Theoretical Mechanics IPSP

Jürgen Vollmer, Universität Leipzig

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book:chap5:5.3_volume_integrals [2022/01/04 17:11] – [5.3.1 Determine volume and mass by volume integrals] abrilbook:chap5:5.3_volume_integrals [2022/01/04 20:29] (current) – abril
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-FIXME draft with missing figures and references :!: 
- 
 ===== 5.3  Volume integrals — A professor falling through Earth  ===== ===== 5.3  Volume integrals — A professor falling through Earth  =====
 <WRAP #section_VolumeMassDensity></WRAP> <WRAP #section_VolumeMassDensity></WRAP>
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 \end{align*} \end{align*}
 In the limit of small volume elements we write this sum as a In the limit of small volume elements we write this sum as a
 +
 <WRAP box round>**Definition 5.2** <wrap em>Volume Integral</wrap> \\  <WRAP box round>**Definition 5.2** <wrap em>Volume Integral</wrap> \\ 
-The //volume integral// $F$ of a function $f(\mathbf q)$ over a region\\ $R \subset \mathbb R^D$ +The //volume integral// $F$ of a function $f(\mathbf q)$ over a region $R \subset \mathbb R^D$ is defined as follows as limit of a sum over the elements of a partition,((Considerable care is taken in calculus courses to explore under which conditions the limit exists and is well-defined. Here, we assume that the function $f$ varies smoothly inside the region. In other words, we assume that for all partition elements the difference  $\lvert f( \mathbf q ) - f( \mathbf q_i )\rvert \lll \lvert f( \mathbf q_i )\rvert$ for all points $\mathbf q \in \Delta V_i$.)) $\{ \Delta V_i\, , \;\; i \in \mathsf{I} \}$ of $R$ and points $\mathbf q_i \in  \Delta V_i$,
-is defined as follows as limit of a sum over the\\ elements of a partition,\footnotemark\  +
-$\{ \Delta V_i\, , \;\; i \in \mathsf{I} \}$ of $R$ and points\\ $\mathbf q_i \in  \Delta V_i$,+
 \begin{align*}  \begin{align*} 
     F     F
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 \end{align*} \end{align*}
 where the boundaries of the integrals must be chosen such that $( q_1, \cdots , q_D) \in R$.  where the boundaries of the integrals must be chosen such that $( q_1, \cdots , q_D) \in R$. 
-</WRAP> 
- 
-<WRAP right 140pt id=??> 
-Considerable care is taken in calculus courses to explore 
-under which conditions the limit exists and is well-defined. 
-Here, we assume that the function $f$ varies smoothly inside the region. 
-In other words, we assume that for all partition elements the difference  
-$\lvert f( \mathbf q ) - f( \mathbf q_i )\rvert \lll \lvert f( \mathbf q_i )\rvert$ 
-for all points $\mathbf q \in \Delta V_i$.} 
 </WRAP> </WRAP>
  
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 <wrap lo>** Remark 5.2. **   <wrap lo>** Remark 5.2. **  
 For the function $f( \mathbf q ) = 1$ the volume integral provides the $D$-dimensional volume of the region $R$. For the function $f( \mathbf q ) = 1$ the volume integral provides the $D$-dimensional volume of the region $R$.
- \manimpossiblecube 
 </wrap> </wrap>
  
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 \end{align*} \end{align*}
 The boundaries of the integral that define the shape of the body have been absorbed into the definition of the density. The boundaries of the integral that define the shape of the body have been absorbed into the definition of the density.
- \manimpossiblecube 
 </wrap> </wrap>
  
 We illustrate the steps taken to evaluate a volume integral by calculating the area and volume of some simple geometric shapes: We illustrate the steps taken to evaluate a volume integral by calculating the area and volume of some simple geometric shapes:
-<WRAP box round>**Example 5.1** <wrap em>Surface areas of rectangles and circles</wrap> \\  
  
