What is the Laplacian of $y^k$ in the upper half plane with hyperbolic metric?

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In summary, the Laplacian is a mathematical operator used to describe second-order spatial variation of a function, commonly in fields such as physics and engineering. The upper half plane, which refers to the region above the x-axis on the complex plane, is important in mathematics as it is used as a model for other mathematical objects. The hyperbolic metric is a way of measuring distances on a hyperbolic surface and differs from the Euclidean metric. The Laplacian of $y^k$ in the upper half plane behaves differently depending on the value of k, with various practical applications in fields such as differential geometry and computer graphics.
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Chris L T521
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I realized that I had posted solutions last night to the POTWs, but forgot to create the new ones last night...I guess that not sleeping well the night before traveling all day can make you do these kinds of things. Anyways, here's this week's problem.

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Problem: Consider the upper half plane with its standard hyperbolic metric $\frac{1}{y^2}(dx^2+dy^2)$. For $k$ a fixed real number, compute the Laplacian of the function $y^k$ relative to this metric.

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Chris L T521 said:
I realized that I had posted solutions last night to the POTWs, but forgot to create the new ones last night...I guess that not sleeping well the night before traveling all day can make you do these kinds of things. Anyways, here's this week's problem.

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Problem: Consider the upper half plane with its standard hyperbolic metric $\frac{1}{y^2}(dx^2+dy^2)$. For $k$ a fixed real number, compute the Laplacian of the function $y^k$ relative to this metric.

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No one answered this week's question. Here's my solution below.

Recall that for any metric $g_{ij}$, the Laplacian of a scalar function $f$ is given by the formula
\[\Delta_g f = \frac{1}{\sqrt{\det g_{ij}}}\partial_i\left(\sqrt{\det g_{ij}}g^{ij}\partial_jf\right)\]
Since we're working with the standard hyperbolic metric $ds^2=E\,dx^2+2F\,dx\,dy + G\,dy^2=\frac{1}{y^2}\,dx^2+\frac{1}{y^2}\,dy^2$, we have
\[g_{ij}=\begin{bmatrix}E & F\\ F & G\end{bmatrix}=\begin{bmatrix}1/y^2 & 0 \\ 0 & 1/y^2\end{bmatrix}\implies g^{ij} = g_{ij}^{-1} = \begin{bmatrix}y^2 & 0\\ 0 & y^2\end{bmatrix}\]
and thus $\det g_{ij}=\dfrac{1}{y^4}\implies \dfrac{1}{\sqrt{\det g_{ij}}}=y^2$. We now have that
\[\begin{aligned}\Delta_g (y^k) &= y^2 \partial_y\left(\frac{1}{y^2}\cdot y^2\cdot ky^{k-1}\right)\\ &= y^2k(k-1)y^{k-2}\\ &= k(k-1)y^k.\end{aligned}\]
Therefore, $\Delta_g y^k = k(k-1)y^k$.
 

FAQ: What is the Laplacian of $y^k$ in the upper half plane with hyperbolic metric?

1. What is the Laplacian and how is it used in mathematics?

The Laplacian is a mathematical operator that is used to describe the second-order spatial variation of a function. It is commonly used in fields such as physics, engineering, and geometry to model various physical phenomena and analyze solutions to differential equations.

2. What is the upper half plane and why is it important in mathematics?

The upper half plane is a geometric concept in mathematics that refers to the region above the x-axis on the complex plane. It is important in mathematics because it is often used as a model for other mathematical objects, such as Riemann surfaces and hyperbolic spaces.

3. What is the hyperbolic metric and how does it differ from the Euclidean metric?

The hyperbolic metric is a way of measuring distances on a hyperbolic surface, which is a type of non-Euclidean geometry. It differs from the Euclidean metric in that it takes into account the curvature of the surface, resulting in different properties and formulas for measuring distances and angles.

4. How does the Laplacian of $y^k$ behave in the upper half plane?

The Laplacian of $y^k$ in the upper half plane with hyperbolic metric is a function that describes the curvature of the surface at each point. Its behavior depends on the value of k, with different values resulting in different patterns of curvature. For example, when k=1, the surface is flat (zero curvature), while when k=2, the surface has constant negative curvature.

5. What are the practical applications of studying the Laplacian of $y^k$ in the upper half plane with hyperbolic metric?

Studying the Laplacian of $y^k$ in the upper half plane with hyperbolic metric has various practical applications in fields such as differential geometry, mathematical physics, and computer graphics. It can be used to model and analyze surfaces with non-Euclidean geometries, such as the surfaces of leaves, crystals, and other natural objects. It also has applications in image processing, shape recognition, and computer vision.

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