Question

I have the vector field F(x,y)= e^x sin(y)i + e^x cos(y)j. It is conservative, and the potential = e^x sin(y)i + e^x sin(y)j . I would like to evaluate the line integral with parameter r(t)=ln(t)i +tj from pi 14 years ago

Answer 1

Give me just a second to work it out and I will tell you :P

Answer 2

Great, thanks!

Answer 3

As far as I know when evaluating line integrals in a vector field the potential is unimportant. What you are looking for is:
\[\int\limits_C \vec{F}(\vec{r}(t)) \cdot \vec{r}'(t)dt\]
So plugging in your parameter in your field you get:
\[\vec{F}(\vec{r}(t))=e^{\ln(t)}\sin(t)\hat{i}+e^{\ln(t)}\cos(t)\hat{j}=t \sin(t)\hat{i}+t \cos(t)\hat{j}\]
Now differentiating your parameter you get:
\[\vec{r}'(t)=\frac{\hat{i}}{t}+\hat{j}\]
Now your dot product is:
\[\sin(t)+t \cos(t)\]
So your integral becomes:
\[\huge\int\limits_{\pi}^{\frac{3\pi}{2}}\sin(t)+t \cos(t) dt=t \sin(t)|_{\pi}^{\frac{3\pi}{2}}\]
I made it big so you could see the limits and everything.
Evaluating that I get negative 3pi/2. :D

Answer 4

Make sense? I'm guessing thats how you did it since we got the same answer xD

Answer 5

I got that too, and as well, I checked it with the fundamental theorem of line integrals. So that would lead me to believe that it is correct. Not a good answer, a GREAT one, thanks for the help!!

Answer 6

You're absolutely welcome :) If you have anymore I'll do what I can to help. I'm on a tutoring listserv at my university so I'm trying to get some experience helping people. I'm glad I could be of help to you and good luck in your course/studies :)