∂y) we have above in (eqn 2)

(∂f/∂y) = (x²yz² + y)

h'(y,z) = y

h(y,z) = ∫ y dy

h(y,z) = (y²/2) + g(z)

Note again that g(z) = constant of integration, since h(y,z), which is a function of y and z has its partial derivative (∂h/∂y) done keeping z constant.

f = (x²y²z²/2) + (x²/2) + h(y,z)

f = (x²y²z²/2) + (x²/2) + (y²/2) + g(z)

Just like above, we then differentiate with respect to z

(∂f/∂z) = (x²y²z) + g'(z)

Comparing again with the expression for (∂f/∂z) from above in (eqn 3)

(∂f/∂z) = (x²y²z)

(x²y²z) + g'(z) = (x²y²z)

g'(z) = 0

(∂g/∂z) = 0

g(z) = ∫ 0 dz

g(z) = 0 + K (where K is the number constant of integration)

Hence,

f = (x²y²z²/2) + (x²/2) + h(y,z)

f = (x²y²z²/2) + (x²/2) + (y²/2) + g(z)

f = (x²y²z²/2) + (x²/2) + (y²/2) + K

f = (1/2) [x²y²z² + x² + y²] + K

Hope this Helps!!!