Divergence and Curl and Their Geometric Interpretations

1Scalar Potentials: Their Gradient Fields and Visualization

2Visualizing Gradient Fields and Laplacian of a Scalar Potential

3Coordinate Transformations in the Vector Analysis Package

4Coordinate Transforms Example: Calculating Distances between Two Cities

5Using Vector Derivative Functions in the Vector Analysis Package

6 A Visualization Example of the CurlThere 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.

1LeavingKansas@x_, y_, z_ , n_D :=zn

Hx^2 + y ^2L Hx^2 + y ^2 + z^2L n28y, -x, 0

Visualize the vector field for n=3, note that the function will be singular near the z-axis

3

Needs@"VectorFieldPlots`"D;

VectorFieldPlot3D@LeavingKansas@x, y, z, 3D,8x, -1, 1

4VectorFieldPlot3D@LeavingKansas@x, y, z, 3D,8x, 0, 1

8Glenda@x, y, z, 1D

: xIx2 + y2 + z2M32

,y

Ix2 + y2 + z2M32,

z

Ix2 + y2 + z2M32>

The above is a vector field that points radially from the origin, with a magnitude that falls off like 1 r 2Visualize the curl for n=1, it will be necessary to "zoom" in to see the field.

9VectorFieldPlot3D@Evaluate@Glenda@x, y, z, 1DD,8x, -0.5, 0.5