Question

Calc 3 help!

Suppose that Pi + Qj is parallel to the tangent vector of a closed curve C.
2 questions about this! Please help!

Answer 1

a) Show that Qi-Pj is perpendicular to the tangent vector.

b) Show that \[\int\limits_{}^{}\int\limits_{D}^{}(\delta P/\delta x + \delta Q/\delta y)dxdy = 0\] where D is the region whose boundary is C

Answer 2

to show that Qi-Pj is perpendicular to Pi+Qj just take the dot product

Answer 3

so it should be 0, right?

what about part b?

Answer 4

still thinking on that part, you sure that's not supposed to be a - sign in the double integral?

Answer 5

Yeah, I wish it was too... But its supposed to be a +

Answer 6

\[\iint\limits_D\left(\frac{\partial P}{\partial x}+\frac{\partial Q}{\partial y}\right)dA=\oint\limits_CQdx-Pdy\]since we have established that Qi-Pj is perpendicular to every point on C, clearly this integral will be zero. I don't see a more rigorous way to prove it just yet.

Answer 7

does that at least make sense though?

Answer 8

still there?

Answer 9

Yeah, I'm back. That makes sense a little bit actually. Thank you!