Showing that Laplace's equation holds

In summary, the conversation discusses a question from the textbook "Bioelectricity: A Quantitative Approach" about the Laplacian of a function r, given by r = sqrt(x^2 + y^2 + z^2). The conversation first presents the attempt at solving the problem, which results in a non-zero value for the Laplacian. However, upon further discussion and examination of the textbook, it is revealed that the original question contains a typo and the correct function should be 1/r. This mistake is acknowledged and the correct answer is confirmed by the expert.
  • #1
Cole A.
12
0

Homework Statement



(From Plonsey, R. and R. C. Barr, "Bioelectricity: A Quantitative Approach")

Show
[tex] \nabla^2 r = 0 [/tex]
given
[tex] r = \sqrt{x^2 + y^2 + z^2}. [/tex]

Homework Equations


[tex] \nabla = \frac{\partial}{\partial x}{\bf i} + \frac{\partial}{\partial y}{\bf j} + \frac{\partial}{\partial z}{\bf k} [/tex] (the "abuse of notation")


The Attempt at a Solution



This is my first attempt at a post on this forum, and I'm embarrassed to say I'm really struggling with this question. (Not a homework problem per se, just an in-text side note). It has been a good while since I've studied vector calculus, so if somebody could point out my error, I would appreciate it.

Beginning with
[tex]
\begin{equation}
r = (x^2 + y^2 + z^2)^{1/2}
\end{equation}
[/tex]
taking the gradient results in
[tex]
\begin{equation}
\begin{aligned}
\nabla r &= \frac{\partial r}{\partial x} {\bf i} + \frac{\partial r}{\partial y} {\bf j} + \frac{\partial r}{\partial z} {\bf k} \\
&= \left[ \frac{x}{(x^2 + y^2 + z^2)^{1/2}} \right]{\bf i} + \left[ \frac{y}{(x^2 + y^2 + z^2)^{1/2}} \right]{\bf j} + \left[ \frac{z}{(x^2 + y^2 + z^2)^{1/2}} \right]{\bf k}.
\end{aligned}
\end{equation}
[/tex]
Now taking the divergence of this gradient [itex] - [/itex] which amounts cumulatively to taking the Laplacian of [itex] r - [/itex] gives
[tex]
\begin{equation}
\begin{aligned}
\nabla \cdot \nabla r &= \nabla^2 r \\
&= \frac{\partial}{\partial x}\left[ \frac{x}{(x^2 + y^2 + z^2)^{1/2}} \right] + \frac{\partial}{\partial y}\left[ \frac{y}{(x^2 + y^2 + z^2)^{1/2}} \right] + \frac{\partial}{\partial z}\left[ \frac{z}{(x^2 + y^2 + z^2)^{1/2}} \right] \\
&= \left[ \frac{1}{(x^2 + y^2 + z^2)^{1/2}} - \frac{x^2}{(x^2 + y^2 + z^2)^{3/2}} \right] + \left[ \frac{1}{(x^2 + y^2 + z^2)^{1/2}} - \frac{y^2}{(x^2 + y^2 + z^2)^{3/2}} \right] + \left[ \frac{1}{(x^2 + y^2 + z^2)^{1/2}} - \frac{z^2}{(x^2 + y^2 + z^2)^{3/2}} \right] \\
&= \left[ \frac{y^2 + z^2}{(x^2 + y^2 + z^2)^{3/2}} \right] + \left[ \frac{x^2 + z^2}{(x^2 + y^2 + z^2)^{3/2}} \right] + \left[ \frac{x^2 + y^2}{(x^2 + y^2 + z^2)^{3/2}} \right] \\
&= \frac{2}{(x^2 + y^2 + z^2)^{1/2}}
\end{aligned}
\end{equation}
[/tex]
which is not zero, although I very much wish it was. (Sorry for the long-winded calculations.)
 
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  • #2
Very good job. You got the right answer. I think the problem is with the original question. The laplacian of r isn't zero. The laplacian of 1/r is zero. Are you sure that's not what it says??
 
  • #3
Ugh, I was starting to suspect I was wasting time because of a typo in the book. I attached the relevant portion of the textbook page as an image, where the above equality is listed along with a couple other vector identities.

Thanks for the extra input (and sanity check).
 

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Last edited:
  • #4
Cole A. said:
Ugh, I was starting to suspect I was wasting time because of a typo in the book. I attached the relevant portion of the textbook page as an image, where the above equality is listed along with a couple other vector identities.

Thanks for the extra input.

It's a typo alright. Pretty bad one too. Jeez. I'm aghast. Shocked, really. That's pretty fundamental, you'd think they would have checked that.
 
Last edited:

Related to Showing that Laplace's equation holds

1. What is Laplace's equation?

Laplace's equation is a second-order partial differential equation that describes the relationship between the values of a scalar function at any point in a region and its values on the boundary of that region.

2. How is Laplace's equation used in science?

Laplace's equation is a fundamental tool in many areas of science, including physics, engineering, and mathematics. It is used to model a wide range of physical phenomena, such as heat conduction, electrostatics, and fluid flow.

3. How can you show that Laplace's equation holds?

To show that Laplace's equation holds, we usually start by assuming that the equation is true and then use mathematical techniques, such as separation of variables or Fourier analysis, to derive the specific solution for a given problem. We then check if the solution satisfies the equation, which confirms that Laplace's equation holds.

4. What are some real-world applications of Laplace's equation?

Laplace's equation has numerous applications in various fields, including the design of electronic circuits, the study of fluid flow in pipes and channels, and the analysis of heat transfer in materials. It is also used in image processing and computer vision to smooth and enhance images.

5. Are there any limitations to Laplace's equation?

Like any mathematical model, Laplace's equation has its limitations. It assumes that the system is linear, homogeneous, and isotropic, which may not always be the case in real-world scenarios. Additionally, it does not account for transient effects or boundary conditions that change over time, making it less suitable for time-dependent problems.

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