Can you prove line integral for vector fields?

Question

As I was learning about the proof of line integral for vector field, there is something I don't understand.

Lets take a vector field F and then look at a point on the curve with tangent T. We are interested in the vector component that has the same direction as the tangent right? But how can dot product guarantee that?

$\vec{F} \cdot \vec{T} = |\vec{F}|\cdot|\vec{T}| \cos(\theta)$

Don't we have to make sure that $\theta$ is giving us the right direction? What if these two vectors were point as opposite directions, the dot product will still give us a negative value (if I'm not mistaken) but we don't want to do that right? Sorry if I was not clear, I'm confused myself.

Answer 1

You're right: when the curve moves in a direction "against" the vector field, the integrand for the line integral is negative. So if it's always "against" the vector field, the total line integral value will be negative. This also means that if $\gamma$ is a curve from $p$ to $q$ and $-\gamma$ is the reverse of the same curve, from $q$ to $p$ along the same points, then their line integrals are exactly opposite:

$$ \int_{-\gamma} \vec{F} \cdot d\vec{s} = -\int_\gamma \vec{F} \cdot d\vec{s} $$

We are interested in the vector component that has the same direction as the tangent right?

More exactly, we're interested in the component along the same line as the vector field at the point. This could be in either direction.