Question #e4aa5

Jan 24, 2017

A point $P$ is defined in spherical coordinated as a function of $\left(r , \theta , \phi\right)$. Where $r$ is the distance from the origin to the point, $\phi$ is the angle that we need to rotate down from the positive $z$-axis to get to the point and $\theta$ is angle we need to rotate around the $z$-axis to get to the point.

As you have rightly posted the figure, infinitesimal volume element $\mathrm{dV}$ in spherical coordinates is given by multiplication of three unit coordinates

1. Infinitesimal radial element $= \mathrm{dr}$
2. Recall formula which relates the arc length $s$ of a circle of radius $r$ to the central angle $\theta$ is given by
$s = r \theta$, where angle is in radians.
Hence, arc length subtended by angle $d \theta$ at a distance $r$ would be $= r d \theta$
3. Consider area element as shown in the figure below

We see that area element is located at a perpendicular distance $= r \sin \theta$ from the $z$-axis.
Using the same formula as used in step 2. above we see that arc length subtended by angle $d \phi$ at a distance of $r \sin \theta$ would be $= r \sin \theta \mathrm{dp} h i$
Therefore we get

$\mathrm{dV} = \mathrm{dr} \cdot r d \theta \cdot r \sin \theta \mathrm{dp} h i$
$\implies \mathrm{dV} = {r}^{2} \sin \theta \cdot \mathrm{dr} \cdot d \theta \cdot \mathrm{dp} h i$

$r$ ranges from $0$ to $r$
$\theta$ from $0$ to $\pi$ and
$\phi$ from $0$ to $2 \pi$

A word of caution : Sometimes $\angle \theta \mathmr{and} \angle \phi$ are interchanged in Physics and Mathematics.
The passage describes the derivation of the infinitesimal volume element in spherical coordinates, including the formulas and reasoning for each component of the volume element. It does not contain a specific mathematical question or a direct answer to a question.
no_question
Given a text passage, determine:
    1. If it contains a mathematical question (including word problems or calculations)
    2. If an answer to that mathematical question is provided in the text

First, describe the content structure you found in 1-2 sentences.

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TYPE: <one of the following values>
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Question #e4aa5

Jan 24, 2017

A point $P$ is defined in spherical coordinated as a function of $\left(r , \theta , \phi\right)$. Where $r$ is the distance from the origin to the point, $\phi$ is the angle that we need to rotate down from the positive $z$-axis to get to the point and $\theta$ is angle we need to rotate around the $z$-axis to get to the point.

As you have rightly posted the figure, infinitesimal volume element $\mathrm{dV}$ in spherical coordinates is given by multiplication of three unit coordinates

1. Infinitesimal radial element $= \mathrm{dr}$
2. Recall formula which relates the arc length $s$ of a circle of radius $r$ to the central angle $\theta$ is given by
$s = r \theta$, where angle is in radians.
Hence, arc length subtended by angle $d \theta$ at a distance $r$ would be $= r d \theta$
3. Consider area element as shown in the figure below

We see that area element is located at a perpendicular distance $= r \sin \theta$ from the $z$-axis.
Using the same formula as used in step 2. above we see that arc length subtended by angle $d \phi$ at a distance of $r \sin \theta$ would be $= r \sin \theta \mathrm{dp} h i$
Therefore we get

$\mathrm{dV} = \mathrm{dr} \cdot r d \theta \cdot r \sin \theta \mathrm{dp} h i$
$\implies \mathrm{dV} = {r}^{2} \sin \theta \cdot \mathrm{dr} \cdot d \theta \cdot \mathrm{dp} h i$

$r$ ranges from $0$ to $r$
$\theta$ from $0$ to $\pi$ and
$\phi$ from $0$ to $2 \pi$

A word of caution : Sometimes $\angle \theta \mathmr{and} \angle \phi$ are interchanged in Physics and Mathematics.
user
The passage describes the derivation of the infinitesimal volume element in spherical coordinates, including the formulas and reasoning for each component of the volume element. It does not contain a specific mathematical question or a direct answer to a question.

TYPE: NO_QUESTION