In vectorial Analysis, the vectorial Laplacian is a differential Opérateur for the vector fields.

Expressions

The Laplacian of a vector field $\backslash \; vec\; A$ is a vector field defined by:

$\backslash \; vec\; \backslash \; Delta\; \backslash \; vec\; has\; =\; \backslash \; overrightarrow\; \backslash \; operatorname\; \{grad\}\; \backslash \; left\; (\backslash \; operatorname\; \{div\}\; \backslash \; vec\; has\; \backslash \; right)\; -\; \backslash \; overrightarrow\; \backslash \; operatorname\; \{belch\}\; \backslash \; left\; (\backslash \; overrightarrow\; \backslash \; operatorname\; \{belch\}\; \backslash \; vec\; has\; \backslash \; right)\; =\; \backslash \; vec\; \backslash \; nabla^2\; \backslash \; vec\; A$

In Coordinated Cartesian, it corresponds to the scalar Laplacian of each component of the vector field:

$\backslash \; vec\; \backslash \; Delta\; \backslash \; vec\; has\; =\; \backslash \; begin\; \{bmatrix\}\; \backslash \; Delta\; A\_x\; \backslash \; \backslash \; \backslash \; Delta\; A\_y\; \backslash \; \backslash \; \backslash \; Delta\; A\_z\; \backslash \; end\; \{bmatrix\}\; =$

\ begin {bmatrix} \ frac {\ partial^2 partial A_x} {\ x^2} + \ frac {\ partial^2 partial A_x} {\ y^2} + \ frac {\ partial^2 partial A_x} {\ z^2} \ \ \ frac {\ partial^2 partial A_y} {\ x^2} + \ frac {\ partial^2 partial A_y} {\ y^2} + \ frac {\ partial^2 partial A_y} {\ z^2} \ \ \ frac {\ partial^2 partial A_z} {\ x^2} + \ frac {\ partial^2 partial A_z} {\ y^2} + \ frac {\ partial^2 A_z} {\partial z^2} \ end {bmatrix} In Coordinated cylindrical, it gives:

$\backslash \; vec\; \backslash \; Delta\; \backslash \; vec\; has\; =\; \backslash \; begin\; \{bmatrix\}\; \backslash \; frac\; \{\backslash \; partial^2\; partial\; A\_r\}\; \{\backslash \; r^2\}\; +\; \backslash \; frac\; \{1\}\; \{r^2\}\; \backslash \; frac\; \{\backslash \; partial^2\; partial\; A\_r\}\; \{\backslash \; \backslash \; theta^2\}\; +\; \backslash \; frac\; \{\backslash \; partial^2\; partial\; A\_r\}\; \{\backslash \; z^2\}\; +\; \backslash \; frac\; \{1\}\; \{R\}\; \backslash \; frac\; \{\backslash \; partial\; A\_r\}\; \{\backslash \; partial\; R\}\; -\; \backslash \; frac\; \{2\}\; \{r^2\}\; \backslash \; frac\; \{\backslash \; partial\; A\_\; \backslash \; partial\; theta\}\; \{\backslash \; \backslash \; theta\}\; -\; \backslash \; frac\; \{A\_r\}\; \{r^2\}\; \backslash \; \backslash \; \backslash frac\; \{\backslash \; partial^2\; A\_\; \backslash \; partial\; theta\}\; \{\backslash \; r^2\}\; +\; \backslash \; frac\; \{1\}\; \{r^2\}\; \backslash \; frac\; \{\backslash \; partial^2\; A\_\; \backslash \; partial\; theta\}\; \{\backslash \; \backslash \; theta^2\}\; +\; \backslash \; frac\; \{\backslash \; partial^2\; A\_\; \backslash \; partial\; theta\}\; \{\backslash \; z^2\}\; +\; \backslash \; frac\; \{1\}\; \{R\}\; \backslash \; frac\; \{\backslash \; partial\; A\_\; \backslash \; theta\}\; \{\backslash \; partial\; R\}\; +\; \backslash \; frac\; \{2\}\; \{r^2\}\; \backslash \; frac\; \{\backslash \; partial\; partial\; A\_r\}\; \{\backslash \; \backslash \; theta\}\; -\; \backslash \; frac\; \{A\_\; \backslash \; theta\}\; \{r^2\}\; \backslash \; \backslash \; \backslash \; frac\; \{\backslash \; partial^2\; partial\; A\_z\}\; \{\backslash \; r^2\}\; +\; \backslash \; frac\; \{1\}\; \{r^2\}\; \backslash \; frac\; \{\backslash \; partial^2\; partial\; A\_z\}\; \{\backslash \; \backslash \; theta^2\}\; +\; \backslash \; frac\; \{\backslash \; partial^2\; partial\; A\_z\}\; \{\backslash \; z^2\}\; +\; \backslash \; frac\; \{1\}\; \{R\}\; \backslash \; frac\; \{\backslash \; partial\; A\_z\}\; \{\backslash \; partial\; R\}\; \backslash \; end\; \{bmatrix\}$