-  -  The surface area of the rectangle $R \subset \mathbb R^2$ with $(x,y) \in R$ iff $0 \leq x \leq a$ and $-b < y < b$ +<WRAP box round #bsp_AreaIntegral>**Example 5.1** <wrap em>Surface areas of rectangles and circles</wrap> \\  
-is+ 
 +**a)** The surface area of the rectangle $R \subset \mathbb R^2$ with $(x,y) \in R$ iff $0 \leq x \leq a$ and $-b < y < b$ is
 \begin{align*}  \begin{align*} 
       \| R \|       \| R \|
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       = 2ab       = 2ab
 \end{align*} \end{align*}
-  -  The surface area of the circle $C$ with center at the origin and radius $R$ is+ 
 +**b)** The surface area of the circle $C$ with center at the origin and radius $R$ is
 \begin{align*}  \begin{align*} 
       \| C \|       \| C \|
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       = \pi \, R^2       = \pi \, R^2
 \end{align*} \end{align*}
-The choice of the integration boundaries is illustrated in \cref{fig:CircleAreaIntegral}.+ 
 +The choice of the integration boundaries is illustrated in [[#fig_CircleAreaIntegral |Figure 5.8]].
 Upon moving to the second line of this equation we substituted $x=R \, \sin\theta$, Upon moving to the second line of this equation we substituted $x=R \, \sin\theta$,
 and in the step to the third line we made use of the $\pi$-periodicity of $\cos^2\theta$. and in the step to the third line we made use of the $\pi$-periodicity of $\cos^2\theta$.
 </WRAP> </WRAP>
  
-<WRAP right id=fig_CircleAreaIntegral> +<WRAP 120pt right #fig_CircleAreaIntegral> 
-{{./Sketch/CircleAreaIntegral.png}}+{{CircleAreaIntegral.png}}
  
-Notations adopted in the surface integral performed in  +Figure 5.8: Notations adopted in the surface integral performed in  
-[[#bsp_AreaIntegral|Example ??]].+[[#bsp_AreaIntegral|Example 5.1 b]].
 </WRAP> </WRAP>
  
  
-<WRAP box round>**Example 5.2** <wrap em>Volume of a sphere</wrap> \\ +<WRAP box round #bsp_VolumeIntegral>**Example 5.2** <wrap em>Volume of a sphere</wrap> \\ 
 The volume of a three-dimensional sphere $S$ with center at the origin and radius $R$ is The volume of a three-dimensional sphere $S$ with center at the origin and radius $R$ is
 \begin{align*}  \begin{align*} 
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 \end{align*} \end{align*}
 Upon moving to the second line we observed that the $y$ and $z$ integrals agreed with the ones performed Upon moving to the second line we observed that the $y$ and $z$ integrals agreed with the ones performed
-to evaluate the are of a circle, \cf\Example{AreaIntegral}b).+to evaluate the are of a circle, cf [[#bsp_AreaIntegral|Example 5.1 b]].
 </WRAP> </WRAP>
  
  
 ==== 5.3.2  Change of variables  ==== ==== 5.3.2  Change of variables  ====
-<WRAP id=ssection_integrationVolume />+<wrap #ssection_integrationVolume></wrap>
  
 The shape of a circle with center at the origin and radius $R$ The shape of a circle with center at the origin and radius $R$
-can much easier be described by polar coordinates rather than Cartesian coordinates: +can much easier be described by polar coordinates rather than Cartesian coordinates((In order to avoid confusion with the radius of the circle  
-\footnote{In order to avoid confusion with the radius of the circle  +the radial coordinate of the polar coordinates is here denoted as $\rho$.)): $\{ (\rho, \theta) \in \mathbb R^+ \times [0,2\pi) \; : \; \rho < R \}$.
-the radial coordinate of the polar coordinates is here denoted as $\rho$.}\   +
-$\{ (\rho, \theta) \in \mathbb R^+ \times [0,2\pi) \; : \; \rho < R \}$.+
 To take advantage of this simplification we have to introduce a transformation of the integration coordinates To take advantage of this simplification we have to introduce a transformation of the integration coordinates
 from Cartesian to polar coordinates. from Cartesian to polar coordinates.
-A heuristic guess based on \cref{fig:PolarIntegrationVolume} suggests+A heuristic guess based on [[#fig_PolarIntegrationVolume|Figure 5.9]] suggests
 that a volume element $\mathrm{d} x\, \mathrm{d} y$ at the position $(x,y)=(\rho\,\cos\theta, \rho\,\sin\theta)$ that a volume element $\mathrm{d} x\, \mathrm{d} y$ at the position $(x,y)=(\rho\,\cos\theta, \rho\,\sin\theta)$
 should be replaced by $\rho \, \mathrm{d}\theta\, \mathrm{d} \rho$. should be replaced by $\rho \, \mathrm{d}\theta\, \mathrm{d} \rho$.
 One readily verifies that this is a reasonable choice by working out the area of the circle with radius $R$: One readily verifies that this is a reasonable choice by working out the area of the circle with radius $R$:
  
