Theoretical Mechanics IPSP

Jürgen Vollmer, Universität Leipzig

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 +[[forcestorques|2. Balancing Forces and Torques]]
 +  * [[  2.1 Motivation and Outline| 2.1 Motivation and outline: forces are vectors ]]
 +  * ** 2.2 Sets **
 +  * [[  2.3 Groups| 2.3 Groups ]]
 +  * [[  2.4 Fields| 2.4 Fields ]]
 +  * [[  2.5 Vector spaces| 2.5 Vector spaces ]]
 +  * [[  2.6 Physics application balancing forces| 2.6.  Physics application: balancing forces]]
 +  * [[  2.7 The inner product | 2.7 The inner product]]
 +  * [[  2.8 Cartesian coordinates | 2.8 Cartesian coordinates]]
 +  * [[  2.9 Cross products --- torques| 2.9 Cross products — torques ]]
 +  * [[ 2.10 Worked example Calder's mobiles| 2.10 Worked example: Calder's mobiles ]]
 +  * [[ 2.11 Problems| 2.11 Problems ]]
 +  * [[ 2.12 Further reading| 2.12 Further reading ]]
 +
 +----
 +
 ===== 2.2 Sets ===== ===== 2.2 Sets =====
  
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 <WRAP lo>**Remark.**  Notations and additional properties:\\ <WRAP lo>**Remark.**  Notations and additional properties:\\
-  * When a set $M$ has a finite number of elements, e.g., $+1$ and $-1$, one can specify the elements by explicitly stating the elements, $M = \{ +1, -1 \}$. In which order they are states does not play a role, +  * When a set $M$ has a finite number of elements, e.g., $+1$ and $-1$, one can specify the elements by explicitly stating the elements, $M = \{ +1, -1 \}$. In which order they are states does not play a role, and it also does not make a difference when elements are provided several times. In other words the set $M$ of cardinality two can be specified by any of the following statements 
-and it also does not make a difference when elements are provided several times. In other words the set $M$ of cardinality two can be specified by any of the following statements +
 \[ \[
 M = \{ -1, +1 \} = \{ +1, -1 \} = \{ -1, 1, 1, 1, \} = \{ -1, 1, +1, -1 \} M = \{ -1, +1 \} = \{ +1, -1 \} = \{ -1, 1, 1, 1, \} = \{ -1, 1, +1, -1 \}
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-<WRAP box round>**Example 2.2** <wrap em>Sets</wrap> +<WRAP box round #bsp_sets >**Example 2.2** <wrap em>Sets</wrap> 
 \\  \\ 
   *  Set of capitals of German states: \\   *  Set of capitals of German states: \\
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 The cardinalities of these sets are \\ The cardinalities of these sets are \\
-\centering $|A_C| = 16$, $|A_L = 30|$, and $|A_M| = 12$.+<wrap center> $|A_C| = 16$, $|A_L = 30|$, and $|A_M| = 12$. </wrap>
 </WRAP> </WRAP>
  
-<WRAP box round>**Example 2.3** <wrap em>Sets of sets</wrap> \\ +<WRAP box round #bsp_setsofsets >**Example 2.3** <wrap em>Sets of sets</wrap> \\ 
 A set can be an element of a set. A set can be an element of a set.
 For instance the set For instance the set
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 In obvious cases we use ellipses such as In obvious cases we use ellipses such as
 $A_L = \{$a, b, c, \dots, z, ä, ö, ü, ß$\}$ $A_L = \{$a, b, c, \dots, z, ä, ö, ü, ß$\}$
-for the set given in \Example{Sets}.+for the set given in  [[#bsp_Sets|Example 2.2]].
 Alternatively, one can provide a set $M$ by specifying the properties $A(x)$ Alternatively, one can provide a set $M$ by specifying the properties $A(x)$
 of its elements $x$ in the following form of its elements $x$ in the following form
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 The properties are separated by commas, and must all be true for all elements of the set. The properties are separated by commas, and must all be true for all elements of the set.
  
