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?
Best Answer
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.