Answer :
Answer:
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!!!
