Find the gradient vector for the scalar function. (That is, find the conservative vector field for the potential function.) $f ( x , y ) = 10 x ^ { 2 } + 8 x y + 3 y ^ { 2 }$
A) $\nabla f ( x , y ) = ( 6 y + 8 x ) \hat { \mathbf { i } } - ( 20 x + 8 y ) \hat { \mathbf { j } }$
B) $\nabla f ( x , y ) = 6 y \hat { \mathbf { i } } - 20 x \hat { \mathbf { j } }$
C) $\nabla f ( x , y ) = 6 y \hat { \mathbf { i } } + 20 x \hat { \mathbf { j } }$
D) $\nabla f ( x , y ) = ( 20 x + 8 y ) \hat { \mathbf { i } } + ( 6 y + 8 x ) \hat { \mathbf { j } }$
E) $\nabla f ( x , y ) = 20 x \hat { \mathbf { i } } + 6 y \hat { \mathbf { j } }$

Question

Quizplus · Accepted Answer

The gradient vector of a scalar function is given by taking the partial derivatives of the function with respect to each variable and setting them as the x and y components of a vector, respectively. Therefore,
$$
abla f(x,y) = \frac{\partial f}{\partial x} \hat{\mathbf{i}} + \frac{\partial f}{\partial y}\hat{\mathbf{j}} = (20x + 8y)\hat{\mathbf{i}} + (6y + 8x)\hat{\mathbf{j}}.$$
Thus, the best choice is A.

Find the Gradient Vector for the Scalar Function f(x,y)=10x2+8xy+3y2f ( x , y ) = 10 x ^ { 2 } + 8 x y + 3 y ^ { 2 }f(x,y)=10x2+8xy+3y2

Find the Gradient Vector for the Scalar Function $f ( x , y ) = 10 x ^ { 2 } + 8 x y + 3 y ^ { 2 }$