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--- book:chap2:2.1_motivation_and_outline [2021/10/07 09:37] – [Outline] jv
+++ book:chap2:2.1_motivation_and_outline [2022/04/01 19:59] (current) – jv
@@ Line 1: / Line 1: @@
+[[forcestorques|2. Balancing Forces and Torques]]
+  * **  2.1 Motivation and outline: forces are vectors **
+  * [[  2.2 Sets| 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 ]]
+----
 ===== 2.1 Motivation and outline: forces are vectors =====
@@ Line 4: / Line 20: @@
 A displacement describes the relative position of two points in space,
 and the velocity can be thought of as a distance divided by the time needed to go from the initial to the final point.
-(A mathematically more thorough definition will be given in \cref{chapter:Newton}.)
+(A mathematically more thorough definition will be given in [[book:chap3:newton|Chapter 3]].)
 For forces it is of paramount importance to indicate in which direction they are acting.
-Similarly, in contrast to speed, a velocity can not be specified in terms of a number with a unit, e.g. $5\,\text{m/s}$.
+Similarly, in contrast to speed, a velocity can not be specified in terms of a number with a unit, e.g. 5 m/s.
 By its very definition one also has to specify the direction of motion.
 Finally, also a displacement involves a length specification and a direction.
-<WRAP 150px right>
+<WRAP 115px left #fig_middlePage >
 {{book:chap2:middle_page_dot.png?direct&400|}}
-The displacement of the red point from the bottom left corner to the the middle of the page
+Figure 2.1: The displacement of the red point from the bottom left corner to the the middle of the page
 can either be specified by the direction $\theta$  and the distance $R$ (polar coordinates, top),
-or by the distances $x$ and $y$ along the sides of the paper (Cartesian coordinates, bottom).}
+or by the distances $x$ and $y$ along the sides of the paper (Cartesian coordinates, bottom).
-\label{figure:middlePage}
 </WRAP>
-<WRAP box round>**Example 1.1** <wrap em>Displacement of a red dot from the lower left corner to the middle of a paper</wrap> \\
-This displacement is illustrated in \cref{figure:middlePage}.
+<WRAP box round>**Example 2.1** <wrap em>Displacement of a red dot from the lower left corner to the middle of a paper</wrap> \\
+This displacement is illustrated in [[#fig_middlePage|Figure 2.1]].
 It can either be specified in terms of the distance $R$ of the point from the corner
 and the angle $\theta$ of the line connecting the points and the lower edge of the paper
@@ Line 30: / Line 47: @@
 In three dimensions, one has to adopt a third direction out of the plane used for the paper, and hence three numbers, to specify a displacements---or indeed any other vector.
 \begin{align*}
 \begin{array}{l|lll}
@@ Line 54: / Line 72: @@
 \end{array}
 \end{align*}
 A basic introduction of mechanics can be given based on this heuristic account of vectors.
 However, for the thorough exposition that serve as a foundation of theoretical physics a more profound mathematical understanding of vectors is crucial.
 Hence, a large part of this chapter will be devoted to mathematical concepts.
@@ Line 68: / Line 88: @@
 or the set of students in my class.
 Mathematical structures refer to sets where the elements obey certain additional properties,
 like in groups and vector spaces.
 They are expressed in terms of //operations// that take one or several elements of the set,
 and return a result that may or may not be part of the given set.
 When an operation $f$ takes an element of a set $A$ and returns another element of $A$ we write
 $f : A \to A$.
-When an operation $\circ$ takes two elements of a set $A$ and returns a single element of $A$ we write
+When an operation $\circ$ takes two elements of a set $A$ and returns a single element of $A$ we write ((Here $A \times A$ is the set, $(a_1, a_2)$, of all pairs of elements $a_1, a_2 \in A$.
-((Here $A \times A$ is the set, $(a_1, a_2)$, of all pairs of elements $a_1, a_2 \in A$.
+Further details will be given in [[book:chap2:2.2_sets#Def_SetOperations|Definition 2.3]] below.)) $\circ : A \times A \to A$.
-Further details will be given in Section \ref{def:SetOperations} below.))
-$\circ : A \times A \to A$.
 Equipped with the mathematical tool of vectors we will explore the physical concepts of forces and torques,
 and how they are balanced in systems at rest.
 ~~DISCUSSION|Questions, Remarks, and Suggestions~~