example: find the conservative vector field for the potential function given by: f(x,y)=sin(3x)cos(4y).

Explanation:

Step1: Recall the formula for conservative vector field

If $f(x,y)$ is a potential function, the conservative vector field $\vec{F}(x,y)=\left\langle\frac{\partial f}{\partial x},\frac{\partial f}{\partial y}\right\rangle$.

Step2: Calculate $\frac{\partial f}{\partial x}$

Using the chain - rule, if $f(x,y)=\sin(3x)\cos(4y)$, then $\frac{\partial f}{\partial x}=\cos(3x)\cdot3\cdot\cos(4y)=3\cos(3x)\cos(4y)$.

Step3: Calculate $\frac{\partial f}{\partial y}$

Using the chain - rule, $\frac{\partial f}{\partial y}=\sin(3x)\cdot(-\sin(4y))\cdot4=- 4\sin(3x)\sin(4y)$.

Answer:

$\vec{F}(x,y)=\langle3\cos(3x)\cos(4y), - 4\sin(3x)\sin(4y)\rangle$