# laplacian

Laplacian of scalar function

## Syntax

``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.

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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])```