1 Answer
The **Laplacian equation** is a second-order partial differential equation given by:
$$
\nabla^2 \phi = 0
$$
where:
* $\nabla^2$ (called the **Laplacian operator**) is the divergence of the gradient of a function.
* $\phi$ is a scalar function (e.g., temperature, electric potential, etc.).
In Cartesian coordinates for a function $\phi(x, y, z)$, the Laplacian is:
$$
\nabla^2 \phi = \frac{\partial^2 \phi}{\partial x^2} + \frac{\partial^2 \phi}{\partial y^2} + \frac{\partial^2 \phi}{\partial z^2}
$$
### Physical Meaning:
The Laplace equation models situations where there is **no net flux** — i.e., the system is in a **steady-state** with **no sources or sinks**. It's commonly used in:
* Electrostatics (electric potential in charge-free regions)
* Fluid dynamics (velocity potential in incompressible, irrotational flow)
* Heat conduction (steady-state temperature distribution)
Would you like a visual of how solutions to the Laplace equation behave in 2D?
$$
\nabla^2 \phi = 0
$$
where:
* $\nabla^2$ (called the **Laplacian operator**) is the divergence of the gradient of a function.
* $\phi$ is a scalar function (e.g., temperature, electric potential, etc.).
In Cartesian coordinates for a function $\phi(x, y, z)$, the Laplacian is:
$$
\nabla^2 \phi = \frac{\partial^2 \phi}{\partial x^2} + \frac{\partial^2 \phi}{\partial y^2} + \frac{\partial^2 \phi}{\partial z^2}
$$
### Physical Meaning:
The Laplace equation models situations where there is **no net flux** — i.e., the system is in a **steady-state** with **no sources or sinks**. It's commonly used in:
* Electrostatics (electric potential in charge-free regions)
* Fluid dynamics (velocity potential in incompressible, irrotational flow)
* Heat conduction (steady-state temperature distribution)
Would you like a visual of how solutions to the Laplace equation behave in 2D?