4. Let S be a sphere of radius r and P a point inside or outside the sphere. Show that 1 √√₂₁ P |dx| S ds {477²/d if Pis outside s where d is the distance from P to the center of the sphere and the integration is over the sphere. (Hint: One possible way to do this is to assume that P is on the z-axis. Then change variables and evaluate. You need to explain why this assumption on P is allowed.)
4. Let S be a sphere of radius r and P a point inside or outside the sphere. Show that 1 √√₂₁ P |dx| S ds {477²/d if Pis outside s where d is the distance from P to the center of the sphere and the integration is over the sphere. (Hint: One possible way to do this is to assume that P is on the z-axis. Then change variables and evaluate. You need to explain why this assumption on P is allowed.)
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Question
![**Problem Statement:**
4. Let \( S \) be a sphere of radius \( r \) and \( P \) a point inside or outside the sphere. Show that
\[
\iint_S \frac{1}{|x-P|} \, dS =
\begin{cases}
4\pi r & \text{if } P \text{ is inside } S \\
4\pi r^2 / d & \text{if } P \text{ is outside } S
\end{cases}
\]
where \( d \) is the distance from \( P \) to the center of the sphere and the integration is over the sphere.
(Hint: One possible way to do this is to assume that \( P \) is on the z-axis. Then change variables and evaluate. You need to explain why this assumption on \( P \) is allowed.)](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F77548912-c51c-4c9d-8b51-f3905a3bec75%2F62b5d122-1264-47ef-bf9c-d9def96d42ac%2Fzymxafp_processed.png&w=3840&q=75)
Transcribed Image Text:**Problem Statement:**
4. Let \( S \) be a sphere of radius \( r \) and \( P \) a point inside or outside the sphere. Show that
\[
\iint_S \frac{1}{|x-P|} \, dS =
\begin{cases}
4\pi r & \text{if } P \text{ is inside } S \\
4\pi r^2 / d & \text{if } P \text{ is outside } S
\end{cases}
\]
where \( d \) is the distance from \( P \) to the center of the sphere and the integration is over the sphere.
(Hint: One possible way to do this is to assume that \( P \) is on the z-axis. Then change variables and evaluate. You need to explain why this assumption on \( P \) is allowed.)
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