Theoretical Mechanics IPSP

Jürgen Vollmer, Universität Leipzig

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book:chap2:2.4_fields [2021/10/26 22:13] – created abrilbook:chap2:2.4_fields [2022/04/01 21:01] (current) – jv
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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| 2.2 Sets ]]
 +  * [[  2.3 Groups| 2.3 Groups ]]
 +  * ** 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 ]]
 +
 +----
 +
 ===== 2.4 Fields ===== ===== 2.4 Fields =====
  
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 For the multiplication of field elements one commonly suppresses the $\cdot$ for the multiplication, For the multiplication of field elements one commonly suppresses the $\cdot$ for the multiplication,
 writing e.g. $a\,b$ rather than $a \cdot b$. writing e.g. $a\,b$ rather than $a \cdot b$.
-  
 </wrap> </wrap>
  
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 </WRAP> </WRAP>
  
-<WRAP box round>**Example 2.13** <wrap em>Complex numbers are a field</wrap> \\+<WRAP third left #fig_complexEuler > 
 +{{:book:chap2:complexeuler.png| Figure 2.8}} 
 + 
 +Figure 2.8 Complex numbers $z$ can be represented as $z=x+\mathrm{i} y$ in a plane 
 +where $(x,y)$ are the Cartesian coordinates of $z$. Alternatively, one can adopt a representation in terms of polar coordinates $z = R \, \mathrm{e}^{\mathrm{i} \varphi}$ where $R = \sqrt{x^2+y^2}$ and $\varphi$ is the angle with respect to the $x$-axis. 
 +</WRAP> 
 + 
 +<WRAP box round #bsp_field_complex_numbers >**Example 2.13** <wrap em>Complex numbers are a field</wrap> \\
 a) The sum of two complex numbers $z_1 = x_1 + \mathrm{i} y_1$ and \\ a) The sum of two complex numbers $z_1 = x_1 + \mathrm{i} y_1$ and \\
 $z_2 = x_2 + \mathrm{i} y_2$ amounts to $z_2 = x_2 + \mathrm{i} y_2$ amounts to
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 One better adopts a representation in terms of polar coordinates, One better adopts a representation in terms of polar coordinates,
 $z_1 = R_1 \, \mathrm{e}^{\mathrm{i} \varphi_1}$ and $z_2 = R_2 \, \mathrm{e}^{\mathrm{i} \varphi_2}$ $z_1 = R_1 \, \mathrm{e}^{\mathrm{i} \varphi_1}$ and $z_2 = R_2 \, \mathrm{e}^{\mathrm{i} \varphi_2}$
-(see [[#complexEuler |Figure 2.8]]) where (cf [[#groupSelftest-EulerRelation |Problem 2.10]])+(see [[#fig_complexEuler|Figure 2.8]]) where (cf [[#quest_groupSelftest-EulerRelation |Problem 2.10]])
 \begin{align*} \begin{align*}
     z_1 \cdot z_2     z_1 \cdot z_2
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 </WRAP> </WRAP>
  
-<WRAP third left #complexEuler> 
-{{:book:chap2:complexeuler.png| Figure 2.8}} 
- 
-Figure 2.8 Complex numbers $z$ can be represented as $z=x+\mathrm{i} y$ in a plane 
-where $(x,y)$ are the Cartesian coordinates of $z$. Alternatively, one can adopt a representation in terms of polar coordinates $z = R \, \mathrm{e}^{\mathrm{i} \varphi}$ where $R = \sqrt{x^2+y^2}$ and $\varphi$ is the angle with respect to the $x$-axis. 
-</WRAP> 
  
 <wrap lo #complexConjugation_remark >**Remark 2.9** [complex conjugation] <wrap lo #complexConjugation_remark >**Remark 2.9** [complex conjugation]
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 ==== 2.4.1 Self Test ==== ==== 2.4.1 Self Test ====
  
-<wrap #groupSelftest-fields >Problem 2.9: ** Checking field axioms **</wrap>\\+<wrap #quest_groupSelftest-fields >Problem 2.9: ** Checking field axioms **</wrap>\\
 Which of the following sets are fields?\\ Which of the following sets are fields?\\
  
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 ----- -----
  
-<wrap #groupSelftest-EulerRelation >Problem 2.10: ** Euler's equation and trigonometric relations **</wrap>\\+<wrap #quest_groupSelftest-EulerRelation >Problem 2.10: ** Euler's equation and trigonometric relations **</wrap>\\
 Euler's equation $\mathrm{e}^{\mathrm{i} x} = \cos x + \mathrm{i} \: \sin x$ Euler's equation $\mathrm{e}^{\mathrm{i} x} = \cos x + \mathrm{i} \: \sin x$
 relates complex values exponential functions and trigonometric functions. relates complex values exponential functions and trigonometric functions.
book/chap2/2.4_fields.1635279223.txt.gz · Last modified: 2021/10/26 22:13 by abril