And to finish, in Coordinated spherical, he is written:

$\backslash \; vec\; \backslash \; Delta\; \backslash \; vec\; has\; =\; \backslash \; begin\; \{bmatrix\}$

\ frac {1} {R} \ frac {\ partial^2 (R \ cdot A_r)}{\ partial r^2} + \ frac {1} {r^2} \ frac {\ partial^2 partial A_r} {\ \ theta^2} + \ frac {1} {r^2 \ cdot \ sin^2 \ theta} \ frac {\ partial^2 partial A_r} {\ \ phi^2} + \ frac {\ cot \ theta} {r^2} \ frac {\ partial partial A_r} {\ \ theta} - \ frac {2} {r^2} \ frac {\ partial A_ \ partial theta} {\ \ theta} - \ frac {2} {r^2 \ cdot \ sin \ theta} \frac {\ partial A_ \ partial phi} {\ \ phi} - \ frac {2A_r} {r^2} - \ frac {2 \ cot \ theta} {r^2} A_ \ theta \ \ \ frac {1} {R} \ frac {\ partial^2 (R \ cdot A_ \ theta)}{\ partial r^2} + \ frac {1} {r^2} \ frac {\ partial^2 A_ \ partial theta} {\ \ theta^2} + \ frac {1} {r^2 \ cdot \ sin^2 \ theta} \ frac {\ partial^2 A_ \ partial theta} {\ \ phi^2} + \ frac {\ cot \ theta} {r^2} \ frac {\ partial A_ \ partial theta} {\ \ theta} - \ frac {2} {r^2} \ frac {\ cot \ theta} {\ sin \ theta} \ frac {\ partial A_ \ phi} {\partial \ phi} + \ frac {2} {r^2} \ frac {\ partial partial A_r} {\ \ theta} - \ frac {A_ \ theta} {r^2 \ cdot \ sin^2 \ theta} \ \ \ frac {1} {R} \ frac {\ partial^2 (R \ cdot A_ \ phi)}{\ partial r^2} + \ frac {1} {r^2} \ frac {\ partial^2 A_ \ partial phi} {\ \ theta^2} + \ frac {1} {r^2 \ cdot \ sin^2 \ theta} \ frac {\ partial^2 A_ \ partial phi} {\ \ phi^2} + \ frac {\ cot \ theta} {r^2} \ frac {\ partial A_ \ partial phi} {\ \ theta} + \ frac {2} {r^2 \ cdot \ sin \ theta} \ frac {\ partial partial A_r} {\ \ phi} + \ frac {2}{r^2} \ frac {\ cot \ theta} {\ sin \ theta} \ frac {\ partial A_ \ partial theta} {\ \ phi} - \ frac {A_ \ phi} {r^2 \ cdot \ sin^2 \ theta} \end{bmatrix}

Applications

The vectorial Laplacian is present:

• in the Poisson's equation for the vectorial versions;
• in mechanics of the viscous fluids where it appears in the Navier-Stokes equations;
• in the equation of Schrödinger.

See too

• vectorial Analysis
• Operator Laplacian
• Laplacian (physical significance)

Random links:

Maurice Koechlin | 1913 in literature | Minamoto No Tameyoshi | Albertini shelves | Veniaminof mount