laplacian

Laplacian of scalar function

Syntax

laplacian(f,x)

laplacian(f)

Description

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

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

f — Input

symbolic expression | symbolic function

Input, specified as a symbolic expression or function.

x — Input

vector of symbolic variables

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

More About

Laplacian of Scalar Function

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

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

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])