Prove the rotational invariance of the Laplace operator

In summary: This conversation discusses the concept of rotation invariance in mathematical equations. In summary, the equations shown involve the Laplacian operator and the rotation matrix, and the conversation explores how rotation invariance is achieved in these equations. It is shown that for a rotation of angle ##\alpha##, the Laplacian remains invariant, and this can be proven using cylindrical or spherical coordinates. The conversation also mentions the concept of inverse rotation, where the inverse of the rotation matrix is equal to its transpose.
  • #1
docnet
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Homework Statement
prove ##\Delta## is rotation invariant.
Relevant Equations
##\Delta##
Screen Shot 2021-01-24 at 10.08.21 PM.png

Hello, please lend me your wisdom.

##\Delta u=\partial_{x1}^2u+\partial_{x2}^2u+...+\partial_{xn}^2u##

##Rx=\left<r_{11}x_1+...r_{1n}x_n+...+r_{n1}x_1+...+r_{nn}x_n\right>##

##(\Delta u)(Rx)=(\partial_{x1}^2u+\partial_{x2}^2u+...+\partial_{xn}^2u)\left<r_{11}x_1+...r_{1n}x_n, ...,r_{n1}x_1+...+r_{nn}x_n\right>####u\circ R=##

##\begin{pmatrix}
(u)r_{11} & ... & (u)r_{1n} \\
... & ... & ... \\
(u)r_{n1} & ... & (u)r_{nn}
\end{pmatrix}##

##u\circ R x=\left<(u)r_{11}x_1+...(u)r_{1n}x_n+...+(u)r_{n1}x_1+...+(u)r_{nn}x_n\right>##

##\Delta u\circ R x= (\partial_{x1}^2+\partial_{x2}^2+...+\partial_{xn}^2)\left<(u)r_{11}x_1+...(u)r_{1n}x_n+...+(u)r_{n1}x_1+...+(u)r_{nn}x_n\right>=##

##(\partial_{x1}^2u+\partial_{x2}^2u+...+\partial_{xn}^2u)\left<r_{11}x_1+...r_{1n}x_n, ...,r_{n1}x_1+...+r_{nn}x_n\right>=(\Delta u)(Rx)##

I think ##u\circ R## does not mean ##(u) (R)## and I just showed the same calculation twice. How do I use the information ##R^{-1}=R^T## to prove this case?
 
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  • #2
Let us take z-axis as axis of rotation of angle ##\alpha## and take cylindrical coordinates or spherical coordinates. For the rotation
[tex]\phi'=\phi + \alpha[/tex]
, we get rotation invariance of Laplacian because
[tex]\frac{\partial^2}{\partial \phi^2}=\frac{\partial^2}{\partial \phi'^2}[/tex]
 
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1. What is the Laplace operator?

The Laplace operator is a mathematical operator used in vector calculus to describe the second-order spatial variation of a function. It is commonly denoted by the symbol ∇2 or Δ.

2. What does it mean for the Laplace operator to be rotationally invariant?

Rotationally invariant means that the Laplace operator remains unchanged under a rotation of the coordinate system. This means that the operator will produce the same result regardless of the orientation of the coordinate axes.

3. Why is proving the rotational invariance of the Laplace operator important?

Proving the rotational invariance of the Laplace operator is important because it allows us to use the operator in a variety of coordinate systems without having to make any adjustments. This makes it a powerful tool in solving problems in physics, engineering, and other fields that involve rotations.

4. How is the rotational invariance of the Laplace operator proven?

The rotational invariance of the Laplace operator can be proven using vector calculus and the properties of rotations. This involves showing that the operator remains unchanged when the coordinates are rotated, and that it produces the same result for a rotated function as it does for the original function.

5. What are some applications of the rotational invariance of the Laplace operator?

The rotational invariance of the Laplace operator has many practical applications, including solving problems in fluid dynamics, electromagnetism, and quantum mechanics. It is also used in image processing and computer graphics to analyze and manipulate images regardless of their orientation.

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