File:Vector Field on a Sphere.png

This image could be re-created using vector graphics as an SVG file. This has several advantages; see Commons:Media for cleanup for more information. If an SVG form of this image is available, please upload it and afterwards replace this template with {{vector version available|new image name}}. It is recommended to name the SVG file “Vector Field on a Sphere.svg”—then the template Vector version available (or Vva) does not need the new image name parameter.

DescriptionVector Field on a Sphere.png	English: The vector field represented by ${\displaystyle \mathbf {f} =2x{\hat {\mathbf {x} }}+y^{2}{\hat {\mathbf {y} }}+z^{2}{\hat {\mathbf {z} }}}$ plotted over the unit sphere. Vectors have been uniformly scaled down to 0.2 times their original length for aesthetics. Originally created as an image for the example problem on the divergence theorem page to replace the less relevant image here.
Date	5 November 2011
Source	Own work
Author	Glosser.ca
Other versions	This math image could be re-created using vector graphics as an SVG file. This has several advantages; see Commons:Media for cleanup for more information. If an SVG form of this image is available, please upload it and afterwards replace this template with {{vector version available|new image name}}. It is recommended to name the SVG file “Vector Field on a Sphere.svg”—then the template Vector version available (or Vva) does not need the new image name parameter.

I, the copyright holder of this work, release this work into the public domain. This applies worldwide. In some countries this may not be legally possible; if so: I grant anyone the right to use this work for any purpose, without any conditions, unless such conditions are required by law.

scaling = 0.2;
stepSize = 1/10;

makeVectors[data_, f_] := Map[{Arrowheads[0.01], Arrow[{#, # + scaling*f @@ #}]} &, data]
upperHemi[x_, y_] := Sqrt[1 - x^2 - y^2]
lowerHemi[x_, y_] := -Sqrt[1 - x^2 - y^2]
f[x_, y_, z_] := {2 x, y^2, z^2}

upperPoints = 
  Flatten[Table[{x, y, upperHemi[x, y]}, {x, -1, 1, 
     stepSize}, {y, -Sqrt[1 - x^2], Sqrt[1 - x^2], stepSize}], 1] ;
lowerPoints = 
  Flatten[Table[{x, y, lowerHemi[x, y]}, {x, -1, 1, 
     stepSize}, {y, -Sqrt[1 - x^2], Sqrt[1 - x^2], stepSize}], 1];

upperVectors = makeVectors[upperPoints, f];
lowerVectors = makeVectors[lowerPoints, f];

Graphics3D[{{Opacity[0.8], Sphere[]}, {Opacity[0.7], 
   upperVectors~Join~lowerVectors}}, ViewPoint -> {Back, Top}, 
 Axes -> True, AxesLabel -> {x, y, z}, LabelStyle -> Directive[Large],
  ImageSize -> {1200, 1200}]