-<WRAP right id=fig_PolarIntegrationVolume> +<WRAP 120pt right #fig_PolarIntegrationVolume> 
-{{./Sketch/PolarIntegrationVolume.png}} +{{PolarIntegrationVolume.png}} 
- +Figure 5.9: Integration volume for polar coordinates.
-Integration volume for polar coordinates.+
 </WRAP> </WRAP>
  
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   = \pi \, R^2   = \pi \, R^2
 \end{align*} \end{align*}
-with a much easier calculation than in \Example{AreaIntegral}b).+with a much easier calculation than in [[#bsp_AreaIntegral|Example 5.1 b]].
 Formally the change of the integration volume is determined by generalizing the substitution rule for integrals, Formally the change of the integration volume is determined by generalizing the substitution rule for integrals,
-as illustrated in  \cref{fig:integral_substitution-rule} for one dimensional integrals.+as illustrated in  [[book:chap3:3.6_problems #fig_integral_substitution-rule|Figure 3.13]] for one dimensional integrals.
 In this rule the derivative $f'(x)$ account for the change of the width of the rectangles that are summed to approximate the integral. In this rule the derivative $f'(x)$ account for the change of the width of the rectangles that are summed to approximate the integral.
 In order to generalize this idea we recall from the discussion of line integrals In order to generalize this idea we recall from the discussion of line integrals
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 </WRAP> </WRAP>
  
-<wrap lo>** Remark 5.4. **  +<WRAP lo>** Remark 5.4. **  
 The determinant of $2\times 2$ and $3\times 3$ matrices takes the form of The determinant of $2\times 2$ and $3\times 3$ matrices takes the form of
 the (sum of) products along the diagonals from left to right the (sum of) products along the diagonals from left to right
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              &\; -a_{11} \, a_{23} \, a_{32} +  a_{22} \, a_{31} \, a_{13} + a_{33} \, a_{12} \, a_{21}               &\; -a_{11} \, a_{23} \, a_{32} +  a_{22} \, a_{31} \, a_{13} + a_{33} \, a_{12} \, a_{21} 
 \end{align*} \end{align*}
-These expressions are entailed by the geometric interpretation of the cross product in \cref{ssec:GeometricCrossProduct}. +These expressions are entailed by the geometric interpretation of the cross product in [[book:chap2:2.9_cross_products_---_torques|Section 2.9.2]]. 
- \manimpossiblecube +</WRAP>
-</wrap>+
  
 Without proof Without proof
 we provide the following general rule for calculating determinants we provide the following general rule for calculating determinants
-<WRAP box round>**Theorem 5.1 <wrap hi>Recursive calculation of determinants</wrap>** \\ + 
 +<WRAP box round>**Theorem 5.1 <wrap em>Recursive calculation of determinants</wrap>** \\ 
 Let $\mathsf A$ be a $D\times D$ matrix with $D \in \mathbb N$ Let $\mathsf A$ be a $D\times D$ matrix with $D \in \mathbb N$
 and entries $a_{ij}$ where $i,j \in \{ 1, \cdots, D \}$. and entries $a_{ij}$ where $i,j \in \{ 1, \cdots, D \}$.
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 Altogether this allows us to identify the factor involved in a change of the integration variables Altogether this allows us to identify the factor involved in a change of the integration variables
 as the Jacobi determinant.  as the Jacobi determinant. 
-<WRAP box round>**Theorem 5.2 <wrap hi>Jacobi matrix and determinant</wrap>** \\ + 
 +<WRAP box round>**Theorem 5.2 <wrap em>Jacobi matrix and determinant</wrap>** \\ 
 We consider a change of integration variables from the coordinates We consider a change of integration variables from the coordinates
 $\mathbf x = (x_1, x_2, \cdots, x_D)$ to $( y_1, y_2, \cdots , y_D)$ $\mathbf x = (x_1, x_2, \cdots, x_D)$ to $( y_1, y_2, \cdots , y_D)$
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 </WRAP> </WRAP>
  
