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 gradient vector 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.

Input Arguments

f

Symbolic expression or symbolic function.

x

Vector with respect to which you compute the Laplacian.

Default: Vector constructed from all symbolic variables found in f. The order of variables in this vector is defined by symvar.

Examples

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

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

Alternatives

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

Δf=(f)

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

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

More About

expand all

Laplacian of a 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:

Δf=i=1n2fixi2

Tips

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

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