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Integral TransformsAdvanced Topic: Generic and Non-Generic Cases

1.5.12 Packages for Symbolic Mathematics

There are many Mathematica packages which implement symbolic mathematical operations. This section describes a few examples drawn from the standard set of packages distributed with Mathematica. As discussed in Section 1.3.10, some copies of Mathematica may be set up so that the functions described here are automatically loaded into Mathematica if they are ever needed.

Vector Analysis

Vector analysis.

This loads the vector analysis package. In some versions of Mathematica, you may not need to load the package explicitly.

In[1]:= <<Calculus`VectorAnalysis`

This specifies that a spherical coordinate system with coordinate names r, theta and phi should be used.

In[2]:= SetCoordinates[Spherical[r, theta, phi]]

Out[2]=

This evaluates the gradient of in the spherical coordinate system.

In[3]:= Grad[r^2 Sin[theta]]

Out[3]=

Solving Inequalities

Solving inequalities.

This loads the inequality solving package. In some versions of Mathematica, you may not need to load the package explicitly.

In[1]:= <<Algebra`InequalitySolve`

The solution to this inequality is a pair of intervals.

In[2]:= InequalitySolve[Abs[x-1](x^2-3) > 3, x]

Out[2]=

Solving Recurrence Relations

Solving recurrence relations.

This loads the recurrence relation solving package. In some versions of Mathematica, you may not need to load the package explicitly.

In[1]:= <<DiscreteMath`RSolve`

This solves the recurrence relation for the factorial function.

In[2]:= RSolve[{a[n]==n a[n-1], a[1]==1}, a[n], n]

Out[2]=

Here is the solution to a slightly more complicated recurrence relation.

In[3]:= RSolve[{a[n]==a[n-1] + 3a[n-2], a[0]==a[1]==1}, a[n], n]

Out[3]=

Integral TransformsAdvanced Topic: Generic and Non-Generic Cases