Suppose that
If , find .
Hint: As a first step, define a path from (0,0,0) to (1, 1, 5) and compute a line integral.
I have no idea how to even set this up! I've been having a lot of troubles with complex line integrals, please help!
Suppose that
If , find .
Hint: As a first step, define a path from (0,0,0) to (1, 1, 5) and compute a line integral.
I have no idea how to even set this up! I've been having a lot of troubles with complex line integrals, please help!
the theorem says
$\displaystyle \int_C \nabla f \cdot d \bold{r} = f(\bold{r}(b)) - f(\bold{r}(a))$
but we also know that $\displaystyle \int_C \bold{F} \cdot d \bold{r} = \int_C \nabla f \cdot d \bold{r}$
Thus you have $\displaystyle \int_C \nabla f \cdot d \bold{r} = f(1,1,5) - f(0,0,0)$
where $\displaystyle C$ is the curve i parameterized for you. you can find the integral on the left (similar to the way you pointed out, except you do not want big F), and you know the value of $\displaystyle f(0,0,0)$ so you can find $\displaystyle f(1,1,5)$
I know I must sound really stupid right now, but I knew all of that, I just been having a heck of a time integrating $\displaystyle \int_C \nabla f \cdot d \bold{r}$. I know that I use the given function and the parameterization you graciously gave me and plug it in, but I just can't integrate the resulting functions!
the curve is $\displaystyle r(t) = \left< t, t, 5t \right>$ for $\displaystyle 0 \le t \le 1$. and so, $\displaystyle r'(t) = \left< 1,1,5 \right>$
so that $\displaystyle \int_C \nabla f \cdot d \bold{r} = \int_0^1 \nabla f(t,t,5t) \cdot r'(t)~dt$
can you finish now? (or did you know that already?)
tell me the integral you ended up with