Chapter 9: Vector Calculus
Section 9.8: Divergence Theorem
Example 9.8.2
Apply the Divergence theorem to the vector field F=x i+y j+z k and R, the surface and interior of a cylinder of height 1, with central axis along the z-axis, and base in the plane z=0.
Solution
Mathematical Solution
The divergence of F:
∇·F=∂xx+∂yy+∂zz=1+1+1=3
Implement the integral of ∇·F over the interior of the cylinder in cylindrical coordinates:
∫02 π∫01∫013 r ⅆz ⅆr ⅆθ = 3 π
To compute the flux through the cylinder, note that there are three boundaries of R, the "wall" of the cylinder, and its circular top and bottom. Hence, there are three flux integrals to calculate.
Flux through the "wall" of the cylinder:
Describe this surface with the position vector R=cos(θ)sin(θ)z so that dσ=Rθ×Rz dA requires the calculation of
|ijk−sinθcosθ0001| = cos(θ)sin(θ)0 = N
and dσ=dA. Using the same parametrization for F, F·N on the wall of the cylinder becomes
cos(θ)sin(θ)z·cos(θ)sin(θ)0 = cos2θ+sin2θ=1
Consequently, the flux through the wall of the cylinder is given by
∫02 π∫011 dz dθ=2 π
Flux through the top of the cylinder:
Take N=k so F·N=z=1 on the top of the cylinder. Use polar coordinates to integrate this over the unit disk, obtaining
∫02 π∫01r dr dθ=π
Flux through the bottom of the cylinder:
Take N=−k so F·N=−z=0 on the bottom of the cylinder. Clearly, the flux integral on this bounding surface is zero.
In the Divergence theorem, the volume integral on the left has value 3 π and the surface flux integral(s) on the right, have values 2 π,π,0, which also sum to 3 π.
Maple Solution - Interactive
The Student VectorCalculus package is needed for calculating the divergence, but it then conflicts with any multidimensional integral set from the Calculus palette. Hence, the Student MultivariateCalculus package is installed to gain Context Panel access to the MultiInt command.
Initialize
Tools≻Load Package: Student Vector Calculus
Loading Student:-VectorCalculus
Tools≻Tasks≻Browse: Calculus - Vector≻Vector Algebra and Settings≻Display Format for Vectors
Press the Access Settings button and select "Display as Column Vector"
Display Format for Vectors
Define the vector field F
Enter the components of F in a free vector. Context Panel: Evaluate and Display Inline
Context Panel: Student Vector Calculus≻Conversions≻To Vector Field
Context Panel: Assign to a Name≻F
x,y,z = →to Vector Field →assign to a nameF
Obtain ∇·F, the divergence of F
Common Symbols palette: Del and dot-product operators
Context Panel: Evaluate and Display Inline
∇·F = 3
Use cylindrical coordinates to obtain the volume integral of the divergence of F
Tools≻Load Package: Student Multivariate Calculus
Loading Student:-MultivariateCalculus
Context Panel: Student Multivariate Calculus≻Integrate≻Iterated Complete the dialogs as per the figures below.
Context Panel: Evaluate Integral
3→MultiInt∫02π∫01∫013rⅆzⅆrⅆθ=3π
Use a task template to obtain the flux of F through the cylinder
Tools≻Tasks≻Browse: Calculus - Vector≻Integration≻Flux≻3-D≻Through a Parametric Surface
Flux through a Parametrically Defined Surface
For the Vector Field:
Select Coordinate SystemCartesian [x,y,z]Cartesian - othercylindricalsphericalbipolarcylindricalbisphericalcardioidalcardioidcylindricalcasscylindricalconicalellcylindricalhypercylindricalinvcasscylindricallogcylindricallogcoshcylindricaloblatespheroidalparaboloidalparacylindricalprolatespheroidalrosecylindricalsixspheretangentcylindricaltangentspheretoroidal
Select Coordinate SystemCartesiancylindricalsphericalbipolarcylindricalbisphericalcardioidalcardioidcylindricalcasscylindricalconicalellcylindricalhypercylindricalinvcasscylindricallogcylindricallogcoshcylindricaloblatespheroidalparaboloidalparacylindricalprolatespheroidalrosecylindricalsixspheretangentcylindricaltangentspheretoroidal
Use a task template to obtain the flux of F through the top of the cylinder
Tools≻Tasks≻Browse: Calculus - Vector≻Integration≻Flux≻3-D≻Through a Surface Defined over a Disk
Flux through a Surface Defined over a Disk
Use the same task template to obtain the flux of F through the bottom of the cylinder
Maple Solution - Coded
Install the Student VectorCalculus package.
withStudent:-VectorCalculus:
Set display of vectors via BasisFormat.
BasisFormatfalse:
Define F via the VectorField command.
F≔VectorFieldx,y,z:
Invoke the Divergence command.
DivergenceF = 3
Use the int command to integrate the divergence of F over R
int3,x,y,z=Region−1..1,−1−x2..1−x2,0..1,output=integral
∫−11∫−−x2+1−x2+1∫013ⅆzⅆyⅆx
int3,x,y,z=Region−1..1,−1−x2..1−x2,0..1 = 3π
Use the Flux command to obtain the flux of F through the cylinder
Use the Vector command to define the cylinder in cylindrical coordinates.
S≔Vector1,θ,z,cylindrical:
FluxF,SurfaceS,θ=0..2 π,z=0..1,output=integral+FluxF,Surfacex,y,1,x,y=Circle0,0,1,r,θ,output=integral+FluxF,Surfacex,y,0,x,y=Circle0,0,1,r,θ,output=integral
∫02π∫01cosθ2+sinθ2ⅆzⅆθ+∫01∫02πrⅆθⅆr+∫01∫02π0ⅆθⅆr
FluxF,SurfaceS,θ=0..2 π,z=0..1+FluxF,Surfacex,y,1,x,y=Circle0,0,1+FluxF,Surfacex,y,0,x,y=Circle0,0,1
3π
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