-<WRAP box round>**Example 2.4** <wrap em>Set definition by property</wrap> \\ +<WRAP box round #bsp_SetOfDigits >**Example 2.4** <wrap em>Set definition by property</wrap> \\ 
 The set of digits $D = \left\{ 1, \, 2, \, 3, \, 4, \, 5, \, 6, \, 7, \, 8, \, 9 \right\}$ The set of digits $D = \left\{ 1, \, 2, \, 3, \, 4, \, 5, \, 6, \, 7, \, 8, \, 9 \right\}$
 can also be defined as follows can also be defined as follows
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 implies $\Rightarrow$, and implies $\Rightarrow$, and
 is equivalent $\Leftrightarrow$ is equivalent $\Leftrightarrow$
-for the relations indicated in \ref{table:Junktors}.+for the relations indicated in [[#tab_Junktors|Table 2.1]].
  
-<WRAP 400px center>+<WRAP 400px center #tab_Junktors >
 ^ $A$ ^  $0$  ^  $0$  ^  $1$  ^  $1$  ^  ^ ^ $A$ ^  $0$  ^  $0$  ^  $1$  ^  $1$  ^  ^
 ^ $B$ ^  $0$  ^  $1$  ^  $0$  ^  $1$  ^  ^ ^ $B$ ^  $0$  ^  $1$  ^  $0$  ^  $1$  ^  ^
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 |$ \lnot A \land B     $  |  0  |  1  |  0  |  0  | not $A$ or $B$ | |$ \lnot A \land B     $  |  0  |  1  |  0  |  0  | not $A$ or $B$ |
 |$ A \land \lnot B     $  |  0  |  0  |  1  |  0  | $A$ and not $B$ | |$ A \land \lnot B     $  |  0  |  0  |  1  |  0  | $A$ and not $B$ |
-List of the results of different junctors acting on two statements $A$ and $B$.+Table 2.1: List of the results of different junctors acting on two statements $A$ and $B$.
 Here $0$ and $1$ indicate that a statement is wrong or right, respectively. Here $0$ and $1$ indicate that a statement is wrong or right, respectively.
 In the rightmost column we state the contents of the expression in the left column in words. In the rightmost column we state the contents of the expression in the left column in words.
 The final three lines provide examples of more complicated expressions. The final three lines provide examples of more complicated expressions.
-<wrap hide>\label{table:Junktors}</wrap> 
 </WRAP> </WRAP>
  
-The definition of the digits in \Example{SetOfDigits} entails that all elements of $D$ are also numbers in $\mathbb{Z}$:+The definition of the digits in [[#bsp_SetOfDigits|Example 2.4]] entails that all elements of $D$ are also numbers in $\mathbb{Z}$:
 we say that $D$ is a subset of $\mathbb{Z}$. we say that $D$ is a subset of $\mathbb{Z}$.
  
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 <WRAP box round>**Example 2.5** <wrap em>Subsets</wrap> \\  <WRAP box round>**Example 2.5** <wrap em>Subsets</wrap> \\ 
  
-  *  The set of month with names that end with 'ber' is a subset of the set $A_M$ of \Example{Sets}+  *  The set of month with names that end with 'ber' is a subset of the set $A_M$ of [[#bsp_Sets|Example 2.2]]:
 \begin{align*} \begin{align*}
       \{ \text{September, October, November, December} \} \subseteq A_M       \{ \text{September, October, November, December} \} \subseteq A_M
 \end{align*} \end{align*}
  
