Question

Determine whether the following vector fields are conservative on R 2 . If the vector field is conservative then find a potential function. F =<e^x cos y, −e^x sin y>.

Answer 1

$$\vec F =<e^x \cos y, −e^x \sin y>$$

$$\dfrac{\partial P}{\partial y}=-e^x\sin y, \dfrac{\partial Q}{\partial x}=-e^x\sin y$$

$$\dfrac{\partial P}{\partial y}=-e^x\sin y=\dfrac{\partial Q}{\partial x}$$

$\vec F$ satisfies the condition $P_y=Q_x.$ Moreover, it is defined on all of $R^2,$ hence it is conservative.

Let us find a potential function $f(x, y)$ for $\vec F .$ We want

$$f_x=P=e^x\cos y$$

$$f_y=Q=-e^x\sin y$$

$$f=\int Pdx=\int e^x\cos ydx=e^x\cos y+g(y)$$

$$f_y=-e^x\sin y+g'(y)$$

Then

$$-e^x\sin y+g'(y)=-e^x\sin y$$

$$=>g'(y)=0=>g(y)=C$$

The potential function is

$$f(x, y)=e^x\cos y+C$$

Choosing the constant $С = 0,$ we obtain the potential function

$$f(x, y)=e^x\cos y$$