The Structure, Stability, and Dynamics
of Self-Gravitating Systems

Interpretation and Submission of Mathematica Expressions

Submission of Mathematica Expressions:

The form of an expression submitted to Mathematica is critical to its proper evaluation, and the following are several tips which will aid one who is not experienced in using this tool:

Useful Mathematica Functions
General Description Mathematica Equivalent
sin(x) Sin[x]
arcsin(x) ArcSin[x]
ex E^x or Exp[x]
ln(x) Log[x]
x1/2 Sqrt[x]

Standard Coordinate Systems
Coordinate System Variables
Bipolar (u, v, z)
Bispherical (u, v, phi)
Cartesian (x, y, z)
ConfocalEllipsoidal (lambda, mu, nu)
ConfocalParaboloidal (lambda, mu, nu)
Conical (lambda, mu, nu)
Cylindrical (r, theta, z)
EllipticCylindrical (u, v, z)
OblateSpheroidal (xi, eta, phi)
ParabolicCylindrical (u, v, z)
Paraboloidal (u, v, phi)
ProlateSpheroidal (xi, eta, phi)
Spherical (r, theta, phi)
Toroidal (u, v, phi)

Interpretation of Mathematica Expressions:

Mathematica outputs mathematical expressions, vectors, and functions in preformatted packages, and the following are several tips to aid one in interpreting the result of one's evaluation:

Example 1:

In order to submit the scalar (x^2 + y^2 + z^2)-1/2, one would enter the Mathematica expression 1/Sqrt[x^2 + y^2 + z^2]. Upon submission of this expression to the Gradient operation, Mathematica will then return an output expression of the form:


           x                     y                     z
{-(-----------------), -(-----------------), -(-----------------)}
     2    2    2 3/2       2    2    2 3/2       2    2    2 3/2
   (x  + y  + z )        (x  + y  + z )        (x  + y  + z )

where the elements of the above list are the components of the vector corresponding to ex, ey, and ez.

Example 2:

In order to submit the vector rer + r*sin(theta)etheta + r*sin(theta)*cos(phi)ephi, one would enter the Mathematica expression {r, r*Sin[theta], r*Sin[theta]*Cos[phi]} while in the Spherical coordinate system. In the Mathematica applications in the Appendices section, submit separately the three components of the vector in the three submit windows. Upon submission of this expression to the Divergence operation, Mathematica will then return an output expression of the form:


               2                 2                          2
Csc[theta] (3 r  Sin[theta] + 2 r  Cos[theta] Sin[theta] - r  Sin[phi] Sin[theta])
----------------------------------------------------------------------------------
                                         2
                                        r