Vector fields, line integrals and surface integrals - Why one measures flux across the boundary and the other along?

If you really want to understand how Green's Theorem works and similar results such as Divergence Theorem and Stokes Theorem, I suggest you look into the Extended Fundamental Theorem of Calculus, which states that $$\int_C dw=\int_{\partial C} w$$ If you want an easy, intuitive approach, I suggest doing pages 63 and 64 this. Or if you want a slightly more rigorous version, I suggest reading through section 8, "Differential Geometry", of this


You are mixing up two different things; the surface integral is not a generalization of the line integral.* This is easiest to see in two dimensions, where everything is an integral along curves and yet you will still find a difference.

For simplicity, let's consider a constant wind field blowing to the right, $\mathbf f(x,y)=(1,0)$. Also consider two curves, $A$ a horizontal line segment from $(0,0)$ to $(1,0)$, and $B$ a vertical line segment $B$ from $(0,0)$ to $(0,1)$. In 2D, given a vector field and a curve there are two different kinds of integral you can consider.

  1. Interpret the curve as a wire on which a bead is threaded. If you move the bead from one end to the other, how much does the wind help or hinder the motion of the bead? This is the usual line integral $\int \mathbf f\cdot\mathrm d\mathbf r$. It is large for curve $A$ and zero for curve $B$.

  2. Interpret the curve as a butterfly net being held stationary while the wind blows through it. How much air passes through it per unit time? This is the flux integral $\int \mathbf f\cdot\mathbf n\,\mathrm d\ell$, where $\mathbf n$ is the unit vector perpendicular to the curve tangent. It is zero for curve $A$ and large for curve $B$.

These two notions generalize to higher dimensions in different ways. The line integral remains an integral over a $1$-dimensional object, i.e. a curve. The flux integral becomes an integral over a over an $(n-1)$-dimensional object, i.e. a surface.

*In an abstract sense one could argue that they are both specializations of the same thing, but that will take us too far into the theory of differential forms.

Tags:

Multivariable Calculus

Greens Theorem

Vector Fields

Line Integrals

Surface Integrals

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