-  *  For the set  $M$  of  \Example{SetsOfSets}  one has+  *  For the set  $M$  of  [[#bsp_SetsOfSets|Example 2.3]]  one has
 \begin{align*} \begin{align*}
       \{ 1 \} \subseteq M \, , \quad        \{ 1 \} \subseteq M \, , \quad 
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 Two sets are the same when they are subsets of each other. Two sets are the same when they are subsets of each other.
-<WRAP box round>**Theorem 2.1 <wrap hi>Äquivalenz von Mengen</wrap>** \\ +<WRAP box round #Thm_SetEquivalence >**Theorem 2.1 <wrap em> Equivalence of Sets</wrap>** \\ 
 Two sets  $A$ and $B$   are //equal// or //equivalent//, iff Two sets  $A$ and $B$   are //equal// or //equivalent//, iff
 \begin{align*} \begin{align*}
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 </WRAP> </WRAP>
  
-<wrap lo>**Anmerkung.** [iff] +<wrap lo>**Remark.** In mathematics "iff" indicates that something holds "if and only if".
-In mathematics ``iff'' indicates that something holds +
-``if and only if''.+
 Observe its use in the following two statements: Observe its use in the following two statements:
 A number is an even number if it is the product of two even numbers. A number is an even number if it is the product of two even numbers.
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 </wrap> </wrap>
  
-<wrap lo>**Anmerkung.** [precedence of operations in logical expressions.] +<wrap lo #rem_LogicalPrecedence >**Remark.** In logical expressions we first evaluate $\in$, $\not\in$ and other set operations that are used to build logical expressions. 
-\label{remark:LogicalPrecedence} +
-In logical expressions we first evaluate $\in$, $\not\in$ and other set operations +
-that are used to build logical expressions. +
 Then we evaluate the junctor $\lnot$ that is acting on a a single logical expression. Then we evaluate the junctor $\lnot$ that is acting on a a single logical expression.
 Finally the other junctors $\land$, $\lor$, $\Rightarrow$, and $\Leftrightarrow$ are evaluated. Finally the other junctors $\land$, $\lor$, $\Rightarrow$, and $\Leftrightarrow$ are evaluated.
-Hence, the brackets are not required in \Thm{SetEquivalence}.+Hence, the brackets are not required in [[#Thm_SetEquivalence|Theorem 2.1]].
 </wrap> </wrap>
  
-=== Proof of \Thm{SetEquivalence} ===+=== Proof of Theorem 2.1 ===
 $A \subseteq B$ implies that $a \in A \Rightarrow a \in B$.  $A \subseteq B$ implies that $a \in A \Rightarrow a \in B$. 
 $B \subseteq A$ implies $b \in B \Rightarrow b \in A$.  $B \subseteq A$ implies $b \in B \Rightarrow b \in A$. 
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 $\rule{3mm}{3mm}$ $\rule{3mm}{3mm}$
  
-The description of sets by properties of its members, \Example{SetOfDigits},+The description of sets by properties of its members, [[#bsp_SetOfDigits|Example 2.4]],
 suggests that one will often be interested in operations on sets. suggests that one will often be interested in operations on sets.
 For instance the odd and even numbers are subsets of the natural numbers. For instance the odd and even numbers are subsets of the natural numbers.
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 <WRAP 200px right> <WRAP 200px right>
-{{ en:book:chap02:mengen_schnitt.png?direct&200 |}}+<wrap #fig_setIntersection >  
 +{{ book:chap2:mengen_schnitt.png?direct&200 |}}
  
-Intersection of two sets. \\ $\quad$ +Figure 2.2: Intersection of two sets. \\ $\quad$
-<wrap hide> +
-\label{figure:setIntersection}+
 </wrap> </wrap>
  
-{{ en:book:chap02:mengen_vereinigung.png?direct&200 |}}+<wrap #fig_setUnion > 
 +{{ book:chap2:mengen_vereinigung.png?direct&200 |}}
  
-Union of two sets. \\ $\quad$ +Figure 2.3: Union of two sets. \\ $\quad$
-<wrap hide> +
-\label{figure:setUnion}+
 </wrap> </wrap>
  
