Calculus

Symbolic differentiation, integration, series operations, limits, and transforms

Using Symbolic Math Toolbox™, you can differentiate and integrate symbolic expressions, perform series expansions, find transforms of symbolic expressions, and perform vector calculus operations by using the listed functions.

When modeling your problem, use assumptions to return the right results. SeeUse Assumptions on Symbolic Variables. To simplify your results, seeSimplify Symbolic Expressions.

Functions

expand all

Limits, Differentiation, and Integration

`limit`	Limit of symbolic expression
`diff`	Differentiate symbolic expression or function
`functionalDerivative`	Functional derivative (variational derivative)
`int`	Definite and indefinite integrals
`vpaintegral`	Numerical integration using variable precision
`changeIntegrationVariable`	Integration by substitution
`integrateByParts`	Integration by parts
`release`	Evaluate integrals

Vector Analysis

`旋度`	Curl of vector field
`divergence`	Divergence of vector field
`gradient`	Gradient vector of scalar function
`hessian`	Hessian matrix of scalar function
`jacobian`	Jacobian matrix
`laplacian`	Laplacian of scalar function
`potential`	Potential of vector field
`vectorPotential`	Vector potential of vector field

Series

Series Expansions

`pade`	Pade approximant
`rsums`	Interactive evaluation of Riemann sums
`series`	Puiseux series
`taylor`	Taylor series

Sums & Products

`cumprod`	Symbolic cumulative product
`cumsum`	Symbolic cumulative sum
`symprod`	Product of series
`symsum`	Sum of series
`vpasum`	Numerical summation using variable precision

Transforms

`fourier`	Fourier transform
`ifourier`	Inverse Fourier transform
`htrans`	Hilbert transform
`ihtrans`	Inverse Hilbert transform
`laplace`	Laplace transform
`ilaplace`	Inverse Laplace transform
`ztrans`	Z-transform
`iztrans`	Inverse Z-transform
`sympref`	Set symbolic preferences

Interactive Tools

`funtool`	Function calculator
`rsums`	Interactive evaluation of Riemann sums
`taylortool`	Taylor series calculator

Topics

Differentiation

Differentiate symbolic expressions and functions.

Create Symbolic Functions

Use symbolic functions that accept symbolic inputs, such asf(x,y).

Integration

Integrate symbolic expressions and functions.

Taylor Series

Taylor series expansion of symbolic expressions and functions.

Fourier and Inverse Fourier Transforms

Fourier and inverse Fourier transforms of symbolic expressions.

Solve Differential Equations Using Laplace Transform

Laplace and Inverse Laplace transforms of symbolic expressions and functions.

Solve Difference Equations Using Z-Transform

z变换和逆的象征性的表达ions and functions.

Symbolic Summation

Sum symbolic vectors, matrices, or symbolic series.

Padé Approximant

Pade approximant of symbolic expressions and functions.

Limits

Limits of symbolic expressions and functions.

Find Asymptotes, Critical, and Inflection Points

Find minima, maxima, and asymptotes by using derivatives and limits.

Functional Derivatives Tutorial

Find functional derivatives, which are the derivative of a functional with respect to a function.

Featured Examples

Learn Calculus in the Live Editor

Learn calculus and applied mathematics using the Symbolic Math Toolbox™. The example shows introductory functions fplot and diff.

Open Live Script

Differentiation

Analytically find and evaluate derivatives using Symbolic Math Toolbox™. In the example you will find the 1st and 2nd derivative of f(x) and use these derivatives to find local maxima, minima and inflection points.

Open Live Script

Integration

Compute definite integrals using Symbolic Math Toolbox™.

Open Live Script

Interactive Calculus in Live Editor

Add interactive controls to solve a calculus problem in a live script.

Open Live Script

Maxima, Minima, and Inflection Points

This demonstration shows how to find extrema of functions using analytical and numerical techniques using the Symbolic Math Toolbox.

Open Live Script

Padé Approximant of Time-Delay Input

Use a Padé approximant in control system theory to model time delays in the response of a first-order system. Time delays arise in systems such as chemical and transport processes where there is a delay between the input and the system response. When these inputs are modeled, they are called dead-time inputs.

Open Live Script