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# laplacian

Laplacian of scalar function

laplacian(f,x)
laplacian(f)

## Description

example

laplacian(f,x) computes the Laplacian of the scalar function or functional expression f with respect to the vector x in Cartesian coordinates.

example

laplacian(f) computes the Laplacian of the scalar function or functional expression f with respect to a vector constructed from all symbolic variables found in f. The order of variables in this vector is defined by symvar.

## Examples

### Compute Laplacian of Symbolic Expression

Compute the Laplacian of this symbolic expression. By default, laplacian computes the Laplacian of an expression with respect to a vector of all variables found in that expression. The order of variables is defined by symvar.

syms x y t
laplacian(1/x^3 + y^2 - log(t))
ans =
1/t^2 + 12/x^5 + 2

### Compute Laplacian of Symbolic Function

Create this symbolic function:

syms x y z
f(x, y, z) = 1/x + y^2 + z^3;

Compute the Laplacian of this function with respect to the vector [x, y, z]:

L = laplacian(f, [x y z])
L(x, y, z) =
6*z + 2/x^3 + 2

## Input Arguments

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Input, specified as a symbolic expression or function.

Input, specified as a vector of symbolic variables. The Laplacian is computed with respect to these symbolic variables.

## More About

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### Laplacian of Scalar Function

The Laplacian of the scalar function or functional expression f with respect to the vector X = (X1,...,Xn) is the sum of the second derivatives of f with respect to X1,...,Xn:

$\Delta f=\sum _{i=1}^{n}\frac{{\partial }^{2}f}{\partial {x}_{i}^{2}}$

## Tips

• If x is a scalar, laplacian(f, x) = diff(f, 2, x).

## Alternatives

The Laplacian of a scalar function or functional expression is the divergence of the gradient of that function or expression:

$\Delta f=\nabla \cdot \left(\nabla f\right)$

Therefore, you can compute the Laplacian using the divergence and gradient functions:

syms f(x, y)
divergence(gradient(f(x, y)), [x y])

## See Also

#### Mathematical Modeling with Symbolic Math Toolbox

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