Find the gradient vector for the scalar function. (That is, find the conservative vector field for the potential function.) $f ( x , y ) = \sin 4 x \cos 6 y$
A) $\nabla f ( x , y ) = - 4 \cos ( 4 x ) \cos ( 6 y ) \hat { \mathbf { i } } + 6 \sin ( 4 x ) \sin ( 6 y ) \hat { \mathbf { j } }$
B) $\nabla f ( x , y ) = 6 \cos ( 4 x ) \cos ( 6 y ) \hat { \mathbf { i } } - 4 \sin ( 4 x ) \sin ( 6 y ) \hat { \mathbf { j } }$
C) $\nabla f ( x , y ) = 4 \cos ( 4 x ) \cos ( 6 y ) \hat { \mathbf { i } } + 6 \sin ( 4 x ) \sin ( 6 y ) \hat { \mathbf { j } }$
D) $\nabla f ( x , y ) = 4 \cos ( 4 x ) \cos ( 6 y ) \hat { \mathbf { i } } - 6 \sin ( 4 x ) \sin ( 6 y ) \hat { \mathbf { j } }$
E) $\nabla f ( x , y ) = - 6 \cos ( 4 x ) \cos ( 6 y ) \hat { \mathbf { i } } - 4 \sin ( 4 x ) \sin ( 6 y ) \hat { \mathbf { j } }$

Question

Quizplus · Accepted Answer

The gradient vector $\nabla f(x, y)$ of a scalar function $f(x, y)$ is found by taking the partial derivatives of $f$ with respect to $x$ and $y$. For $f(x, y) = \sin(4x)\cos(6y)$, the partial derivative with respect to $x$ is $4\cos(4x)\cos(6y)$, and the partial derivative with respect to $y$ is $-6\sin(4x)\sin(6y)$. These form the components of the gradient vector, leading to the correct answer.

Find the Gradient Vector for the Scalar Function f(x,y)=sin⁡4xcos⁡6yf ( x , y ) = \sin 4 x \cos 6 yf(x,y)=sin4xcos6y

Find the Gradient Vector for the Scalar Function $f ( x , y ) = \sin 4 x \cos 6 y$