Interpretation of vector line integrals

Question

I recently came across two types of line integrals which I had not previously come across before and am unsure as to what their interpretation would be:

Integrating a scalar field with vector line element, e.g. $\exists \phi:\mathbb{R}^{3}\to\mathbb{R}$: $$\int_{\gamma}\phi\:\mathrm{d}\vec{\ell}$$

And:

Integrating the cross product of a vector field with a vector line element, e.g. $\exists \vec{\psi}:\mathbb{R}^{3}\to\mathbb{R}^{3}$: $$\int_{\gamma}\vec{\psi}\times \mathrm{d}\vec{\ell}$$

I would appreciate any example as to use/interpretation of these types of line integrals.

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Addendum: I should probably add that I came across these line integrals in a simple "evaluate the following" setting in a Physics-based assigment.

Answer 1

The geometrical or physical interpretation of these integrals has to come from the people who set them up. Nevertheless, you have to be aware that $$\vec A:=\int\nolimits_\gamma \phi\ \vec{d\ell},\qquad \vec B:=\int\nolimits_\gamma \vec\psi\times \vec{d\ell}\tag{1}$$ can be considered as a shorthand notation for a limit of certain Riemann sums. By interpreting these sums physically one obtains an intuitive idea of the intended "limit quantities" $\vec A$ and $\vec B$.

For both integrals we are given a curve $$\gamma:\quad t\mapsto{\bf x}(t)\qquad(a\leq t\leq b)$$ embedded in a region $\Omega\subset{\mathbb R}^3$. In addition we are given for $\vec A$ a scalar field (e.g., a temperature distribution) $\phi:\ \Omega\to{\mathbb R}$, and for $\vec B$ a vector field (e.g., a force field, or a flow field) $\vec\psi:\ \Omega\to{\mathbb R}^3$.

Choosing a fine partition $$a=t_0<t_1<t_2<\ldots<t_N=b\qquad (N\gg1)$$ we obtain a finite sequence $${\bf x}_k:=\gamma(t_k)\qquad(0\leq k\leq N)$$ of "subsequent points ${\bf x}_k$ along $\gamma\>$". By definition of the integrals $(1)$ one then has $$\vec A\doteq\sum_{k=1}^N \phi({\bf x}_k)({\bf x}_k-{\bf x}_{k-1}),\qquad \vec B\doteq\sum_{k=1}^N \psi({\bf x}_k)\times({\bf x}_k-{\bf x}_{k-1})\ .$$ When $\phi$ and $\vec\psi$ are sufficiently smooth one may consider $\phi$ and $\vec\psi$ as constant on the segments $[{\bf x}_{k-1},{\bf x}_k]$. It is then up to the beholder to give some physical interpretation to the sums appearing here.

Answer 2

The easiest (though not the most elegant) definition of $\int_\gamma \phi\, d\vec{\ell}$ is via coordinates: the integral is a vector with components $\int_\gamma \phi\, dx$, $\int_\gamma \phi\, dy$, $\int_\gamma \phi\, dz$.

The same idea applies to the integral $\int_\gamma \vec{\psi}\times\,d\vec{\ell}$. The components of the resulting vector are $v_x=\int_\gamma (\psi_y\,dz - \psi_z\,dy)$, $v_y=\int_\gamma (\psi_z\,dx - \psi_x\,dz)$, $v_z=\int_\gamma (\psi_x\,dy - \psi_y\,dx)$.

The more elegant and geometrical approach uses the concept of a vector-valued differential form. You'll find a nice treatment of this in Gravitation, by Misner, Thorne, and Wheeler, sec. 14.5. Once you have a meaning for $d\vec{\ell}$, it's a short step to $\int_\gamma \phi\,d\ell$, likewise $\int_\gamma \vec{\psi}\times\,d\vec{\ell}$.

Turning to a very rough-and-ready intuitive approach, it's best to focus on $d\ell$ first and add in the other complications later. Chop the path $\gamma$ into many short ("infinitesimal") directed segments. Each one can be regarded as a vector $\Delta \vec{\ell}_i$. Taking the vector sum, $\sum_{i=1}^n \Delta\vec{\ell}_i$, you just get the vector from the starting point of $\gamma$ to its endpoint.

Now form the similar vector sum: $$\sum_{i=1}^n \phi(p_i)\Delta\vec{\ell}_i$$ where $p_i$ is a point on $\gamma$ between the start and end of $\Delta\vec{\ell}_i$. Taking finer and finer divisions, this sum approaches $\int_\gamma \phi\,d\vec{\ell}$.

You can interpret the sum $\sum_{i=1}^n \phi(p_i)\Delta\vec{\ell}_i$ geometrically: by stringing the vectors $\phi(p_i)\Delta\vec{\ell}_i$ together ("tail-to-head" vector addition), we get another (polygonal) path, starting at the same point as the start point of $\gamma$, but with a different endpoint. Taking limits, we get a tranformed curve, call it $\hat{\phi}(\gamma)$; the integral $\int_\gamma \phi\,d\vec{\ell}$ is the vector from the startpoint to the endpoint of $\hat{\phi}(\gamma)$.

Of course, in a rigorous treatment, we'd need various assumptions about continuity, etc.

Notice that this all depends on being able to move vectors around freely; i.e., given vectors $\vec{v}_i$ with tails at different basepoints, we haven't worried about adding them together. This is OK for $\mathbb{R}^3$, but becomes an issue on a arbitrary manifold. Gravitation discusses this in detail.