-<WRAP box round>**Example 5.3** <wrap em>Integration volumes</wrap> \\ +<WRAP box round #bsp_IntegrationVolumes>**Example 5.3** <wrap em>Integration volumes</wrap> \\ 
  
-  -  //polar coordinates//+**a)** //polar coordinates//
 $(x,y) = \rho  \: (\cos\theta, \: \sin\theta)$ \\ transform as $(x,y) = \rho  \: (\cos\theta, \: \sin\theta)$ \\ transform as
 \begin{align*}  \begin{align*} 
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                      = \rho \, \mathrm{d}\rho \, \mathrm{d} \theta                      = \rho \, \mathrm{d}\rho \, \mathrm{d} \theta
 \end{align*} \end{align*}
-  -  //cylindrical coordinates//+ 
 +**b)** //cylindrical coordinates//
 $(x,y,z) = (\rho \cos\theta, \: \rho \sin\theta, z)$\\ transform as $(x,y,z) = (\rho \cos\theta, \: \rho \sin\theta, z)$\\ transform as
 \begin{align*}  \begin{align*} 
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                      = \rho \, \mathrm{d}\rho \, \mathrm{d} \theta \, \mathrm{d} z                      = \rho \, \mathrm{d}\rho \, \mathrm{d} \theta \, \mathrm{d} z
 \end{align*} \end{align*}
-  -  \text{//spherical coordinates// + 
 +**c)** //spherical coordinates// 
 $(x,y,z) = \rho \: (\sin\theta \cos\phi, \, \sin\theta \sin\phi, \, \cos\theta)$}\\ transform as $(x,y,z) = \rho \: (\sin\theta \cos\phi, \, \sin\theta \sin\phi, \, \cos\theta)$}\\ transform as
 \begin{align*}  \begin{align*} 
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 a physics professor and its environment. a physics professor and its environment.
 In the absence of interaction with other matter the professor will freely fall towards the center of Earth, In the absence of interaction with other matter the professor will freely fall towards the center of Earth,
-accelerated by a force that arises as sum of the mass elements constituting Earth (see\cref{fig:FallingThroughEarth}).+accelerated by a force that arises as sum of the mass elements constituting Earth (see [[#fig_FallingThroughEarth |Figure 5.10]]).
 For the professor of mass $m$ at position $\mathbf q_P$ and the mass element at position $\mathbf q_e$ this force amounts to For the professor of mass $m$ at position $\mathbf q_P$ and the mass element at position $\mathbf q_e$ this force amounts to
 $\mathbf F(\mathbf q_P, \mathbf q_e) = -\nabla ( m \, \rho(\mathbf q_e) \, G ) / \left\lvert \mathbf q_P - \mathbf q_e \right\rvert$. $\mathbf F(\mathbf q_P, \mathbf q_e) = -\nabla ( m \, \rho(\mathbf q_e) \, G ) / \left\lvert \mathbf q_P - \mathbf q_e \right\rvert$.
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 Then, the force on the professor takes the form Then, the force on the professor takes the form
  
-<WRAP right id=fig_FallingThroughEarth> +<WRAP 120pt right #fig_FallingThroughEarth> 
-{{./Sketch/FallingThroughEarth.png}} +{{FallingThroughEarth.png}} 
- +Figure 5.10: Initially positioned at the upper right (yellow), the professor will fall down (red),
-Initially positioned at the upper right (yellow), the professor will fall down (red),+
 and eventually pop out at the other side and return (green). and eventually pop out at the other side and return (green).
 </WRAP> </WRAP>
  
