Find the gradient vector field of the scalar function $f$. (That is, find the conservative vector field $\mathbf { F }$ for the potential function $f$ of $\mathbf { F }$.)
$$f ( x , y , z ) = 7 x y ^ { 2 } + 2 y z ^ { 2 }$$
A) $\mathbf { F } ( x , y , z ) = \left( 7 y ^ { 2 } \right) \mathbf { i } + \left( 2 z ^ { 2 } \right) \mathbf { j } + ( 4 y z ) \mathbf { k }$
B) $\mathbf { F } ( x , y , z ) = ( 14 x y ) \mathbf { i } + ( 14 x y ) \mathbf { j } + \left( 2 z ^ { 2 } \right) \mathbf { k }$
C) $\mathbf { F } ( x , y , z ) = \left( 7 y ^ { 2 } \right) \mathbf { i } + \left( 14 x y + 2 z ^ { 2 } \right) \mathbf { j } + ( 4 y z ) \mathbf { k }$
D) $\mathbf { F } ( x , y , z ) = \left( 7 y ^ { 2 } \right) \mathbf { i } + ( 14 x y ) \mathbf { j } + ( 4 y z ) \mathbf { k }$

Question

Quizplus · Accepted Answer

The gradient vector field of a scalar function $f$ is given by
$$\nabla f = \frac{\partial f}{\partial x}\mathbf{i} + \frac{\partial f}{\partial y}\mathbf{j} + \frac{\partial f}{\partial z}\mathbf{k}.$$
In this case, we have
$$\frac{\partial f}{\partial x} = 7y^2,\quad \frac{\partial f}{\partial y} = 14xy,\quad \text{and}\quad \frac{\partial f}{\partial z} = 4yz.$$
Therefore, the gradient vector field of $f$ is given by
$$\mathbf{F}(x,y,z) = \left(7y^2\right)\mathbf{i} + \left(14xy+2z^2\right)\mathbf{j} + \left(4yz\right)\mathbf{k}.$$

Find the Gradient Vector Field of the Scalar Function fff (That Is, Find the Conservative Vector Field

Find the Gradient Vector Field of the Scalar Function $f$ (That Is, Find the Conservative Vector Field