Question

Prove

Answer 1

Prove that a vector field \[F(x,y) = P(x,y)i+Q(x,y)j\] is conservative if \[\frac{ \partial p }{ \partial y } = \frac{ \partial Q }{ \partial x }\] by constructing the potential function, and showing that F(x,y) is the gradient of the function you constructed.

What I thought for this was, I just make up my own question sort of, letting P and Q being arbitrary functions, but I don't think that's right? @rational

Answer 2

Maybe I'm just reading it wrong, doesn't seem to hard.

Answer 3

is that "if" a one way conditional if or is it an iff (biconditional) ?

Answer 4

By symmetry of the question, iff. :P

Answer 5

Yeah!

Answer 6

No it's one way condition

Answer 7

\(\Rightarrow\)
Let \(f(x,y)\) be the potential function and \(P(x,y)=f_x\) and \(Q(x,y)=f_y\)
The conservative vector field is then \[F=\langle P,Q\rangle\]

Clearly \(P_y=f_{xy}=f_{yx}=Q_x\)

Answer 8

I think that should do for forward direction

Answer 9

\(\Leftarrow\)
Let the vector field be
\[F=\langle P,Q\rangle\]
such that \(P_y = Q_x\), we need to show that \(F\) is conservative

hmm this looks tricky, lets see..

Answer 10

To show that F is conservative, we need to prove that a potential function exists.

Answer 11

Clairaut's theorem?

Answer 12

\[\frac{ \partial P }{ \partial y } = \frac{ \partial ^2 f }{ \partial y \partial x } = \frac{ \partial ^2 f }{ \partial x \partial y } = \frac{ \partial Q }{ \partial x }\]

Answer 13

@dan815 @Kainui