Question

I don't suppose anyone could assist me with the following problem (noted in the url): http://1337.is/~gaulzi/tex2png/view.php?png=201103301610252648.png

and if you are familiar with latex, then u can use this http://1337.is/~gaulzi/tex2png
to convert from tex2png... if you like that better

Answer 1

What is the path C1?

Answer 2

the parametrization that is given, is used for the path C1

Answer 3

oh..ok, overlooked it, sorry...^-^

Answer 4

what course is it?

Answer 5

im not sure what the english name for it is but if i translate it correctly it is Mathematical Analysis II

Answer 6

ok, where are you? It has been a while since I did path integral

Answer 7

do you know the conservative vector field?

Answer 8

well I think I can do it, but you need to know the "fundamental theorem of line integrals"

Answer 9

let \[P=2xsiny \implies \partial P/\partial y=2xcosy\]
and \[Q=x^2cosy-3y^2 \implies \partial Q/\partial x=2xcosy\]
we can see that the partial derivatives are the same, which means that the function is conservative (by the test of a conservative field), and there exist, by definition a potential function Φ such that:
\[\partial \phi/ \partial x =P \rightarrow (1) , \partial \phi/\partial y = Q \rightarrow (2)\]

Answer 10

integrate (1) with respect to x, you get:
\[\phi(x,y)=x^2siny+g(y) \rightarrow(3)\]
to find g(y), we differentiate equation (3) w.r.t y, we get:
\[\partial \phi/ \partial y=x^2cosy +g'(y) \rightarrow (4)\]but this should be the same as Q (as we said above), that's
\[x^2cosy+g'(y)= Q =x^2cosy -3y^2 \implies g'(y)=-3y^2\]
then g(y)= -y^3+c ( we can take c to be zero )
substitute g(y)=-y^3 in equation (3),
\[\phi(x,y)=x^2siny-y^3\]

Answer 11

from r(t) in your question we have two parameter functions,
x(t)=tcost , y(t)=t^2sint..
by the fundamental theorem of line integrals, the value of the integral
I=Φ(B)-Φ(A), where in our case, B=(x(2pi),y(2pi)), A(x(0), y(0))
\[x(2\pi)=2\pi , y(2\pi)= 0 , x(0)=0, y(0)=0\]

\[Φ(2\pi,0)= 0 , Φ(0,0)=0\]
the value of the line integral over C1 is
\[Φ(2\pi,0)-Φ(0,0)=0\]

Answer 12

the result does not seem right to me :( .. but that's what I got