For a vector field $\vec F$ and a curve $C$ with length $c$, if $\vec F$ is tangent to $C$ at every point, then the line integral of $\vec F$ along $C$ is:
$$\int_C \vec F \cdot d\vec r = \int_C ( ||\vec F|| *||d\vec r||)= ||\vec F|| *\int_C||d\vec r|| =||\vec F|| *c$$
I use $\cdot$ to mean the dot product and $*$ to mean multiplication.
I'm not sure about this reasoning. I don't think I'm using integrals correctly.
I get to $||\vec F|| * ||d\vec r||$ because $\vec F$ is always tangent to $C$, which means $\cos\theta$ in the dot product formula is $1$. I then take $||\vec F||$ out of the integral, since it's a scalar.
The next step is the dubious one: I have an integral $\int_C||d\vec r||$. Is this technically a valid integral? I don't know if I'm allowed to play around with the differential in the integral like that.
Furthermore, would it even be correct to say that $\int_C||d\vec r|| = $ (length of the curve $C$) ?
Any help is greatly appreciated!