Symbolic computation (CAS)

What is it?

Formulas

$$\frac{\dd}{\dd x}(u\cdot v) \;\rightsquigarrow\; u'\cdot v + u\cdot v'$$

Why does it matter?

The mathematics behind it

• Differentiation rules★★★★★fundamental

  Each rule is a rewrite rule on the expression tree; the hard part is simplifying the result.

• Antiderivatives and indefinite integrals★★★★★fundamental

  The Risch algorithm decides whether an elementary antiderivative exists and finds it — a landmark of computer algebra.

• Derivative★★★★★frequent

  Symbolic differentiation is the textbook example of term rewriting on expression trees.

• Fundamental theorem of calculus★★★★★frequent

  CAS evaluate definite integrals by finding an antiderivative (Risch algorithm) and applying Barrow's rule.

• Taylor and Maclaurin series★★★★★frequent

  CAS compute series expansions (series(f, x, 0, n)) to simplify, take limits and approximate.

• Real numbers★★★★★frequent

  Computer algebra systems keep $\sqrt 2$ or $\pi$ exact as symbols instead of approximating them.

• Functions★★★★★frequent

  A CAS represents functions as expression trees that can be simplified, differentiated and integrated.

• Integration by parts★★★★★frequent

  A standard heuristic of symbolic integrators (LIATE ordering).