17Calculus - Laplacian
17Calculus
Laplacian
Recommended Books on Amazon (affiliate links) | ||
|---|---|---|
 |
 |
 |
Laplacian Resources
Playlists and Videos |
Related Topics and Links |
|---|
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.
Really UNDERSTAND Calculus
Log in to rate this page and to see it's current rating.
To bookmark this page, log in to your account or set up a free account.
Search Practice Problems
Do you have a practice problem number but do not know on which page it is found? If so, enter the number below and click 'page' to go to the page on which it is found or click 'practice' to be taken to the practice problem.
|
| |
I recently started a Patreon account to help defray the expenses associated with this site. To keep this site free, please consider supporting me. |
|---|
| Support 17Calculus on Patreon |
|
|---|