-{{ en:book:chap02:mengen_differenz.png?direct&200 |}}+<wrap #fig_setDifference >  
 +{{ book:chap2:mengen_differenz.png?direct&200 |}}
  
-Difference of two sets. \\ $\quad$ +Figure 2.4: Difference of two sets. \\ $\quad$
-<wrap hide> +
-\label{figure:setDifference}+
 </wrap> </wrap>
  
-{{ en:book:chap02:mengen_komplement.png?direct&150 |}}+<wrap #fig_setComplement > 
 +{{ book:chap2:mengen_komplement.png?direct&150 |}}
  
-Complement of a set. \\ $\quad$ +Figure 2.5: Complement of a set. \\ $\quad$
-<wrap hide> +
-\label{figure:setComplement}+
 </wrap> </wrap>
 </WRAP> </WRAP>
  
  
-<WRAP box round>**Definition 2.3** <wrap em>Set Operations</wrap> \\ +<WRAP box round #def_setoperations >**Definition 2.3** <wrap em>Set Operations</wrap> \\ 
 For two sets  $M_1$ and $M_2$  we define the following operations: For two sets  $M_1$ and $M_2$  we define the following operations:
  
-  *  //Intersection//: +  *  //Intersection//:  $M_1 \bigcap M_2 = \left\{ m   \; \vert \;  m \in M_1 \land m \in M_2 \right\}$, 
-  +  *  //Union//:  $M_1 \bigcup M_2 = \left\{ m   \; \vert \;  m \in M_1 \lor m \in M_2 \right\}$, 
-$M_1 \bigcap M_2 = \left\{ m   \; \vert \;  m \in M_1 \land m \in M_2 \right\}$, +  *  //Difference//:  $M_1 \backslash M_2 = \left\{ m   \; \vert \;  m \in M_1 \land m \notin M_2 \right\}$, 
-  *  //Union//: +  *  The //complement// of a set $M$ in a //universe// $U$ is defined for subsets $M \subseteq U$ as $M^{C} = \left\{ m \in U   \; \vert \;  m \notin M \right\} = U \backslash M$. 
-   +  *  The //Cartesian product of two sets $M_1$ and $M_2$// is defined as the set of ordered pairs $(a, \, b)$ of elements $a \in M_1$ and \\ $b \in M_2$:  $\displaystyle M_1 \times M_2 = \left\{ (a, \, b)   \; \vert \;  a \in M_1, \, b \in M_2 \right\} $.
-$M_1 \bigcup M_2 = \left\{ m   \; \vert \;  m \in M_1 \lor m \in M_2 \right\}$, +
-  *  //Difference//:  +
-  +
-$M_1 \backslash M_2 = \left\{ m   \; \vert \;  m \in M_1 \land m \notin M_2 \right\}$, +
-  *  The //complement// of a set $M$ in a //universe// $U$ is defined for subsets $M \subseteq U$ +
-as   +
-$M^{C} = \left\{ m \in U   \; \vert \;  m \notin M \right\} = U \backslash M$. +
-  *  } +
-The //Cartesian product of two sets $M_1$ and $M_2$// is defined as the set of ordered pairs +
-$(a, \, b)$ of elements $a \in M_1$ and\\ +
-$b \in M_2$: +
- $\displaystyle +
-M_1 \times M_2 = \left\{ (a, \, b)   \; \vert \;  a \in M_1, \, b \in M_2 \right\} +
-$.+
  
 A graphical illustration of the operations is provided in A graphical illustration of the operations is provided in
-\cref{figure:setIntersection,figure:setUnion,figure:setDifference,figure:setComplement}.+[[#fig_setIntersection|Figure 2.2]], [[#fig_setUnion|Figure 2.3]], [[#fig_setDifference|Figure 2.4]], [[#fig_setComplement|Figure 2.5]].
 </WRAP> </WRAP>
  
