Section 6.10 Divergence Theorem
6.10 Divergence Theorem
In this section, we study Divergence Theorem. Recall that the flux form of Green’s theorem states that
\[\int\int_{D}\text{div}\overrightarrow{F}dA=\int_{C}\overrightarrow{F}\cdot\overrightarrow{N}ds.\] The divergence theorem is a version of Green’s theorem in one higher dimension. If we think of divergence as a derivative of sorts, then the divergence theorem relates a triple integral of derivative div[latex]\overrightarrow{F}[/latex] over a solid to a flux integral of [latex]\overrightarrow{F}[/latex] over the boundary of the solid. More specifically, the divergence theorem relates a flux integral of vector field [latex]\overrightarrow{F}[/latex] over a closed surface [latex]S[/latex] to a triple integral of the divergence of [latex]\overrightarrow{F}[/latex] over the solid enclosed by [latex]S[/latex].
Theorem: The Divergence Theorem
Let [latex]S[/latex] be a piecewise, smooth closed surface that encloses solid [latex]E[/latex] in space. Assume that [latex]S[/latex] is oriented outward, and let [latex]\overrightarrow{F}[/latex] be a vector field with continuous partial derivatives on an open region containing [latex]E[/latex] . Then
\[\int\int\int_{E}\text{div}\overrightarrow{F}dV=\int\int_{S}\overrightarrow{F}\cdot dS\]
\[=\int\int_{S}\overrightarrow{F}\cdot\overrightarrow{N}dS\]
\[=\int\int_{D}\overrightarrow{F}\cdot(\overrightarrow{s_{u}}\times\overrightarrow{s_{v}})dA.\]
Example 1: Find [latex]\int\int_{S}\overrightarrow{F}\cdot dS[/latex], where [latex]\overrightarrow{F}=\lt ze^{y},xy,z^{2}\gt[/latex] and [latex]S[/latex] is the surface, oriented outward, bounded by [latex]x^{2}+z^{2}=4[/latex], [latex]y=1[/latex] and [latex]y=-1[/latex].
Exercise 1: Find [latex]\int\int_{S}\overrightarrow{F}\cdot dS[/latex], where [latex]\overrightarrow{F}=\lt yz+\text{cos}(z),e^{x+z}+y,z^{2}\gt[/latex] and [latex]S[/latex] is the surface, oriented outward, bounded by [latex]x^{2}+y^{2}=4[/latex], [latex]z=0[/latex] and [latex]z=1[/latex].
Example 2: Find [latex]\int\int_{S}\overrightarrow{F}\cdot dS[/latex], where [latex]\overrightarrow{F}=\lt x^{3}+z^{3},x^{3}+y^{3},z^{3}+x^{3}\gt[/latex] and [latex]S[/latex] is the surface of sphere radius [latex]3[/latex], oriented outward.
Exercise 2: Find [latex]\int\int_{S}\overrightarrow{F}\cdot dS[/latex], where [latex]\overrightarrow{F}=\lt x^{3}+y^{3},z^{3}+y^{3},z^{3}+y^{3}\gt[/latex] and [latex]S[/latex] is the surface of sphere radius [latex]2[/latex], oriented outward.
Example 3: Find [latex]\int\int_{S}\overrightarrow{F}\cdot dS[/latex], where [latex]\overrightarrow{F}=\lt xz,x^{2}+z^{2},z^{2}\gt[/latex] and [latex]S[/latex] is the surface, oriented outward, bounded by [latex]x^{2}+z^{2}=9[/latex], [latex]y=0[/latex] and [latex]y=z+1[/latex].
Exercise 3: Find [latex]\int\int_{S}\overrightarrow{F}\cdot dS[/latex], where [latex]\overrightarrow{F}=\lt yz,x^{2}+y^{2},yz\gt[/latex] and [latex]S[/latex] is the surface, oriented outward, bounded by [latex]x^{2}+y^{2}=4[/latex], [latex]z=0[/latex] and [latex]z=x-2[/latex].
Example 4: Find [latex]\int\int_{S}\overrightarrow{F}\cdot dS[/latex] where [latex]\overrightarrow{F}=\lt \frac{1}{3}x^{3}+y^{3},\frac{1}{3}\text{sin}(z)^{3}+x^{2},\frac{1}{3}z^{3}+y^{4}\gt[/latex] and [latex]S[/latex] is the surface bounded by [latex]y=4-x^{2}-z^{2}[/latex] with [latex]y\geq0[/latex], oriented outward.
Group work:
1. Find [latex]\int\int_{S}\overrightarrow{F}\cdot dS[/latex] where [latex]\overrightarrow{F}=\lt xe^{y},z-e^{y},-xy\gt[/latex] and [latex]S[/latex] is the ellipsoid [latex]x^{2}+4y^{2}+5z^{2}=20[/latex] oriented outward.
2. Find [latex]\int\int_{S}\overrightarrow{F}\cdot dS[/latex] where [latex]\overrightarrow{F}=\lt 2x^{3}+y^{3},\frac{1}{3}\text{tan}(z)^{3}+y^{3},3y^{2}z\gt[/latex] and [latex]S[/latex] is the surface bounded by [latex]z=9-x^{2}-y^{2}[/latex] with [latex]z\geq0[/latex], oriented outward.
3. Find [latex]\int\int_{S}\overrightarrow{F}\cdot dS[/latex], where [latex]\overrightarrow{F}=\lt xy,-y^{2}+z^{2},3yz\gt[/latex] and [latex]S[/latex] is the surface, oriented outward, bounded by [latex]x^{2}+y^{2}=25[/latex], [latex]z=0[/latex] and [latex]z=y+5[/latex].