Divergence and Curl and Their Geometric Interpretations
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1 Divergence and Curl and Their Geometric Interpretations 1 Scalar Potentials and Their Gradient and Laplacian Fields 2 Coordinate Transformations in the Vector Analysis Package 3 Using Vector Derivative Functions in the Vector Analysis Package 4 A Visualization Example of the Curl There is a very useful free software tool for solving minimal surface (and many other) variational problems called Surface Evolver by Ken Brakke. To use Surface Evolver to greatest possible advantage, a user should be adept at using results from vector analysis. Mathematica's Vector Analysis package is very helpful aid for developing powerful Evolver codes. The following example is extracted from the Surface Evolver manual. 40 LeavingKansas@x_, y_, z_, n_d := z n ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ ÅÅÅÅÅÅ Hx^2 + y^2l Hx^2 + y^2 + z^2l n 8y, -x, 0< 2 41 LeavingKansas@x, y, z, 3D y z ÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄ ÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄ Hx 2 + y 2 L Hx 2 + y 2 + z 2 L, - x z ÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄ ÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄ, 0= 3ê2 Hx 2 + y 2 L Hx 2 + y 2 + z 2 3ê2 L 42 << Graphics`PlotField3D`
2 2 Lecture-13.nb Visualize the vector field for n=3, note that the function will be singular near the z-axis 43 y, z, 3D, 8x, -1, 1<, 8y, -1, 1<, 8z, -.5,.5<, VectorHeads Ø True, ColorFunction Ø *.66DL &L, PlotPoints Ø 15, ScaleFactor Ø 0.5D Power::infy : Infinite expression ÅÅÅÅÅ 1 encountered. More 0 ::indet : Indeterminate expression 0 ComplexInfinity encountered. More ::indet : Indeterminate expression 0 ComplexInfinity encountered. More ::indet : Indeterminate expression 0 ComplexInfinity encountered. More General::stop : Further output of ::indet will be suppressed during this calculation. More Power::infy : Infinite expression ÅÅÅÅÅ 1 encountered. More 0 Power::infy : Infinite expression ÅÅÅÅÅ 1 encountered. More 0 General::stop : Further output of Power::infy will be suppressed during this calculation. More
3 Lecture-13.nb 3 Ö Graphics3D Ö
4 4 Lecture-13.nb We could make the function better behaved along the z-axis by brute force: y_, z_, n_d := Module A 8CindRadsq = x^2 + y^2<, CindRadsq = If@CindRadsq 10-4, 10-4, CindRadsq, CindRadsqD; z n ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ ÅÅÅÅÅÅÅÅÅÅ ÅÅÅÅÅÅ CindRadsq HCindRadsq + z^2l n 8y, -x, 0< E 2 PlotVectorField3D@LeavingKansasNicely@x, y, z, 3D, 8x, -1, 1<, 8y, -1, 1<, 8z, -.5,.5<, VectorHeads Ø True, ColorFunction Ø HHHue@# *.66DL &L, PlotPoints Ø 15, ScaleFactor Ø 0.5D
5 Lecture-13.nb 5 Ö Graphics3D Ö
6 6 Lecture-13.nb Or simply by avoiding the axis altogether and using the symmetry of the field 46 y, z, 3D, 8x,.01, 1<, 8y,.01, 1<, 8z,.01,.5<, VectorHeads Ø True, ColorFunction Ø *.66DL &L, PlotPoints Ø 15, ScaleFactor Ø 0.5D
7 Lecture-13.nb 7
8 8 Lecture-13.nb Ö Graphics3D Ö Calculate the curl of the function using the VectorAnalysis package--note that the coordinate system is specified as cartesian. For the particular case of n=3: 47 y, z, 3D, y, zdd êê Simplify 3 x z ÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄ, 3 y z ÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄ, 3 z ÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄ = Hx 2 + y 2 + z 2 L 5ê2 Hx 2 + y 2 + z 2 L 5ê2 Hx 2 + y 2 + z 2 L 5ê2 Define a new vector function for the curl for general n 48 Glenda@x_, y_, z_, n_d := Simplify@Curl@LeavingKansas@x, y, z, nd, Cartesian@x, y, zddd Demonstrate the assertion that the curl has a fairly simple form and is sphericaly symmetric for n=1 49 Glenda@x, y, z, nd 9n x z -1+n Ix 2 + y 2 + z 2 M -1- ÄÄÄÄÄ n 2, n y z -1+n Ix 2 + y 2 + z 2 M -1- ÄÄÄÄÄ n 2, n z n Ix 2 + y 2 + z 2 M -1- ÄÄÄÄÄ n 2 = 50 Glenda@x, y, z, 1D x 9 ÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄ Hx 2 + y 2 + z 2 L, y ÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄ 3ê2 Hx 2 + y 2 + z 2 L, z ÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄ 3ê2 Hx 2 + y 2 + z 2 L = 3ê2 The above is a vector field that points radially from the origin, with a magnitude that falls off like 1 ê r 2 Visualize the curl for n=3 51 PlotVectorField3D@Evaluate@Glenda@x, y, z, 3DD, 8x, 0,.5<, 8y, 0,.5<, 8z, 0.1,.5<, VectorHeads Ø True, ColorFunction Ø HHHue@# *.66DL &L, PlotPoints Ø 7D
9 Lecture-13.nb 9 Ö Graphics3D Ö
10 10 Lecture-13.nb Demonstrate that the divergence of the curl vanishes for the above function independent of n 52 DivCurl = Div@Glenda@x, y, z, nd, Cartesian@x, y, zdd 2 J-1 - n ÄÄÄÄÄ 2 J-1 - n ÄÄÄÄÄ 53 Simplify@DivCurlD 0 2 N n x2 z -1+n Ix 2 + y 2 + z 2 M -2- n ÄÄÄÄÄ J-1 - ÄÄÄÄÄ n 2 N n y2 z -1+n Ix 2 + y 2 + z 2 M -2- n ÄÄÄÄÄ N n z1+n Ix 2 + y 2 + z 2 M -2- n ÄÄÄÄÄ n z -1+n Ix 2 + y 2 + z 2 M -1- ÄÄÄÄÄ n 2 + n 2 z -1+n Ix 2 + y 2 + z 2 M -1- ÄÄÄÄÄ n 2
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