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. See Use Assumptions on Symbolic Variables. To simplify your results, see Simplify Symbolic Expressions.

Functions

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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

`curl`	Curl of symbolic vector field
`divergence`	Divergence of symbolic vector field
`gradient`	Gradient vector of symbolic scalar field
`hessian`	Hessian matrix of symbolic scalar function
`jacobian`	Jacobian matrix of symbolic function
`laplacian`	Laplacian of symbolic field
`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`	Symbolic sum of series
`vpasum`	Numerical summation using variable precision (Since R2020b)

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

Partial Differential Equations

`pdeCoefficients`	Extract coefficients of partial differential equation (Since R2021a)
`pdeCoefficientsToDouble`	Convert symbolic PDE coefficients to `double` format (Since R2021a)

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 for analytical calculations.
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 of RLC Circuit Using Laplace Transform
Solve differential equations of an RLC circuit by using Laplace and inverse Laplace transforms.
Solve Difference Equations Using Z-Transform
Z-Transforms and inverses of symbolic expressions 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
This example shows how to use functional derivatives in Symbolic Math Toolbox using the context of a wave equation.

Related Information

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.

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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.

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Integration

Compute definite integrals using Symbolic Math Toolbox™.

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Interactive Calculus in Live Editor

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

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Maxima, Minima, and Inflection Points

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

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Find Extremum of Multivariate Function and Its Approximation

Use symbolic matrix variables to find the extremum of a multivariate function and its approximation.

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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.

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Heating of Finite Slab

Find the temperature distribution of a one-dimensional finite slab by solving the differential equation using the method of separation of variables.

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Solve Partial Differential Equation of Nonlinear Heat Transfer

Solve a partial differential equation (PDE) of nonlinear heat transfer in a thin plate.

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Derive Equations of Motion and Simulate Cart-Pole System

Derive the equations of motion for the cart-pole system using Symbolic Math Toolbox™ and then simulate the cart-pole system using the ode45 solver. In the later sections of the example, you explore how to derive the equations in other forms that can be used to numerically simulate the system (and validate the results) using different tools, such as Simulink®, Simscape™ Multibody™, and Robotics System Toolbox™.

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Finite Element Formulation for Timoshenko Beam Problem

Apply the finite element method (FEM) to solve a Timoshenko beam problem, using both linear and quadratic basis functions for analysis. The Timoshenko beam theory is a 1-D problem that reduces the complex 3-D problem of beam deformation to a set of 1-D differential equations along the length of the beam. In contrast to the Euler–Bernoulli beam theory, which does not consider shear deformation, the Timoshenko beam theory accounts for both shear deformation and rotational bending effects. The Timoshenko beam theory is generally more accurate for short, thick beams where shear deformation cannot be neglected. Using a cantilever beam and a beam fixed at both ends, this example discusses the analytical solutions of the Timoshenko beam problem and compares them with the FEM solutions.

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