# Laplacian cylindrical coordinates

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• LagrangeEuler
In summary, Laplacian cylindrical coordinates are a type of coordinate system used to describe points in three-dimensional space. They differ from Cartesian coordinates in that they use a radial distance, azimuthal angle, and height from the origin to describe a point's position. The Laplacian operator in cylindrical coordinates is a mathematical operator that represents the sum of second partial derivatives. These coordinates have various applications in fields such as fluid mechanics, electromagnetism, and quantum mechanics. To convert between Laplacian cylindrical coordinates and Cartesian coordinates, specific equations can be used.
LagrangeEuler
Laplacian in cylindrical coordinates is defined by

$$\Delta=\frac{\partial^2}{\partial \rho^2}+\frac{1}{\rho}\frac{\partial}{\partial \rho}+\frac{1}{\rho^2}\frac{\partial^2}{\partial \varphi^2}+\frac{\partial^2}{\partial z^2}$$
I am confused. I I have spherical symmetric function f(r) then
$$\Delta f(r)=\frac{d^2}{dr^2}f(r)+\frac{2}{r}\frac{d}{dr}f(r)$$
If I worked on function ##f(r)## with Laplacian in cylindrical coordinates. I suppose that $$f(r)=f(\rho)$$ but then factor ##2## is problem.

I think it is ##f(r)=f(\sqrt{\rho^2+z^2})##.

Of course. Thanks.

## What are Laplacian cylindrical coordinates?

Laplacian cylindrical coordinates are a type of coordinate system used in mathematics and physics to describe points in three-dimensional space. They consist of a radial distance, an azimuthal angle, and a height or vertical distance from the origin.

## How do Laplacian cylindrical coordinates differ from Cartesian coordinates?

In Laplacian cylindrical coordinates, the position of a point is described by its distance from the origin, its angle from a reference direction, and its height. In Cartesian coordinates, the position of a point is described by its distance along the x, y, and z axes.

## What is the Laplacian operator in cylindrical coordinates?

The Laplacian operator in cylindrical coordinates is a mathematical operator that represents the sum of the second partial derivatives with respect to the radial distance, azimuthal angle, and height. It is often denoted as ∇² or Δ.

## What are some real-world applications of Laplacian cylindrical coordinates?

Laplacian cylindrical coordinates are commonly used in fields such as fluid mechanics, electromagnetism, and quantum mechanics. They can be used to describe the motion of particles in a fluid, the behavior of electromagnetic fields around cylindrical objects, and the quantum states of particles in a cylindrical potential well.

## How do I convert between Laplacian cylindrical coordinates and Cartesian coordinates?

To convert from Laplacian cylindrical coordinates to Cartesian coordinates, use the following equations:
x = r cos(θ)
y = r sin(θ)
z = z
To convert from Cartesian coordinates to Laplacian cylindrical coordinates, use the following equations:
r = √(x² + y²)
θ = arctan(y/x)
z = z

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