-\begin{align} \label{eq:professor-totForce}+<wrap #eq_professor-totForce></wrap> 
 +\begin{align}
   \mathbf F_{\text{tot}}   \mathbf F_{\text{tot}}
-  &= - \int_{\mathbb R^3}  \mathrm{d}^3 q \: \nabla \frac{ m \, \rho(\mathbf q_e) \, G }{ \left\lvert \mathbf q_P - \mathbf q_e \right\rvert }+  &= - \int_{\mathbb R^3}  \mathrm{d}^3 q \: \nabla \frac{ m \, \rho(\mathbf q_e) \, G }{ \left\lvert \mathbf q_P - \mathbf q_e \right\rvert } \tag{5.3.1}
   \\   \\
-  &= -   m \, \rho \, G \; \nabla  \int_{\text{Earth}}  \mathrm{d}^3 q \; \frac{1}{ \sqrt{ q_P^2 + q_e^2 - 2\, q_P \, q_e \, \cos\theta } }+  &= -   m \, \rho \, G \; \nabla  \int_{\text{Earth}}  \mathrm{d}^3 q \; \frac{1}{ \sqrt{ q_P^2 + q_e^2 - 2\, q_P \, q_e \, \cos\theta } } \tag{5.3.2}
 \end{align} \end{align}
 where $\theta$ is the angle between the two vectors $\left\lvert \mathbf q_P \right\rvert$ and $\left\lvert \mathbf q_e \right\rvert$, where $\theta$ is the angle between the two vectors $\left\lvert \mathbf q_P \right\rvert$ and $\left\lvert \mathbf q_e \right\rvert$,
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 $g = MG/R = 4\pi\,\rho\,R^2\,G/3$. $g = MG/R = 4\pi\,\rho\,R^2\,G/3$.
 The professor moves under the influence of a //harmonic// central force, The professor moves under the influence of a //harmonic// central force,
-as studied in\cref{quest:CoordTrafos,quest:ODE-12,quest:Conservation-08}! +as studied in 
-After a while (\cf\cref{quest:volIntegral-professor}) he reappears at the very same spot where he started,\+[[book:chap4:4.6_the_center_of_mass_cm_inertial_frame #quest_CoordTrafos |Problem 4.19]], 
 +[[book:chap4:4.9_solving_odes_by_coordinate_transformations #quest_ODE-12 |Problem 4.21]] and 
 +[[book:chap4:4.10_problems #quest_Conservation-08 |Problem 4.30]]! 
 + 
 +After a while (cf [[book:chap5:5.7_problems #quest_volIntegral-professor |Problem 5.12]]) he reappears at the very same spot where he started,
 except that Earth moved on while he was under way. except that Earth moved on while he was under way.
  
  
 ==== 5.3.4 Self Test ==== ==== 5.3.4 Self Test ====
- 
-----  
  
 <wrap #quest_volIntegral-parallelogram > Problem 5.5: </wrap>** Area of a parallelogram ** <wrap #quest_volIntegral-parallelogram > Problem 5.5: </wrap>** Area of a parallelogram **
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 describes a sphere of radius  $R$. describes a sphere of radius  $R$.
 The volume  $V$ of a solid of revolution are given by the integral The volume  $V$ of a solid of revolution are given by the integral
-\begin{align}\label{eq:revolutionSolidVolume} + 
-    V = \pi \; \int \mathrm{d} x \: \left( f(x) \right)^2+<wrap #eq_revolutionSolidVolume></wrap> 
 +\begin{align} 
 +    V = \pi \; \int \mathrm{d} x \: \left( f(x) \right)^2 \tag{5.3.3}
 \end{align} \end{align}
  
-  -  Sketch the function  $f(x) = \sqrt{R^2 - x^2}$ +  -  Sketch the function  $f(x) = \sqrt{R^2 - x^2}$ and verify that the solid of revolution is indeed a sphere. 
-and verify that the solid of revolution is indeed a sphere. +  -  Determine the volume of the sphere based on the given equation Compare you calculation and the result to the calculation given in [[#bsp_VolumeIntegral |Example 5.2]]. 
-  -  Determine the volume of the sphere based on the given equation. +  -  Show that the volume integral for a solid of revolution provides [[#eq_revolutionSolidVolume |Equation 5.3.3]]  when one adopts cylindrical coordinates.
-Compare you calculation and the result to the calculation given in \Example{VolumeIntegral}. +
-  -  Show that the volume integral for a solid of revolution provides  +
-\cref{eq:revolutionSolidVolume}  +
-when one adopts cylindrical coordinates.+
  
    
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 \\ \\
 Determine the Jacobi matrix and its determinant for the transformation Determine the Jacobi matrix and its determinant for the transformation
-from Cartesian to spherical coordinates, \cf\Example{IntegrationVolumes}c).+from Cartesian to spherical coordinates, cf [[#bsp_IntegrationVolumes |Example 5.3 c)]].
  
  
 ~~DISCUSSION~~  ~~DISCUSSION~~ 
  
book/chap5/5.3_volume_integrals.1641312662.txt.gz · Last modified: 2022/01/04 17:11 by abril