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17Calculus - Laplacian

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Laplacian

Laplacian Resources

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Laplace Operator - Wikipedia

Laplace Equation - Wikiversity

Laplacian Operator

The Laplacian or Laplace Operator is a differential operator.   It is the divergence of the gradient vector of a scalar function.
The notation is \( \nabla^2 = \nabla \cdot \nabla \).   You may also see it written with the symbol \(\Delta\).   Although we prefer the notation \( \nabla^2 \), we also use \(\Delta\) since it is widely used elsewhere.
The result of the Laplacian is a scalar function.   Notice that the gradient is a vector field but the divergence of a vector field is a scalar function.

Laplacian Equations

Now let's look at the equations for the Laplacian in 3-space.   We adapted these equations from Wikipedia.

Laplacian Equation - Cartesian Coordinates
\(f(x,y,z)\)

\[ \Delta f = \frac{\partial^2 f}{\partial x^2} + \frac{\partial^2 f}{\partial y^2} + \frac{\partial^2 f}{\partial z^2} \]

Laplacian Equation - Cylindrical Coordinates
\(f(r, \theta, z)\)

\[ \Delta f = \frac{1}{r} \frac{\partial}{\partial r} \left( r \frac{\partial f}{\partial r} \right) + \frac{1}{r^2} \frac{\partial^2 f}{\partial \theta^2} + \frac{\partial^2 f}{\partial z^2} \]

Laplacian Equation - Spherical Coordinates
\(f(\rho, \theta, \varphi)\)

\[ \Delta f = \frac{1}{\rho^2} \frac{\partial}{\partial \rho} \left( \rho^2\frac{\partial f}{\partial r} \right) + \frac{1}{\rho^2 \sin \varphi} \frac{\partial}{\partial \varphi} \left( \sin\varphi \frac{\partial f}{\partial \varphi} \right) + \frac{1}{\rho^2 \sin^2 \varphi} \frac{\partial^2 f}{\partial \theta^2} \]

Comment on Notation - Wikipedia uses the spherical coordinates \( f(r, \varphi, \theta) \).   We have chosen to be consistent with the notation on this site.   This is an example of making sure you watch your context.   As usual, check with your instructor to see what they expect.

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