Laplacian operator in different coordinates

In summary, to write the Laplacian operator in spherical coordinates and cylindrical coordinates from a Cartesian basis, you can use substitution and the rules of change of variables in partial differentials. In general, the Laplacian operator can be represented as a vector using the norms of the coordinates and their corresponding partial derivatives.
  • #1
captain
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how do you write the laplacian operator in spherical coordinates and cylindrical coordinates from a cartesian basis?
 
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  • #2
captain said:
how do you write the laplacian operator in spherical coordinates and cylindrical coordinates from a cartesian basis?

Cylindrical: Use the substitution [tex]r=\sqrt{x^2+y^2}[/tex] and [tex]\theta = \tan^{-1} \frac{y}{x}[/tex] assuming this is valid on this region.

This leads to,
[tex]\nabla^2 u = \frac{\partial ^2 u}{\partial r^2} +\frac{1}{r} \frac{\partial u}{\partial r} + \frac{1}{r^2
}\frac{\partial ^2 u}{\partial \theta ^2} + \frac{\partial ^2 u}{\partial z^2}=0[/tex]
For the most part [tex]z[/tex] coordinate is not taken and that term vanished.

Spherical: Using Spherical Coordinate substitutions:
[tex]\nabla^2 u = \frac{1}{r^2} \left\{ \frac{\partial (r^2u_r)}{\partial r}+\csc^2 \theta \frac{\partial ^2 y}{\partial \theta^2}+\csc \theta \frac{\partial (\sin \phi u_{\phi})}{\partial \phi} \right\} = 0[/tex]
 
  • #3
in general (which is something you learn in vector analysis for physicists):
[tex]\nabla^2=(\frac{h_3h_2}{h_1}\frac{\partial}{\partial u_1}\frac{\partial h_1}{\partial u_1},\frac{h_3h_1}{h_2}\frac{\partial}{\partial u_2}\frac{\partial h_2}{\partial u_2},\frac{h_1h_2}{h_3}\frac{\partial}{\partial u_3}\frac{\partial h_3}{\partial u_3})[/tex] or something like this.
where:
r=xi+yj+zk
and h_i=|dr/du_i|
i.e you take the norm of the vector.
 
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  • #4
Cheat sheet (well not really cheating, unless you like deriving these things)
 
  • #5
Simply use the rules of change of variables in partial differentials. For example

[tex] \frac{\partial}{\partial x}=\frac{\partial r}{\partial x}\frac{\partial}{\partial r} +... [/tex]

and then sub everyting in terms of the spherical coordinates. Then compute the 2-nd partial wrt to x and the same for y and z.
 
  • #6
MathematicalPhysicist said:
in general (which is something you learn in vector analysis for physicists):
[tex]\nabla^2=(\frac{h_3h_2}{h_1}\frac{\partial}{\partial u_1}\frac{\partial h_1}{\partial u_1},\frac{h_3h_1}{h_2}\frac{\partial}{\partial u_2}\frac{\partial h_2}{\partial u_2},\frac{h_1h_2}{h_3}\frac{\partial}{\partial u_3}\frac{\partial h_3}{\partial u_3})[/tex] or something like this.
where:
r=xi+yj+zk
and h_i=|dr/du_i|
i.e you take the norm of the vector.

Either I am confused at the moment, or it is not right. In general case it should be

[tex]\nabla^2 \Phi= \frac{1}{h_1h_2h_3} \left[ \frac{\partial}{\partial u_1} \left( \frac{h_2h_3}{h_1} \frac{\partial \Phi}{\partial u_1} \right) +

\frac{\partial}{\partial u_2} \left( \frac{h_3h_1}{h_2}\frac{\partial \Phi}{\partial u_2} \right)

+\frac{\partial}{\partial u_3} \left(\frac{h_1h_2}{h_3}\frac{\partial \Phi}{\partial u_3}\right)

\right]
[/tex]

Source: Hobson, Mathematical methods for Physics and Engineering, pg. 374.
 

FAQ: Laplacian operator in different coordinates

What is the Laplacian operator in different coordinates?

The Laplacian operator is a mathematical tool used in vector calculus to describe the second order spatial variation of a scalar field. It is represented by the symbol ∇2 and is also known as the vector Laplacian or the Laplace-Beltrami operator.

How is the Laplacian operator defined in different coordinate systems?

In Cartesian coordinates, the Laplacian operator is defined as the sum of the second partial derivatives of a function with respect to each variable. In cylindrical coordinates, it involves an additional term involving the radial coordinate, and in spherical coordinates, it includes terms involving both the radial and angular coordinates.

What are the applications of the Laplacian operator in different coordinate systems?

The Laplacian operator has various applications in physics, engineering, and mathematics. It is used to describe the behavior of electric and magnetic fields, heat flow, fluid dynamics, and the motion of particles in a gravitational field. It is also used in image processing and machine learning algorithms.

What are the advantages of using different coordinate systems for the Laplacian operator?

Different coordinate systems offer advantages in solving specific problems. For example, cylindrical coordinates are useful for studying problems with circular symmetry, while spherical coordinates are better for problems with spherical symmetry. In some cases, using a particular coordinate system can simplify the Laplacian operator and make the calculations easier.

Are there any limitations to using the Laplacian operator in different coordinates?

While the Laplacian operator is a powerful tool, it may not always be applicable in all situations. For example, in non-Cartesian coordinates, the Laplacian operator may not have a simple form, making it more difficult to work with. Additionally, certain boundary conditions may require alternative approaches for solving problems involving the Laplacian operator.

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