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

question 36

Multiple Choice

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) \(\mathbf { F } ( x , y , z ) = \left( 7 y ^ { 2 } \right) \mathbf { i } + ( 14 x y ) \mathbf { j } + ( 4 y z ) \mathbf { k }\)