What quantity does a line integral represent?

Question

I'm currently trying to wrap my head around line integrals, Green's theorem, and vector fields and I'm having a bit of difficulty understanding what a line integral represents geometrically.

Is it basically the arc length of a curve, for a scalar field?

And then when you bring the concept into a vector field, then what does it represent?

Answer 1

For example if the integrand is a function of point mass, then the line integral over a curve will give you the mass of that curve. Also, when you integrate over a vector field, it physically represents the work done by the field on a particle that moves along a path.

A line integral can help you to calculate the area of a fence or curtain that's below the curve, in $\mathbb R^3.$

Rodolfo Llinás says that we learn when we put things into context. I believe him, and these interpretations of line integrals are very classical.

Answer 2

There are at least two worthwhile interpretations of a line integral in two-dimensions.

First, $\int_C (\vec{F} \cdot T)ds = \int_C Pdx+Qdy$ measures the work or circulation of the vector field along the oriented curve $C$. This integral is largest when the vector field aligns itself along the tangent direction of $C$. As this relates to Green's Theorem we obtain the usual form of Green's Theorem which is identified with the $z$-component of the curl a bit later in the course. For $C = \partial R$ $$ \int_{\partial R} (\vec{F} \cdot T)ds = \iint_R (\nabla \times \vec{F})_z dA $$

Second, $\int_C (\vec{F} \cdot n)ds = \int_C Pdy-Qdx$ measures the flux of the vector field emitted through the oriented curve $C$. This integral is largest when the vector field aligns itself along the normal direction of $C$. As this relates to Green's Theorem we obtain the so-called divergence-form of Green's Theorem which is related to the Divergence Theorem in due course. $$ \int_{\partial R} (\vec{F} \cdot n)ds = \iint_R (\nabla \cdot \vec{F}) dA $$

If you want to read more, one source would be pages 357-367 of my multivariable calculus notes which were heavily influenced by Taylor's excellent calculus text.