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

question 45

Multiple Choice

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) \(\mathbf { F } ( x , y , z ) = \left( 15 x y ^ { 2 } \right) \mathbf { i } + \left( 15 x y ^ { 2 } \right) \mathbf { j } + \left( 3 z ^ { 2 } \right) \mathbf { k }\)

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