Find the gradient vector field of the scalar function F. (That is, find the conservative vector field F for the potential function f of F.) $f ( x , y , z ) = 5 x y ^ { 3 } + 3 y z ^ { 2 }$
A) $$\mathbf { F } ( x , y , z ) = \left( 5 y ^ { 3 } \right) \mathbf { i } + \left( 15 x y ^ { 2 } \right) \mathbf { j } + ( 6 y z ) \mathbf { k }$$
B) $\mathbf { F } ( x , y , z ) = \left( 5 y ^ { 3 } \right) \mathbf { i } + \left( 15 x y ^ { 2 } + 3 z ^ { 2 } \right) \mathbf { j } + ( 6 y z ) \mathbf { k }$
C) $\mathbf { F } ( x , y , z ) = \left( 5 y ^ { 3 } \right) \mathbf { i } + \left( 3 z ^ { 2 } \right) \mathbf { j } + ( 6 y z ) \mathbf { k }$
D) $\mathbf { F } ( x , y , z ) = \left( 15 x y ^ { 2 } \right) \mathbf { i } + \left( 15 x y ^ { 2 } \right) \mathbf { j } + \left( 3 z ^ { 2 } \right) \mathbf { k }$

Question

Quizplus · Accepted Answer

We need to find the gradient vector field of the given scalar function F.

Gradient of F:

∇F = (∂F/∂x) i + (∂F/∂y) j + (∂F/∂z) k

Now,

∂F/∂x = 5y^3
∂F/∂y = 15x y^2 + 3z^2
∂F/∂z = 6yz

Therefore,

∇F = (5y^3) i + (15x y^2 + 3z^2) j + (6yz) k

Hence, the correct option is B.

Find the Gradient Vector Field of the Scalar Function F f(x,y,z)=5xy3+3yz2f ( x , y , z ) = 5 x y ^ { 3 } + 3 y z ^ { 2 }f(x,y,z)=5xy3+3yz2

Find the Gradient Vector Field of the Scalar Function F $f ( x , y , z ) = 5 x y ^ { 3 } + 3 y z ^ { 2 }$