Maxima provides a compact, symbolic workflow for standard calculus operations—differentiation, integration, limits, summation/product manipulation, series expansion, algebraic equation solving, and differential equation solving—supporting both routine computation and physics-oriented analytical work.

Differentiation and Physical Interpretation

Differentiation in Maxima is performed with diff, yielding first or higher derivatives with respect to a chosen variable. In applications, derivatives encode local change, slopes, and time-evolution quantities such as velocity and acceleration, and they appear centrally in higher-order differential models.

  • First derivative: diff(f, x); computes $\frac{df}{dx}$.
  • $n$th derivative: diff(f, x, n); computes $\frac{d^n f}{dx^n}$.
diff(x^3 + sin(x), x);
diff(exp(x)*cos(x), x, 2);

Integration, Limits, and Discrete Operators

Integration via integrate supports both indefinite and definite forms, including improper integrals such as $\int_0^\infty f(x)\,dx$, which commonly arise in normalization and expectation-value calculations. Limits computed by limit are essential for testing continuity, diagnosing singular behavior, and extracting asymptotic structure. Discrete calculus and special-function manipulations frequently use finite sums and products.

  • Indefinite integral: integrate(f, x); returns $\int f\,dx$.
  • Definite integral: integrate(f, x, a, b); returns $\int_a^b f\,dx$.
  • Limit: limit(f, x, a); returns $\lim_{x\to a} f(x)$.
  • Summation: sum(f, k, m, n); returns $\sum_{k=m}^{n} f$.
  • Product: product(f, k, m, n); returns $\prod_{k=m}^{n} f$.
integrate(x^2, x);
integrate(exp(-x), x, 0, inf);
limit(sin(x)/x, x, 0);
sum(k^2, k, 1, n);
product(k, k, 1, n);

Series Expansion and Algebraic Solving

Local approximation and perturbative analysis often rely on series expansions. Maxima computes Taylor expansions about a point $x=a$ up to a specified order. Algebraic solving with solve provides closed-form roots and symbolic conditions that are routinely used for stationary points and analytical parameter relations.

  • Taylor series: taylor(f, x, a, n); expands $f$ about $x=a$ through order $n$.
  • Algebraic solve: solve(eq, x); solves an algebraic equation for $x$.
taylor(exp(x), x, 0, 5);
solve(x^2 - 5*x + 6 = 0, x);

Differential Equations and a Minimal Working Session

Differential equations are solved symbolically with ode2, which is particularly useful for first-order models and canonical linear dynamics encountered in equations of motion, growth–decay systems, and wave or dynamical formulations.

  • Differential equation solve: ode2(eq, y(x)); solves for $y(x)$ under the given equation.
ode2('diff(y,x) + y = 0, y, x);

A minimal working session that resets the environment and exercises common operations is:

kill(all)$
diff(x^4, x);
integrate(x^2, x);
limit((sin(x))/x, x, 0);
sum(k, k, 1, 5);
taylor(log(1+x), x, 0, 4);
solve(x^2 - 1 = 0, x);