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 ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ ÅÅÅÅÅÅÅÅÅÅÅÅ Hx^2 + y^2L Hx^2 + y^2 + z^2L n ÅÅÅÅ 2 8y, -x, 0< 41 LeavingKansas@x, y, z, 3D 9 yz 3 ÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄ ÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄ ÄÄÄÄÄÄÄÄÄÄÄÄÄ Ä Hx 2 + y 2 LHx 2 + y 2 + z 2 L 3ê2 , - xz 3 ÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄ ÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄ ÄÄÄÄÄÄÄÄÄÄÄÄÄ Ä Hx 2 + y 2 LHx 2 + y 2 + z 2 L 3ê2 ,0= 42 << Graphics`PlotField3D`
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Divergence and Curl and Their Geometric Interpretationspruffle.mit.edu/3.016-2011/pdf/L13/Lecture-13-4.pdf · Divergence and Curl and Their Geometric Interpretations 1 Scalar Potentials
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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.
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 Curl@LeavingKansas@x, y, z, 3D, Cartesian@x, y, zDD êê Simplify
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, zDDDDemonstrate 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 Ix2 + y2 + z2 M-1- nÄÄÄÄÄÄ2 , n y z-1+n Ix2 + y2 + z2 M-1- nÄÄÄÄÄÄ2 , n zn Ix2 + y2 + z2 M-1- nÄÄÄÄÄÄ2 =