Chapter 9: Vector Calculus
Section 9.8: Divergence Theorem
Example 9.8.7
Apply the Divergence theorem to the vector field F=x y3i+y z j+x2z k and R, the region bounded by the paraboloid z=x2+y2 and the paraboloid z=2−x2−y2.
Solution
Mathematical Solution
The divergence of F:
∇·F=∂xx y3+∂yy z+∂zx2z=y3+z+x2
Implement the integral of ∇·F over the interior of R:
∫−11∫−1−x21−x2∫x2+y22−x2−y2y3+z+x2 ⅆz ⅆy ⅆx = 76 π
To compute the flux through R, note that there are two boundaries, the lower paraboloid z=x2+y2, and the upper paraboloid z=2−x2−y2. The intersection of these two bounding surfaces is a circle of radius 1 lying in the plane z=1.
To compute the flux through the lower paraboloid, note that on it
F·N dσ
=x y3y zx2z·2 x2 y−1 11+4 x2+4 y21+4 x2+4 y2 dA
=2 x2y3+2 y2z−x2z dA
=2 x2y3+2 y2−x2 x2+y2 dA
If this be integrated over the unit disk, the result is
∫−11∫−1−x21−x22 x2y3+2 y2−x2 x2+y2 dy dx=π6
On the upper paraboloid,
=x y3y zx2z·2 x2 y1 11+4 x2+4 y21+4 x2+4 y2 dA
=2 x2y3+2 y2z+x2z dA
=2 x2y3+2 y2+x2 2− x2− y2 dA
Its integral over the unit disk is
∫−11∫−1−x21−x22 x2y3+2 y2−x2 2− x2− y2 dy dx=π
The total flux is then 16+1 π=76 π, the same value obtained for the volume integral of the divergence, as predicted by the Divergence theorem.
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 y3,y z,x2z = →to Vector Field →assign to a nameF
Obtain ∇·F, the divergence of F, and represent it parametrically
Common Symbols palette: Del and dot-product operators Context Panel: Evaluate and Display Inline
Context Panel: Assign to a Name≻divF
∇·F = y3+x2+z→assign to a namedivF
Obtain the volume integral of the divergence of F
Tools≻Load Package: Student Multivariate Calculus
Loading Student:-MultivariateCalculus
Context Panel: Assign to a Name≻Y
1−x2→assign to a nameY
Write the name given to the divergence. Context Panel: Evaluate and Display Inline
Context Panel: Student Multivariate Calculus≻Integrate≻Iterated Complete the dialogs as per the figures below.
Context Panel: Evaluate Integral
divF = y3+x2+z→MultiInt∫−11∫−−x2+1−x2+1∫x2+y2−x2−y2+2y3+x2+zⅆzⅆyⅆx=76π
There are two parts to the boundary of R, the lower paraboloid z=x2+y2 and the upper paraboloid z=2−x2−y2. The intersection of these two bounding surfaces is a circle of radius 1 lying in the plane z=1. For the flux through the lower paraboloid, use a task template.
Tools≻Tasks≻Browse: Calculus - Vector≻Integration≻Flux≻3-D≻Through a Surface Defined over a Disk
Flux through a Surface Defined over a Disk
For the Vector Field:
Select Coordinate SystemCartesian [x,y,z]Cartesian - othercylindricalsphericalbipolarcylindricalbisphericalcardioidalcardioidcylindricalcasscylindricalconicalellcylindricalhypercylindricalinvcasscylindricallogcylindricallogcoshcylindricaloblatespheroidalparaboloidalparacylindricalprolatespheroidalrosecylindricalsixspheretangentcylindricaltangentspheretoroidal
For the "open" surface z=x2+y2, Maple uses the normal Rx×Ry, where R=x i+y j+x2+y2 k is a position-vector description of this paraboloid. This gives a normal that is upward, and hence inward for the closed region R. Because R is a closed region, the normal should be outward, and hence downward; the flux through the lower paraboloid is actually π/6.
For the flux through the upper paraboloid can be obtained with the same task template.
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 y3,y z,x2z:
Obtain ∇·F, the divergence of F
Invoke the Divergence command.
divF≔DivergenceF
y3+x2+z
Use the int command to integrate the divergence of F over R
intdivF,x,y,z=Region−1..1,−1−x2..1−x2,x2+y2..2−x2−y2,output=integral
∫−11∫−−x2+1−x2+1∫x2+y2−x2−y2+2y3+x2+zⅆzⅆyⅆx
intdivF,x,y,z=Region−1..1,−1−x2..1−x2,x2+y2..2−x2−y2 = 76π
Use the Flux command to obtain the flux of F through the lower paraboloid
FluxF,Surfacex,y,x2+y2,x=−1..1,y=−1−x2..1−x2,output=integral
∫−11∫−−x2+1−x2+1−2x2y3−2y2x2+y2+x2x2+y2ⅆyⅆx
FluxF,Surfacex,y,x2+y2,x=−1..1,y=−1−x2..1−x2 = −16π
For the "open" surface z=x2+y2, Maple uses the normal Rx×Ry, where R=x i+y j+x2+y2 k is a position-vector description of the paraboloid. This gives a normal that is upward, and hence inward for the closed region R. Because R is a closed region, the normal should be outward, and hence downward; the flux through the paraboloid is actually π/6.
The flux on the upper paraboloid can also be obtained with the Flux command.
FluxF,Surfacex,y,2−x2−y2,x,y=Circle0,0,1,r,θ,output=integral = ∫01∫02π−r32cosθ4sinθr3−2cosθ2sinθr3−r2cosθ2+2cosθ2+2r2−4ⅆθⅆrFluxF,Surfacex,y,2−x2−y2,x,y=Circle0,0,1,r,θπ
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