Skip to content

Laplacian

Status: Stable

documented, exercised by the test suite and/or worked examples, with no known limitations recorded.

Description

Laplacian[f, {x1, ..., xn}]

gives the Laplacian D[f,{x1,2}] + ... + D[f,{xn,2}]; for an array f the scalar Laplacian is applied to each component (same dimensions).

Laplacian[f, {x1, ..., xn}, chart]

gives the Laplacian of a scalar in chart ("Cartesian", "Polar", "Cylindrical", "Spherical").

Examples (8)

Every input below was run against the current Mathilda build and its output recorded.

Basic examples (8)

In[1]:= Grad[Sin[x^2 + y^2], {x, y}]
Out[1]= {2 x Cos[x^2 + y^2], 2 y Cos[x^2 + y^2]}

In[2]:= Grad[{x y, y z, z x}, {x, y, z}]
Out[2]= {{y, x, 0}, {0, z, y}, {z, 0, x}}

In[3]:= Div[{x^2, y^2, z^2}, {x, y, z}]
Out[3]= 2 x + 2 y + 2 z

In[4]:= Curl[{y, -x}, {x, y}]
Out[4]= -2

In[5]:= Laplacian[x^2 + y^2 + z^2, {x, y, z}]
Out[5]= 6

In[6]:= Div[{r Sin[t], -r Cos[t]}, {r, t}, "Polar"]
Out[6]= 3 Sin[t]

In[7]:= -Grad[k q/r, {r, t, p}, "Spherical"]
Out[7]= {(k q)/r^2, 0, 0}

In[8]:= Laplacian[Sin[r^2], {r, t}, "Polar"] // Simplify
Out[8]= 4 - 4 r^2

Implementation notes

Attributes: Protected.

References

See also: Grad, Div, Curl, D