Fundamental Theorem of Line Integrals Semi Proof

Question

Here's a question I'm working on that I have little understanding of. I'm new to line integrals and vector fields, but here it goes:

Suppose $C$ is a continuous differential curve, and $r(t)$, $a\leq t \leq b$ is a vector function, show
$\int_{c}r\bullet dr = \frac{1}{2}[||r(b)||^2-||r(a)||^2]$

And as a hint I'm being told that the function $f(x) = \frac{1}{2}||x||^2$ should be used to find $\nabla f$, and that the fundamental theorem of line integrals should be used.

Since I'm new to this, where I'm stuck is finding out $\nabla f$. Is it just $<||x||>$ ? I don't see how this can be used to show the statement above.

Anyway, thanks.

Answer 1

Note that,

$$ \frac12 \| \vec{r} \|^2 = \frac12 ( x^2+y^2+z^2), $$

so that,

$$ \nabla \frac12 \| \vec{r} \|^2 = \frac12 < 2x, 2y, 2z> = <x,y,z> = \vec{r}.$$

Where each component of $\nabla \frac12 \| \vec{r} \|^2$ is computed with the appropriate partial derivative operation.