Engineering Math Basics

Table of Contents

Engineering Math Basics

1 Derivatives

1.1 Rules

Power rule: $$\frac{d}{dx}\left(x^a\right)=a\cdot x^{a-1}$$ Derivative of a constant: $$\frac{d}{dx}\left(a\right)=0$$ Sum difference rule: $$\left(f\pm g\right)'=f'\pm g'$$ Constant out: $$\left(a\cdot f\right)'=a\cdot f'$$ Product rule: $$\left(f\cdot g\right)'=f'\cdot g+f\cdot g'$$ Quotient rule: $$\left(\frac{f}{g}\right)'=\frac{f'\cdot g-g'\cdot f}{g^2}$$ Chain rule: $$\frac{df\left(u\right)}{dx}=\frac{df}{du}\cdot \frac{du}{dx}$$

1.2 Common

$$\frac{d}{dx}\left(\ln \left(x\right)\right)=x^{-1}$$ $$\frac{d}{dx}\left(\ln \left(\left|x\right|\right)\right)=x^{-1}$$ $$\frac{d}{dx}\left(e^x\right)=e^x$$ $$\frac{d}{dx}\left(\log _a\left(x\right)\right)=\frac{1}{x\ln \left(a\right)}$$ $$\frac{d}{dx}\left(\sin \left(x\right)\right)=\cos \left(x\right)$$ $$\frac{d}{dx}\left(\cos \left(x\right)\right)=-\sin \left(x\right)$$ $$\frac{d}{dx}\left(\tan \left(x\right)\right)=\sec ^2\left(x\right)$$

1.3 Trigonometric

$$\frac{d}{dx}\left(\sec \left(x\right)\right)=\frac{\tan \left(x\right)}{\cos \left(x\right)}$$ $$\frac{d}{dx}\left(\csc \left(x\right)\right)=\frac{-\cot \left(x\right)}{\sin \left(x\right)}$$ $$\frac{d}{dx}\left(\cot \left(x\right)\right)=-\frac{1}{\sin ^2\left(x\right)}$$

2 Integrals

2.1 Indefinite Integral Rules

Power rule: $$\int x^adx=\frac{x^{a+1}}{a+1}, \ \quad \ a\ne -1$$ Integration by parts: $$\int \ uv'=uv-\int \ u'v$$ Integral of a constant: $$\int f\left(a\right)dx=x\cdot f\left(a\right)$$ Take the constant out: $$\int a\cdot f\left(x\right)dx=a\cdot \int f\left(x\right)dx$$ Sum rule: $$\int f\left(x\right)\pm g\left(x\right)dx=\int f\left(x\right)dx\pm \int g\left(x\right)dx$$ Add a constant to the solution: $$\mathrm{If \ }\frac{dF\left(x\right)}{dx}=f\left(x\right)\mathrm{ \ then \ }\int f\left(x\right)dx=F\left(x\right)+C$$ Integral substitution: $$\int f\left(g\left(x\right)\right)\cdot g'\left(x\right)dx=\int f\left(u\right)du, \ \quad u=g\left(x\right)$$

2.2 Common

Assume definite integrals can be simply reversed from the previously listed derivatives chart. For example: $\int e^xdx=e^x$ and $\int x^{-1} dx=\ln \left(x\right)$.

$$\int |x|dx=\frac{x\sqrt{x^2}}{2}$$ $$\int \sin \left(x\right)dx=-\cos \left(x\right)$$ $$\int \cos \left(x\right)dx=\sin \left(x\right)$$

2.3 Trigonometric

$$\int \sec ^2\left(x\right)dx=\tan \left(x\right)$$ $$\int \csc ^2\left(x\right)dx=-\cot \left(x\right)$$ $$\int \frac{1}{\sin ^2\left(x\right)}dx=-\cot \left(x\right)$$ $$\int \frac{1}{\cos ^2\left(x\right)}dx=\tan \left(x\right)$$

3 Laplace Transform Pairs

$$X \left( s \right) = \int_{-\infty}^{\infty} x \left( t \right) e^{-st} dt$$

$$f \left( t \right)$$	$$F= \left( s \right)$$
$$n\delta \left( t \right)$$	$$n$$
$$u \left( t \right)$$	$$\frac{1}{s}$$
$$e^{-at}$$	$$\frac{1}{s+a}$$
$$t$$	$$\frac{1}{s^2}$$
$$t^n$$	$$\frac{n!}{s^{n+1}}$$
$$te^{-at}$$	$$\frac{1}{ \left( s+a \right) ^2}$$
$$t^ne^{-at}$$	$$\frac{n!}{ \left( s+a \right) ^{n+1}}$$
$$\sin \left( \omega t \right)$$	$$\frac{ \omega }{s^2+ \omega ^2}$$
$$\cos \left( \omega t \right)$$	$$\frac{s}{s^2+ \omega ^2}$$
$$\sin \left( \omega t + \theta \right)$$	$$\frac{s\sin\theta+ \omega \cos\theta}{s^2+ \omega ^2}$$
$$\cos \left( \omega t + \theta \right)$$	$$\frac{s\cos\theta- \omega \sin\theta}{s^2+ \omega ^2}$$
$$e^{-at} \sin \left( \omega t \right)$$	$$\frac{ \omega }{ \left( s+a \right) ^2+ \omega ^2}$$
$$e^{-at} \cos \left( \omega t \right)$$	$$\frac{s+a}{ \left( s+a \right) ^2+ \omega ^2}$$
$$\frac{ d^{n}f \left( t \right) }{dt^n}$$	$$s^nF \left( s \right) -s^{n-1}f \left( 0 \right) -s^{n-2} f' \left( 0 \right) -f^{ \left( n-1 \right) } \left( 0 \right)$$
$$\int_0^t f \left( t \right) dt$$	$$\frac{1}{s} F \left( s \right)$$

