Engineering Basics

Table of Contents

Engineering 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 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:

$$\omega_0 = \frac{2 \pi}{T}$$ $$a_0 = \frac{1}{T} \int_{-T/2}^{T/2} x \left( t \right) dt$$

$n$ is 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.

and for integer $n > 0$:

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

5 Fourier Transform

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

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