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 The set of non-female participants is  $P \backslash F$. The set of non-female participants is  $P \backslash F$.
 The set of heterosexual couples in the class is a subset of the Cartesian product $F \times M$.  The set of heterosexual couples in the class is a subset of the Cartesian product $F \times M$. 
-Furthermore, the union $F \bigcup M$ is a proper subset of$P$,  +Furthermore, the union $F \bigcup M$ is a proper subset of $P$, when there is a participant who is neither female nor male.
-when there is a participant who is neither female nor male.+
 </WRAP> </WRAP>
  
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 <WRAP box round>**Example 2.7** <wrap em>Logical quantors and properties of set elements</wrap> \\  <WRAP box round>**Example 2.7** <wrap em>Logical quantors and properties of set elements</wrap> \\ 
-Let $|m|$ denote the number of days in a month $a \in A_M$  +Let $|m|$ denote the number of days in a month $a \in A_M$ (Refer to [[#bsp_Sets|Example 2.2]]).
-(\cf\Example{Sets}).+
 Then the following statements are true: Then the following statements are true:
 There is exactly one month that has exactly $28$ days:  There is exactly one month that has exactly $28$ days: 
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 Many sets of numbers that are of interest in physics have infinitely many elements. Many sets of numbers that are of interest in physics have infinitely many elements.
-We construct them in \cref{table:SetsOfNumbers} based on the natural numbers+We construct them in [[#tab_SetsOfNumbers|Table 2.2]] based on the natural numbers
 \begin{align*} \begin{align*}
   \mathbb N   \mathbb N
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 are handy when it comes to problems involving three-dimensional rotations. are handy when it comes to problems involving three-dimensional rotations.
 In any case one needs intervals of numbers. In any case one needs intervals of numbers.
-<WRAP center>+<WRAP center #tab_SetsOfNumbers >
 ^ name            ^ symbol        ^ description ^ ^ name            ^ symbol        ^ description ^
 | natural numbers | $\mathbb{N}$  | $\left\{ 1, \, 2, \, 3, \, \ldots \right\}$ | | natural numbers | $\mathbb{N}$  | $\left\{ 1, \, 2, \, 3, \, \ldots \right\}$ |
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 | negative numbers         | $-\mathbb{N}$  | $\left\{ -n   \; \vert \;  n \in \mathbb{N} \right\}$ | | negative numbers         | $-\mathbb{N}$  | $\left\{ -n   \; \vert \;  n \in \mathbb{N} \right\}$ |
 | even numbers             | $2 \mathbb{N}$ | $\left\{ 2\,n   \; \vert \;  n \in \mathbb{N} \right\}$ | | even numbers             | $2 \mathbb{N}$ | $\left\{ 2\,n   \; \vert \;  n \in \mathbb{N} \right\}$ |
-| odd numbers              | $2 \mathbb{N} - 1$   | $\left\{ 2\,n -1  \; \vert \;  n \in \mathbb{N} \right\} |+| odd numbers              | $2 \mathbb{N} - 1$   | $\left\{ 2\,n -1  \; \vert \;  n \in \mathbb{N}\right\}$ |
 | integer numbers        | $\mathbb{Z}$   | $\left( -\mathbb{N} \right) \bigcup \mathbb{N}_0$ | | integer numbers        | $\mathbb{Z}$   | $\left( -\mathbb{N} \right) \bigcup \mathbb{N}_0$ |
 | rational numbers       | $\mathbb{Q}$   | $\left\{ \frac{p}{q}   \; \vert \;  p \in \mathbb{Z}, \, q \in \mathbb{N} \right\}$ | | rational numbers       | $\mathbb{Q}$   | $\left\{ \frac{p}{q}   \; \vert \;  p \in \mathbb{Z}, \, q \in \mathbb{N} \right\}$ |
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 | complex numbers        | $\mathbb{C}$   | $\mathbb{R}+ \text{i}\mathbb{R},$ where $\text{i} = \sqrt{-1}$ | | complex numbers        | $\mathbb{C}$   | $\mathbb{R}+ \text{i}\mathbb{R},$ where $\text{i} = \sqrt{-1}$ |
  
