Manipulate[expr, {u, u_min, u_max}] generates a version of expr with controls added to allow interactive manipulation of the value of u. Manipulate[expr, {u, u_min, u_max, ...

ParametricPlot[{f_x, f_y}, {u, u_min, u_max}] generates a parametric plot of a curve with x and y coordinates f_x and f_y as a function of u. ParametricPlot[{{f_x, f_y}, ...

This gives a result for the integral of x^n that is valid for almost all values of n. For the special case of x^-1, however, the correct result is different. The overall goal ...

In some cases it is useful to split the differential system into subsystems and solve each subsystem using appropriate integration methods. Recombining the individual ...

Turning conditions into numbers. Boole[expr] is a basic function that turns True and False into 1 and 0. It is sometimes known as the characteristic function or indicator ...

The Mathematica compiler can run computations in parallel. It does this by threading a compiled function over lists of data in parallel. A first step is to create a compiled ...

CirculantGraph[n, j] gives the circulant graph with n vertices and jump j C_n (j).CirculantGraph[n, {j_1, j_2, ...}] gives the circulant graph with n vertices and jumps j_1, ...

CompleteGraph[n] gives the complete graph with n vertices K_n.CompleteGraph[{n_1, n_2, ..., n_k}] gives the complete k-partite graph with n_1 + n_2 + \[CenterEllipsis] + n_k ...

CompleteKaryTree[n] gives the complete binary tree with n levels.CompleteKaryTree[n, k] gives the complete k-ary tree with n levels.

DeBruijnGraph[m, n] gives the n-dimensional De Bruijn graph with m symbols.DeBruijnGraph[m, n, type] gives the De Bruijn graph with connectivity given by type.