# 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

expand all

 `limit` Limit of symbolic expression `diff` Differentiate symbolic expression or function `functionalDerivative` Functional derivative `int` Definite and indefinite integrals `vpaintegral` Numerical integration using variable precision `changeIntegrationVariable` Integration by substitution `integrateByParts` Integration by parts `release` Evaluate integrals
 `curl` 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 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
 `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
 `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 as `f(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-Transforms and inverses of symbolic expressions and functions.

Symbolic Summation

Sum symbolic vectors, matrices, or symbolic series.

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

#### Mathematical Modeling with Symbolic Math Toolbox

Get examples and videos