- Understand the divergence theorem
- Calculate flux using the divergence theorem
- Apply the divergence theorem to physical problems
Overview of Theorems
Before examining the divergence theorem, it is helpful to begin with an overview of the versions of the Fundamental Theorem of Calculus we have discussed. Each theorem relates an integral of a “derivative” over a domain to an integral over the boundary of that domain.
- The Fundamental Theorem of Calculus:
[latex]\large{\displaystyle\int_a^bf^\prime(x)dx=f(b)-f(a)}[/latex].
This theorem relates the integral of derivative [latex]f'[/latex] over line segment [latex][a,b][/latex] along the [latex]x[/latex]-axis to a difference of [latex]f[/latex] evaluated on the boundary. - The Fundamental Theorem for Line Integrals:
[latex]\large{\displaystyle\int_C\nabla f\cdot d{\bf{r}}=f(P_1)-f(P_0)}[/latex].
where [latex]P_0[/latex] is the initial point of [latex]C[/latex] and [latex]P_1[/latex] is the terminal point of [latex]C[/latex]. The Fundamental Theorem for Line Integrals allows path [latex]C[/latex] to be a path in a plane or in space, not just a line segment on the [latex]x[/latex]-axis. If we think of the gradient as a derivative, then this theorem relates an integral of derivative [latex]\nabla f[/latex] over path [latex]C[/latex] to a difference of [latex]f[/latex] evaluated on the boundary of [latex]C[/latex]. - Green’s theorem, circulation form:
[latex]\large{\displaystyle\iint_D(Q_x-P_y)dA=\displaystyle\int_C{\bf{F}}\cdot d{\bf{r}}}[/latex].
Since [latex]Q_x-P_y=\text{curl }{\bf{F}}\cdot{\bf{k}}[/latex] and curl is a derivative of sorts, Green’s theorem relates the integral of derivative curl[latex]{\bf{F}}[/latex] over planar region [latex]D[/latex] to an integral of [latex]{\bf{F}}[/latex] over the boundary of [latex]D[/latex]. - Green’s theorem, flux form:
[latex]\large{\displaystyle\iint_D(P_x+Q_y)dA=\displaystyle\int_C{\bf{F}}\cdot{\bf{N}}ds}[/latex].
Since [latex]P_x+Q_y=\text{div }{\bf{F}}[/latex] and divergence is a derivative of sorts, the flux form of Green’s theorem relates the integral of derivative div [latex]{\bf{F}}[/latex] over planar region [latex]D[/latex] to an integral of [latex]{\bf{F}}[/latex] over the boundary of [latex]D[/latex]. - Stokes’ theorem:
[latex]\large{\displaystyle\iint_S\text{curl }{\bf{F}}\cdot ds=\displaystyle\int_C{\bf{F}}\cdot d{\bf{r}}}[/latex]
If we think of the curl as a derivative of sorts, then Stokes’ theorem relates the integral of derivative curl[latex]{\bf{F}}[/latex] over surface [latex]S[/latex] (not necessarily planar) to an integral of [latex]{\bf{F}}[/latex] over the boundary of [latex]S[/latex].