I'm trying to understand the geometric intuition behind the definition of the line integrals over vector fields. The definition is given below:

Definition: Let $\vec{F}$ be a continuous vector field defined on a smooth curve $\gamma$ given by a vector function $r(t)$. Then the line integral of $\vec{F}$ along $\gamma$ is

$$\int_{\gamma}\vec{F}\cdot d\vec{r}=\int_{\gamma} \vec{F}\cdot\vec{T} ds$$

Where $T$ is the unit tangent.

So the line integral of the vector field $\vec{F}$ along $\gamma$ is defined as line integral over a scalar field. The geometric interpretation of this one can be found here.

So using the geometric interpretation of line integrals over scalar fields, I'm trying to understand this one over vector fields.

In the definition above the scalar product $\vec{F}\cdot \vec{T}$ is a function $\alpha(x,y)$ which takes a point in the curve $\gamma$ and gives out a point with $\alpha(x,y)=|\vec{T}|$ (Since $T$ is a unit vector, $\vec{F}\cdot \vec{T}$ is the length of the projection of the vector $\vec{F}(x,y)$ over the tangent).

So using the geometric interpretation of the line integral over scalar fields, is the integral $\int_{\gamma}\vec{F}\cdot d\vec{r}$ the area below the curve $\alpha$? If yes, why is this geometric relevant?

  • $\begingroup$ The vector field is some sort of force. And if you are moving with the field the force is at your back (or you are moving down hill) and you are gaining energy, and if the force is in your face you pay energy to get anywhere, and if you move orthogonal to the field it costs you nothing. If the force is conservative, then it doesn't matter what path you take. And if it is not, then there is some sort of vortex (curl) and there will be paths that are more with the flow and paths that fight the flow. $\endgroup$ – Doug M May 4 '17 at 6:16
  • $\begingroup$ @DougM Thank you for your comment, but I know the classical physical motivation of this integral, I'm looking for a mathematical geometrical motivation. $\endgroup$ – user42912 May 4 '17 at 6:19
  • 1
    $\begingroup$ en.m.wikipedia.org/wiki/Line_integral this has several gifs that help to illustrate their points on how the line integral of scalar and vector fields generalize the one dimensional Riemann integral. Unfortunately the second one is a little fast $\endgroup$ – Triatticus May 4 '17 at 7:48

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