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!!!

(1, –1), (16, –4), (36, –6)

step-by-step explanation:

given that the function is

this can also be written as

we have to graph this function and check which list of points lie on the graph

see attachment for graph:

we find that the points lying are (1,-1) (16,-4) and (36,-6)

since square root is there we find that x can only be posiitve but y can be since square root both positive and negative.

its adding 5.5 each time

step-by-step explanation: