What is the fundamental solution of Laplace's equation in 3D?
2 Answers
The fundamental solution of Laplace's equation in 3D is a basic solution to the equation that represents how a point source affects the potential field. Laplace’s equation in 3D is given by:
\[
\nabla^2 \phi = 0
\]
where \(\phi\) is the scalar potential, and \(\nabla^2\) (the Laplacian operator) in three dimensions is:
\[
\nabla^2 \phi = \frac{\partial^2 \phi}{\partial x^2} + \frac{\partial^2 \phi}{\partial y^2} + \frac{\partial^2 \phi}{\partial z^2}
\]
### Fundamental Solution
The fundamental solution (also called the Green's function) is the response to a Dirac delta source, meaning a concentrated point source located at some position, say the origin, \(\mathbf{r} = (0, 0, 0)\). For a point source located at the origin in 3D space, the fundamental solution of Laplace’s equation satisfies:
\[
\nabla^2 \phi(\mathbf{r}) = -\delta(\mathbf{r})
\]
where \(\mathbf{r} = (x, y, z)\) is the position vector, and \(\delta(\mathbf{r})\) is the Dirac delta function, which ensures that the source is concentrated at the origin.
To find the fundamental solution, let's consider a potential \(\phi\) that depends only on the radial distance \(r = |\mathbf{r}| = \sqrt{x^2 + y^2 + z^2}\) from the origin, meaning \(\phi(\mathbf{r}) = \phi(r)\).
1. In spherical coordinates, the Laplacian operator for a radially symmetric function \(\phi(r)\) simplifies to:
\[
\nabla^2 \phi(r) = \frac{1}{r^2} \frac{d}{dr} \left( r^2 \frac{d\phi}{dr} \right)
\]
2. We are looking for a solution to:
\[
\frac{1}{r^2} \frac{d}{dr} \left( r^2 \frac{d\phi}{dr} \right) = 0
\]
This is the Laplace equation in spherical symmetry.
3. To solve this equation, first integrate with respect to \(r\):
\[
r^2 \frac{d\phi}{dr} = C_1
\]
where \(C_1\) is a constant of integration.
4. Integrating again:
\[
\frac{d\phi}{dr} = \frac{C_1}{r^2}
\]
\[
\phi(r) = \frac{C_1}{r} + C_2
\]
where \(C_2\) is another constant of integration.
For the fundamental solution, we are interested in the behavior at large distances (where the potential tends to zero), so we set \(C_2 = 0\). Thus, the potential simplifies to:
\[
\phi(r) = \frac{C_1}{r}
\]
5. To determine \(C_1\), we use the fact that the potential must satisfy:
\[
\nabla^2 \phi = -\delta(\mathbf{r})
\]
The Dirac delta function essentially acts as a point source, and by solving the Poisson equation with a unit source strength, we find that \(C_1 = \frac{1}{4\pi}\).
Therefore, the fundamental solution of Laplace's equation in 3D is:
\[
\phi(r) = \frac{1}{4\pi r}
\]
### Summary
The fundamental solution to Laplace's equation in 3D for a point source at the origin is:
\[
\phi(\mathbf{r}) = \frac{1}{4\pi |\mathbf{r}|} = \frac{1}{4\pi \sqrt{x^2 + y^2 + z^2}}
\]
This solution describes the potential field created by a point source located at the origin in a three-dimensional space. It decays with the inverse of the distance from the source, which is characteristic of potential fields governed by Laplace’s equation in 3D.
\[
\nabla^2 \phi = 0
\]
where \(\phi\) is the scalar potential, and \(\nabla^2\) (the Laplacian operator) in three dimensions is:
\[
\nabla^2 \phi = \frac{\partial^2 \phi}{\partial x^2} + \frac{\partial^2 \phi}{\partial y^2} + \frac{\partial^2 \phi}{\partial z^2}
\]
### Fundamental Solution
The fundamental solution (also called the Green's function) is the response to a Dirac delta source, meaning a concentrated point source located at some position, say the origin, \(\mathbf{r} = (0, 0, 0)\). For a point source located at the origin in 3D space, the fundamental solution of Laplace’s equation satisfies:
\[
\nabla^2 \phi(\mathbf{r}) = -\delta(\mathbf{r})
\]
where \(\mathbf{r} = (x, y, z)\) is the position vector, and \(\delta(\mathbf{r})\) is the Dirac delta function, which ensures that the source is concentrated at the origin.
To find the fundamental solution, let's consider a potential \(\phi\) that depends only on the radial distance \(r = |\mathbf{r}| = \sqrt{x^2 + y^2 + z^2}\) from the origin, meaning \(\phi(\mathbf{r}) = \phi(r)\).
