Question

Trying to compute line integrals for C1 and C2 based from parameters. Image of my attempt attached. I'm pretty sure I'm doing f(x,y) wrong, but not why.

Answer 1

the given vector field is not conservative so you can't find a potential function

whats the complete question ?

Answer 2

It says given this vector field:\[F(x,y)=(x^2+y^2)\text i+9xy \ j \] compute the line integrals:\[\int\limits_{c1}F\cdot dr\]\[\int\limits_{c2}F\cdot dr\]paramaterized by, respectively:\[ r(t)=i t+j t^2\]\[ r(t)=i t+j t\]for \[0\leq t\leq 1\]

Answer 3

So I had that it wasn't conservative, but I where I think I'm messing up is getting the c1 and c2, which I believe is to show that they're different.

Answer 4

may i know why are you integrating the components of force in your attached screenshot ?

Answer 5

That was to get the potential function into which I would put the values of the ending and starting points so that I could get: \[\int\limits_{c1}F\cdot dr= F(r(b))-F(r(b))\]

Answer 6

That should be lowercase f on the right hand side of the equation.

Answer 7

but there won't be a potential function here because the curl is not 0

Answer 8

you should test whether the curl is 0 or not before attempting to find a potential function

Answer 9

So when they ask for: \[\int\limits_{C1} F \cdot dr\]given the parameterization of: \[r(t)=t\space i+t^2 \space j\]am I just computing the integral directly?

Answer 10

\[F(x,y)=(x^2+y^2)\text i+9xy \ j \]
\[r(t)=i t+j t^2 , ~0\le t\le 1 \]
x = t , dx = dt
y = t^2 , dy = 2tdt
\[\int\limits_{c1}F\cdot dr = \int\limits_{0}^1 (t^2+t^4) + (9t^3)2t~dt \]

Answer 11

Aaaaaah... :)

Answer 12

Yes, you just use the given parameterization for second path also. Nothing much to do here as the field is not conservative.. just evaluate the line integral straight

Answer 13

Amazing how much more sense that makes when you see someone do it right. ;) Thanks!

Answer 14

np:)