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Calculus Packages

Mathematica has extensive built-in support for calculus, including integration, differentiation, differential equation solving, and limits. The Calculus packages extend this functionality by providing functions for finding complete integrals of differential equations, computing Padé approximations, and doing vector operations in various three-dimensional coordinate systems. The package VariationalMethods is useful for solving for a function extremizing a definite integral functional, a problem that appears in many areas of physics and engineering.

This causes each Calculus package to load as needed.

In[1]:= <<Calculus`

The function VariationalBound can be used to give an upper bound on the first eigenvalue associated with the vibrating string problem. Here assume that the string has unit length, density, and tension. The integral is minimized over all approximating eigenfunctions having the same form as the specified trial function.

For a vibrating string with a fixed end at , a trial function is picked such that . The parameter is chosen to satisfy a normalization condition.

In[2]:= (phitrial = a x^k;
eq = (Integrate[phitrial^2, {x, 0, 1},
Assumptions -> {k > 1/2}] == 1);
phitrial = (phitrial /. Solve[eq, a])[[1]])


The bound is , close to the precise value .

In[3]:= VariationalBound[D[phi[x], x]^2,
phi[x], {x, 0, 1}, phitrial, {k}]