4 Fourier Analysis

4.1 Fourier Series

The fourier series represents periodic signals as a sum of sinusoids.

$$x \left( t \right) = A_0 + \sum_{n = 1}^{\infty} A_n \cos \left( n \omega_0 t \right) + B_n \sin \left( n \omega_0 t \right)$$

where:

$$\omega_0 = \frac{2 \pi}{T}$$ $$A_0 = \frac{1}{T} \int_{-T/2}^{T/2} x \left( t \right) dt$$

$n$: The harmonic number, starting from 1.
- Harmonics are integer multiples of the fundamental frequency ($n = 1$).
  - If the fundamental frequency is $f_0$, the first harmonic would be $2f_0$, the second harmonic $3f_0$, and so on.
$T$: The signal time period (time for 1 full cycle).

and for integer $n > 0$:

$$A_n = \frac{2}{T} \int_{-T/2}^{T/2} x \left( t \right) \cos \left( n \omega_0 t \right) dt$$ $$B_n = \frac{2}{T} \int_{-T/2}^{T/2} x \left( t \right) \sin \left( n \omega_0 t \right) dt$$

Amplitude or Fourier Magnitude is given by:

$$| C_n | = \sqrt{A_n^2 + B_n^2}$$

4.2 Fourier Transform

The Fourier transform represents a signal as a sum of sinusoid's of continuous frequencies, allowing for the analysis of non-periodic signals in the frequency domain.

$$X \left( j \omega \right) = \int_{-\infty}^{\infty} x \left( t \right) e^{-j \omega t} dt$$ $$x \left( t \right) = \frac{1}{2 \pi} \int_{-\infty}^{\infty} X \left( j \omega \right) e^{j \omega t} dt$$

4.2.1 Discrete Fourier Transform (DFT)

$$X \left( k \right) = \sum_{n = 0}^{N - 1} x \left( n \right) \cdot e^{-j \frac{2 \pi k n}{N}}$$ $$x \left( f \right)_{normalized, \space single \space ended} = \frac{2}{N} X \left( k \right)$$

where:

$N$: The total number of discrete samples.
$x \left( n \right)$: The signal sample value at index $n$.
$n$: The time-domain sample index.
- It runs from $0$ to $N-1$, covering all the $N$ samples in the input signal.
$k$: The frequency-domain index. Like $n$, it runs from $0$ to $N-1$.
- Each value of $k$ corresponds to a specific frequency in the transformed domain. The frequency bin is given by the sampling frequency $f_s$:
  - $f_k = \frac{k}{N} f_s$.
$e^{-i 2 \pi \frac{kr}{N}}$: The complex exponential term used to extract the frequency components.
- This is effectively creating Fourier series sines and cosines at various frequencies.
- See the section below for Euler's Formula.
$\frac{2}{N}$: A normalization factor ($2 \times \frac{1}{N}$) to adjust magnitude to the true magnitude at a given frequency bin.
- Due to the symetry at the Nyquist limit ($\frac{f_s}{2}$) a product of 2 ($\times 2$) is used to obtain the single ended result.
  - Following the Nyquist-Shannon sampling theorem, the sampling frequency ($f_{s}$) must be greater than or equal to twice the maximum signal frequency ($2 f_{signal}$) to prevent aliasing.
- To find the true magnitude of the frequency the result is divided by the sample count ($\times \frac{1}{N}$).
$\Delta t$: The time step, given by the following:
- $\Delta t = \frac{T}{N}$.

Euler's Formula:

$$e^{jx} = \cos x + j \sin x$$

This is used to find the frequency components, representing the sinusoidal representation at a given frequency.
- Here $x$ represents the larger complex exponential in the DFT.
Thus, the final result is a real number and imaginary number:

$$x(k) = A_k + jB_k$$

Magnitude and Phase angle of the final x(k) result is given by:

$$C_n = \sqrt{A_k^2 + B_k^2}$$ $$\theta = \tan ^{-1} \left( \frac{B_k}{A_k} \right)$$

For more information:

YouTube, Simon Xu: Discrete Fourier Transform - Simple Step by Step.
YouTube, MATLAB: Understanding the Discrete Fourier Transform and the FFT.