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mechanical/mmme1026_maths_for_engineering.md
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@ -146,15 +146,23 @@ $$e^{i\theta} = \cos\theta + i\sin\theta$$
**Note**: $$\bar z = r\cos\theta - ir\sin\theta = re^{-i\theta}$$ **Note**: $$\bar z = r\cos\theta - ir\sin\theta = re^{-i\theta}$$
<details>
<summary>
### Example 1 ### Example 1
Write $z = -1 + i$ in exponential form Write $z = -1 + i$ in exponential form
</summary>
> $\arg z = \frac {3\pi} 4$ > $\arg z = \frac {3\pi} 4$
> $|z| = \sqrt 2$ > $|z| = \sqrt 2$
> >
> So $z = \sqrt2e^{i\frac{3\pi} 4}$ > So $z = \sqrt2e^{i\frac{3\pi} 4}$
</details>
<details>
<summary>
### Example 2 ### Example 2
The equations for a mechanical vibration problem are found to have the following mathematical The equations for a mechanical vibration problem are found to have the following mathematical
@ -162,6 +170,8 @@ solution:
$$z(t) = \frac{e^{i\omega t}}{\omega_0^2-\omega^2 + i\gamma}$$ $$z(t) = \frac{e^{i\omega t}}{\omega_0^2-\omega^2 + i\gamma}$$
</summary>
where $t$ represents time and $\omega$, $\omega_0$ and $\gamma$ are all positive real physical where $t$ represents time and $\omega$, $\omega_0$ and $\gamma$ are all positive real physical
constants. constants.
Although $z(t)$ Although $z(t)$
@ -200,6 +210,8 @@ b. Hence find the constants $b$ and $\varphi$ such that
y(t) &= \frac 1 a \sin(\omega t - \delta) \\ y(t) &= \frac 1 a \sin(\omega t - \delta) \\
> \end{align*} > \end{align*}
</details>
## Products of Complex Numbers ## Products of Complex Numbers
Suppose we have 2 complex numbers: Suppose we have 2 complex numbers:
@ -243,10 +255,14 @@ r^n(\cos\theta +i\sin\theta)^n &= r^n(\cos{n\theta} + i\sin{n\theta}) \\
(\cos\theta +i\sin\theta)^n &= (\cos{n\theta} + i\sin{n\theta}) (\cos\theta +i\sin\theta)^n &= (\cos{n\theta} + i\sin{n\theta})
\end{align*} \end{align*}
<details>
<summary>
### Example 1 ### Example 1
Write $1+i$ in polar form and use de Moivre's theorem to calculate $(1+i)^{15}$. Write $1+i$ in polar form and use de Moivre's theorem to calculate $(1+i)^{15}$.
</summary>
> \begin{align*} > \begin{align*}
r &= |1+i| = \sqrt2 \\ r &= |1+i| = \sqrt2 \\
\theta &= \arg{1+i} = \frac \pi 4 \\ \theta &= \arg{1+i} = \frac \pi 4 \\
@ -258,7 +274,10 @@ Write $1+i$ in polar form and use de Moivre's theorem to calculate $(1+i)^{15}$.
&= 2^7 (1 - i) \\ &= 2^7 (1 - i) \\
&= 128 - 128i &= 128 - 128i
> \end{align*} > \end{align*}
</details>
<details>
<summary>
### Example 2 ### Example 2
Use de Moivre's theorem to show that Use de Moivre's theorem to show that
@ -269,6 +288,8 @@ Use de Moivre's theorem to show that
\sin{2\theta} &= 2\sin\theta\cos\theta \sin{2\theta} &= 2\sin\theta\cos\theta
\end{align*} \end{align*}
</summary>
> Let $n=2$: > Let $n=2$:
> \begin{align*} > \begin{align*}
@ -277,11 +298,17 @@ Use de Moivre's theorem to show that
\text{Imaginary part: } 2\sin\theta\cos\theta &= \sin{2\theta} \text{Imaginary part: } 2\sin\theta\cos\theta &= \sin{2\theta}
> \end{align*} > \end{align*}
</details>
<details>
<summary>
### Example 3 ### Example 3
Given that $n \in \mathbb{N}$ and $\omega = -1 + i$, show that Given that $n \in \mathbb{N}$ and $\omega = -1 + i$, show that
$w^n + \bar{w}^n = 2^{\frac n 2 + 1}\cos{\frac{3n\pi} 4}$ with Euler's formula. $w^n + \bar{w}^n = 2^{\frac n 2 + 1}\cos{\frac{3n\pi} 4}$ with Euler's formula.
</summary>
> \begin{align*} > \begin{align*}
r &= \sqrt{2} \\ r &= \sqrt{2} \\
\arg \omega = \theta &= \frac 3 4 \pi \\ \arg \omega = \theta &= \frac 3 4 \pi \\
@ -292,14 +319,20 @@ $w^n + \bar{w}^n = 2^{\frac n 2 + 1}\cos{\frac{3n\pi} 4}$ with Euler's formula.
&= 2^{\frac n 2 + 1}\cos{\frac {3n\pi} 4} &= 2^{\frac n 2 + 1}\cos{\frac {3n\pi} 4}
> \end{align*} > \end{align*}
</details>
## Complex Roots of Polynomials ## Complex Roots of Polynomials
<details>
<summary>
### Example ### Example
Which complex numbers $z$ satisfy Find which complex numbers $z$ satisfy
$$z^3 = 8i$$ $$z^3 = 8i$$
</summary>
> 1. Write $8i$ in exponential form, > 1. Write $8i$ in exponential form,
> >
> $|8i| = 8$ and $\arg{8i} = \frac \pi 2$ > $|8i| = 8$ and $\arg{8i} = \frac \pi 2$
@ -342,3 +375,5 @@ $$z^3 = 8i$$
> Some of these complex roots may be real numbers. > Some of these complex roots may be real numbers.
> >
> 4. There are three solutions > 4. There are three solutions
</details>