Theoretical Mechanics IPSP

Jürgen Vollmer, Universität Leipzig

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book:chap2:2.4_fields

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book:chap2:2.4_fields [2021/10/26 23:21] – jvbook:chap2:2.4_fields [2024/11/09 10:12] (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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 </WRAP> </WRAP>
  
-<WRAP box round>**Example 2.13** <wrap em>Complex numbers are a field</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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     & = ( x_1 + \mathrm{i} y_1 ) \cdot ( x_2 + \mathrm{i} y_2 )     & = ( x_1 + \mathrm{i} y_1 ) \cdot ( x_2 + \mathrm{i} y_2 )
     \\     \\
-    & = ( x_1 \, x_2 + \mathrm{i} y_1 \, x_2 + \mathrm{i} y_1 \, x_2 + \mathrm{i}^2 \, y_1\, y_2)+    & = x_1 \, x_2 + x_1\, (\mathrm{i} y_2) + (\mathrm{i} y_1) \, x_2 + (\mathrm{i} y_1)\, (\mathrm{i} y_2)
     \\     \\
-    &= ( x_1\, x_2 - y_1\,y_2 ) + \mathrm{i} \, (y_1 \, x_2 + x_1 \, y_2)+    &= ( x_1\, x_2 - y_1\,y_2 ) + \mathrm{i} \, (x_1 \, y_2 + y_1 \, x_2)
 \end{align*} \end{align*}
 Checking the group axioms based on this representation of the complex numbers is tedious. Checking the group axioms based on this representation of the complex numbers is tedious.
 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 [[#fig_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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 ==== 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.1635283277.txt.gz · Last modified: 2021/10/26 23:21 by jv