Zela101
  • Zela101
How is this F conservative?
Mathematics
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SOLVED
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katieb
  • katieb
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Zela101
  • Zela101
\(\large F(x, y) = ye^{xy} i + xe^{xy} j\); P (−1, 1), Q(2, 0) The integration of either of the equations are not identical BUT each consist of xs and ys. \(\Large \frac{\partial \phi}{\partial x}=ye^{xy}\) ---> \(\phi=y^2e^{xy}+K(y)\) \(\Large \frac{\partial \phi}{\partial y}=xe^{xy}\) ---> \(\phi=x^2e^{xy}+K(x)\)
Zela101
  • Zela101
@ganeshie8
Zela101
  • Zela101
The answer has \(\large \phi=e^{xy}\)

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ganeshie8
  • ganeshie8
\[\large F(x, y) = ye^{xy} i + xe^{xy} j\] \(M = ye^{xy}\) \(N = xe^{xy}\) \(N_x = e^{xy} + xye^{xy}\) \(M_y = e^{xy} + xye^{xy}\) \(\text{curl}(F) = N_x-M_y = 0\) Since the curl is 0, the vector field is conservation and consequently there exists a potential function \(\phi\)
Miracrown
  • Miracrown
Very neat
ganeshie8
  • ganeshie8
\(\large F(x, y) = ye^{xy} i + xe^{xy} j\); 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
Zela101
  • Zela101
Oh, lol i took the derivatives sorry. \(\color{red}{\phi=e^{xy}+K}\)
Zela101
  • Zela101
I know where my mistake was lol i need some sleep xD
ganeshie8
  • ganeshie8
:) I think you're trying to work it using a line integral...
Zela101
  • Zela101
to be honest, i have no idea why i did that
ganeshie8
  • ganeshie8
It is a very nice method too
Zela101
  • Zela101
Thanks again :)
Zela101
  • Zela101
I did several problems where i integrated it, but when it came to this question i took the derivative for some stupid reason.

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