From 1321fdadc8f67e0e37a8ec4c30cc746a49c3a6fd Mon Sep 17 00:00:00 2001 From: Alvie Rahman Date: Thu, 14 Oct 2021 15:33:38 +0100 Subject: [PATCH] maybe fix summary tag usage --- mechanical/mmme1026_maths_for_engineering.md | 13 +++++++++++++ 1 file changed, 13 insertions(+) diff --git a/mechanical/mmme1026_maths_for_engineering.md b/mechanical/mmme1026_maths_for_engineering.md index 292cde5..fb2e1c1 100755 --- a/mechanical/mmme1026_maths_for_engineering.md +++ b/mechanical/mmme1026_maths_for_engineering.md @@ -40,8 +40,13 @@ $$\bar{z} = z -iy$$ - Multiply numerator and denominator by the conjugate of the denominator +
+ + #### Example + + > \begin{align*} z_1 &= 5 + i \\ z_2 &= 1 -i \\ @@ -51,6 +56,8 @@ $$\bar{z} = z -iy$$ &= \frac{4 + 6i}{2} = 2 + 3i > \end{align*} +
+ ### Algebra and Conjugation When taking complex conjugate of an algebraic expresion, we can replace $i$ by $-i$ before or after @@ -148,6 +155,7 @@ $$e^{i\theta} = \cos\theta + i\sin\theta$$
+ ### Example 1 Write $z = -1 + i$ in exponential form @@ -163,6 +171,7 @@ Write $z = -1 + i$ in exponential form
+ ### Example 2 The equations for a mechanical vibration problem are found to have the following mathematical @@ -257,6 +266,7 @@ r^n(\cos\theta +i\sin\theta)^n &= r^n(\cos{n\theta} + i\sin{n\theta}) \\
+ ### Example 1 Write $1+i$ in polar form and use de Moivre's theorem to calculate $(1+i)^{15}$. @@ -278,6 +288,7 @@ Write $1+i$ in polar form and use de Moivre's theorem to calculate $(1+i)^{15}$.
+ ### Example 2 Use de Moivre's theorem to show that @@ -302,6 +313,7 @@ Use de Moivre's theorem to show that
+ ### Example 3 Given that $n \in \mathbb{N}$ and $\omega = -1 + i$, show that @@ -325,6 +337,7 @@ $w^n + \bar{w}^n = 2^{\frac n 2 + 1}\cos{\frac{3n\pi} 4}$ with Euler's formula.
+ ### Example Find which complex numbers $z$ satisfy