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--- book:chap2:2.4_fields [2021/10/26 23:23] – jv
+++ book:chap2:2.4_fields [2024/11/09 10:12] (current) – jv
@@ Line 1: / Line 1: @@
+[[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 =====
@@ Line 62: / Line 78: @@
 </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 \\
 $z_2 = x_2 + \mathrm{i} y_2$ amounts to
@@ Line 78: / Line 94: @@
     & = ( 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*}
 Checking the group axioms based on this representation of the complex numbers is tedious.