Different methods of calculating line integral?

Question

I remember the way that it was taught when I was taking Multivariable Calculus was to parameterize the curve and vector field. This also seems to be the method I've found online.

However, in my current class to solve such a problem, we simply take the dot product of the vector field with the differential length, which if it's in rectangular coordinates is: xdx+ydy+z*dz.

What's the difference between the two methods? Why do you need to unnecessarily parameterize everything if you can simply use the second method?

Answer 1

There is no difference computationally, just presentation. Given a vector field $F$ and a path $\gamma$, the vector-line integral or work integral is given by:

$$\int_C F \cdot d\textbf{r} = \int_{a}^b F(\textbf{r}) \cdot \textbf{r}' \ dt$$

Notice that if you let $\textbf{r} = (u(t),y(t)) = (x,y)$ then $\textbf{r}' = (dx,dy) = (u'(t) dt, y'(t) dt)$. Letting $F = (F^1,F^2)$ we have:

$$\int_{C} F \cdot d \textbf{r} = \int_{C} F^1 dx + F^2 dy$$