-Summary of important sets of numbers. +Table 2.2: Summary of important sets of numbers.
-</wrap> +
-\label{table:SetsOfNumbers} +
-</wrap>+
 </WRAP> </WRAP>
  
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 ==== 2.2.2 Self Test ==== ==== 2.2.2 Self Test ====
  
-Problem 2.1: <wrap hide>\label{quest:setSelftest-SetRelations}</wrap> +<WRAP #quest_setSelftest-SetRelations > 
 +Problem 2.1: 
 ** Relations between sets ** ** Relations between sets **
  
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 </WRAP> </WRAP>
 </WRAP> </WRAP>
 +</WRAP>
 +
  
 ---- ----
  
-Problem 2.2: <wrap hide>\label{quest:setSelftest-intervals}</wrap>+<WRAP #quest_setSelftest-intervals > 
 +Problem 2.2: 
 ** Intervals ** ** Intervals **
  
   -  Provide $\lbrack 1; 17 \rbrack \cap \rbrack 0; 5 \lbrack$ as a single interval.   -  Provide $\lbrack 1; 17 \rbrack \cap \rbrack 0; 5 \lbrack$ as a single interval.
   -  Provide $[-1, 4] \backslash [1, 2[$ as union of two intervals.   -  Provide $[-1, 4] \backslash [1, 2[$ as union of two intervals.
 +</WRAP>
  
 ---- ----
  
-Problem 2.3: <wrap hide>\label{quest:setSelftest-numbers}</wrap>+<WRAP #quest_setSelftest-numbers > 
 +Problem 2.3: 
 ** Sets of numbers ** ** Sets of numbers **
  
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   -  $\lbrace 6 \cdot z \vert z \in \mathbb{Z} \rbrace \subset \lbrace 2 \cdot z \vert z \in \mathbb{Z} \rbrace$.   -  $\lbrace 6 \cdot z \vert z \in \mathbb{Z} \rbrace \subset \lbrace 2 \cdot z \vert z \in \mathbb{Z} \rbrace$.
   -  $\left\lbrace 2 \cdot z \vert z \in \mathbb{Z} \right\rbrace \cap \left\lbrace 3 \cdot z \vert z \in \mathbb{Z} \right\rbrace = \left\lbrace 6 \cdot z \vert z \in \mathbb{Z} \right\rbrace$.   -  $\left\lbrace 2 \cdot z \vert z \in \mathbb{Z} \right\rbrace \cap \left\lbrace 3 \cdot z \vert z \in \mathbb{Z} \right\rbrace = \left\lbrace 6 \cdot z \vert z \in \mathbb{Z} \right\rbrace$.
-  -  Let $T(a)$ be the set of numbers that divide $a$. Then +  -  Let $T(a)$ be the set of numbers that divide $a$. Then  $\quad\displaystyle  \forall a, b \in \mathbb{N}  \; \vert  \; T(a) \cup T(b) = T(a \cdot b) $. 
-\begin{align*} +
-      \forall a, b \in \mathbb{N}  \; \vert \;\quad T(a) \cup T(b) = T(a \cdot b)  +
-\end{align*}+
 Example:  $T(2) = \{1,2\}$, $T(3) = \{1,3\}$, and $T(6) = \{1,2,3,6\}$.  Example:  $T(2) = \{1,2\}$, $T(3) = \{1,3\}$, and $T(6) = \{1,2,3,6\}$. 
 +</WRAP>
  
 ~~DISCUSSION|Questions, Remarks, and Suggestions~~ ~~DISCUSSION|Questions, Remarks, and Suggestions~~
book/chap2/2.2_sets.1635182794.txt.gz · Last modified: 2021/10/25 19:26 by jv