1. In spherical coordinates, the Laplacian operator for a radially symmetric function \(\phi(r)\) simplifies to:
\[
\nabla^2 \phi(r) = \frac{1}{r^2} \frac{d}{dr} \left( r^2 \frac{d\phi}{dr} \right)
\]
2. We are looking for a solution to:
\[
\frac{1}{r^2} \frac{d}{dr} \left( r^2 \frac{d\phi}{dr} \right) = 0
\]
This is the Laplace equation in spherical symmetry.
3. To solve this equation, first integrate with respect to \(r\):
\[
r^2 \frac{d\phi}{dr} = C_1
\]
where \(C_1\) is a constant of integration.
4. Integrating again:
\[
\frac{d\phi}{dr} = \frac{C_1}{r^2}
\]
\[
\phi(r) = \frac{C_1}{r} + C_2
\]
where \(C_2\) is another constant of integration.
For the fundamental solution, we are interested in the behavior at large distances (where the potential tends to zero), so we set \(C_2 = 0\). Thus, the potential simplifies to:
\[
\phi(r) = \frac{C_1}{r}
\]
5. To determine \(C_1\), we use the fact that the potential must satisfy:
\[
\nabla^2 \phi = -\delta(\mathbf{r})
\]
The Dirac delta function essentially acts as a point source, and by solving the Poisson equation with a unit source strength, we find that \(C_1 = \frac{1}{4\pi}\).
Therefore, the fundamental solution of Laplace's equation in 3D is:
\[
\phi(r) = \frac{1}{4\pi r}
\]
### Summary
The fundamental solution to Laplace's equation in 3D for a point source at the origin is:
\[
\phi(\mathbf{r}) = \frac{1}{4\pi |\mathbf{r}|} = \frac{1}{4\pi \sqrt{x^2 + y^2 + z^2}}
\]
This solution describes the potential field created by a point source located at the origin in a three-dimensional space. It decays with the inverse of the distance from the source, which is characteristic of potential fields governed by Laplace’s equation in 3D.
Laplace's equation in three dimensions is given by:
\[ \nabla^2 u(x, y, z) = 0, \]
where \(\nabla^2\) is the Laplacian operator, and \(u(x, y, z)\) is the function we want to find.
The fundamental solution to Laplace's equation in 3D is a specific function \(u(x, y, z)\) that satisfies the equation with a Dirac delta function as the source term. In other words, it's the solution to the equation:
\[ \nabla^2 G(x, y, z) = \delta(x, y, z), \]
where \(G(x, y, z)\) is the fundamental solution and \(\delta(x, y, z)\) is the Dirac delta function centered at the origin.
In three dimensions, the fundamental solution \(G(x, y, z)\) is given by:
\[ G(x, y, z) = -\frac{1}{4\pi r}, \]
where \(r\) is the Euclidean distance from the origin, defined as:
\[ r = \sqrt{x^2 + y^2 + z^2}. \]
To verify that this function is indeed the fundamental solution, we can check that it satisfies the Laplace equation when the Dirac delta function is considered. For \(r \neq 0\), the Laplacian of \(-\frac{1}{4\pi r}\) is zero, which means that it satisfies Laplace's equation in regions where \(r \neq 0\). At \(r = 0\), the behavior of the Laplacian corresponds to a point source, represented by the Dirac delta function.
In summary, the fundamental solution of Laplace's equation in 3D is:
\[ G(x, y, z) = -\frac{1}{4\pi \sqrt{x^2 + y^2 + z^2}}. \]
\[ \nabla^2 u(x, y, z) = 0, \]
where \(\nabla^2\) is the Laplacian operator, and \(u(x, y, z)\) is the function we want to find.
The fundamental solution to Laplace's equation in 3D is a specific function \(u(x, y, z)\) that satisfies the equation with a Dirac delta function as the source term. In other words, it's the solution to the equation:
\[ \nabla^2 G(x, y, z) = \delta(x, y, z), \]
where \(G(x, y, z)\) is the fundamental solution and \(\delta(x, y, z)\) is the Dirac delta function centered at the origin.
In three dimensions, the fundamental solution \(G(x, y, z)\) is given by:
\[ G(x, y, z) = -\frac{1}{4\pi r}, \]
where \(r\) is the Euclidean distance from the origin, defined as:
\[ r = \sqrt{x^2 + y^2 + z^2}. \]
To verify that this function is indeed the fundamental solution, we can check that it satisfies the Laplace equation when the Dirac delta function is considered. For \(r \neq 0\), the Laplacian of \(-\frac{1}{4\pi r}\) is zero, which means that it satisfies Laplace's equation in regions where \(r \neq 0\). At \(r = 0\), the behavior of the Laplacian corresponds to a point source, represented by the Dirac delta function.
In summary, the fundamental solution of Laplace's equation in 3D is:
\[ G(x, y, z) = -\frac{1}{4\pi \sqrt{x^2 + y^2 + z^2}}. \]