Question

How is this F conservative?

Answer 1

\(\large F(x, y) = ye^{xy} i + xe^{xy} j\);
P (−1, 1), Q(2, 0)

The integration of either of the equations are not identical BUT each consist of xs and ys.
\(\Large \frac{\partial \phi}{\partial x}=ye^{xy}\) ---> \(\phi=y^2e^{xy}+K(y)\)
\(\Large \frac{\partial \phi}{\partial y}=xe^{xy}\) ---> \(\phi=x^2e^{xy}+K(x)\)

Answer 2

@ganeshie8

Answer 3

The answer has \(\large \phi=e^{xy}\)

Answer 4

\[\large F(x, y) = ye^{xy} i + xe^{xy} j\]

\(M = ye^{xy}\)
\(N = xe^{xy}\)

\(N_x = e^{xy} + xye^{xy}\)
\(M_y = e^{xy} + xye^{xy}\)

\(\text{curl}(F) = N_x-M_y = 0\)

Since the curl is 0, the vector field is conservation and consequently there exists a potential function \(\phi\)

Answer 5

Very neat

Answer 6

\(\large F(x, y) = ye^{xy} i + xe^{xy} j\);
P (−1, 1), Q(2, 0)

The integration of either of the equations are not identical BUT each consist of xs and ys.
\(\Large \frac{\partial \phi}{\partial x}=ye^{xy}\) ---> \(\color{red}{\phi=y^2e^{xy}+K(y)}\)


could you double check above integration part

Answer 7

Oh, lol i took the derivatives sorry.
\(\color{red}{\phi=e^{xy}+K}\)

Answer 8

I know where my mistake was lol i need some sleep xD

Answer 9

:) I think you're trying to work it using a line integral...

Answer 10

to be honest, i have no idea why i did that

Answer 11

It is a very nice method too

Answer 12

Thanks again :)

Answer 13

I did several problems where i integrated it, but when it came to this question i took the derivative for some stupid reason.