Balder W. Holst

Hello there!

My name is Balder and this is my personal website. Here i write down what i learn to maybe help others some day. For now i will just act like people are reading my articles, and make them for fun.

This website has three main sections:

I write these articles for fun, and even though i try to make sure that they are correct, i am not a perfect human and there are sure to be mistakes. Below is a list of most recent articles in all categories.

Recent Posts

Differentialregning
Differntialregning Noter list from #differentialer sort file.name
0001/01/01 · Notes
Differentialregning - Basis
Differentialregning - Basis Definition af et Differentiale Vi skal finde hældningen af sekanten, der går gennem de to punkter. Til dette kan vi bruge topunktsformlen. $$a=\frac{y_2-y_1}{x_2-x_{1}}= \frac{\Delta y}{\Delta x}$$ Vi kan nu sættespunkterne $(x_0,f(x_0))$ og $(x_0+h,f(x_0+h))$ ind i formlen $$a=\frac{f(x_0+h)-f(x_0)}{\bcancel{x_0}+h-\bcancel{x_0}} \arrows a=\frac{f(x_0+h)-f(x_0)}{h}$$ For at finde hældningen i det første punkt $(x_0,f(x_0))$, skal vi minimere størrelsen af $h$. Det gør vi med en grænseværdi. Derfor er $f'(x)$ (den differentierede funktoin) defineret på denne måde:
0001/01/01 · Notes
Differentialregning Regneregler
Differntialregning Regneregler page=25#page=25" Kvotientreglen $$\frac{d}{dx} \frac{f(x)}{g(x)} = \frac{f'(x) \cdot g(x) - f(x) \cdot g'(x)}{g(x)^2}$$ Flere Trigonometriregler $$\arcsin'(x) = \frac{1}{\sqrt{1-x^2}}$$ $$\arccos'(x)=-\frac{1}{\sqrt{1-x^2}}$$ $$\arctan'(x)=\frac{1}{1+x^2}$$
0001/01/01 · Notes
Differentiation af funktioner med to variable
Differentiation af Funktioner af flere Variable / Partielle Afledte Lad $f(x,y)$ være en funktion, så er førsteordens partielle afledte: $$ \begin{align} \frac{\partial f}{\partial x} &= f_{1}(x,y) = \frac{f(x+h_{1} \cdot y)-f(x,y)}{h}\ \frac{\partial f}{\partial x} &= f_{2}(x,y) = \frac{f(x+h_{2} \cdot y)-f(x,y)}{h} \end{align} $$ Partiels afledt: en funktion der er differentieret enten med hensyn til $x$ eller $y$. Praksis $$ \begin{align} \frac{\partial f}{\partial x} &\s \to \s \begin{cases} \text{Opfat $x$ som \emph{variabel}} \ \text{Opfat $y$ som \textbf{konstant}} \end{cases} \s \to \s \text{Differentier i forhold til $x$} \ \ \frac{\partial f}{\partial y} &\s \to \s \begin{cases} \text{Opfat $x$ som \textbf{konstant}} \ \text{Opfat $y$ som \emph{variabel}} \end{cases} \s \to \s \text{Differentier i forhold til $y$} \end{align} $$ Skrivemåde Differentieret med hensyn til $x$: $f_x'(x,y)$ (hældningen i $x$-retningen given $x$ og $y$)
0001/01/01 · Notes
Digital Realisation Structures
Digital Realisation Structures A representation of a digital filter that could be implemented in a programming language. See slides. $$ \begin{align} H(z) &= \frac{Y(z)}{X(z)} = \frac{ \sum_{i=0}^{N} a_{i}z^{-i} }{ 1 + \sum_{i=1}^{N} b_{i} z^{-i} } \ \ y(n) &= \sum_{i=0}^{N} a_{i} \cdot x(n-i) - \sum_{i=1}^{N}b_{i} \cdot y(n-1) \end{align} $$ $n$: time step $x(n)$: input in the discrete time domain $y(n)$: output in discrete time domain Implementing a Transfer Function Seperate $X(z)$ and $Y(z)$ terms the their own sides of the equation.
0001/01/01 · Notes
Dioder
Dioder Halvleder $$I_{D} = I_{S} \left(e^{\frac{V_{D}}{n \cdot V_{T}}} -1\right) \s \text{hvor} \s V_{T} = \frac{k \cdot T}{q}$$ $I_{D}$ : Strømmen gennem dioden $I_{S}$ : Geometrisk Konstant $k$ : Bolzmanns Konstant $T$ : Temperaturen i *kelvin* $q$ : Elementarladningen
0001/01/01 · Notes
Dirac Delta Function
Dirac Delta Function A signal with infinite magnitude over an infinitely small time span. Its integral is equal to $1$. All signals can be described as a weighted average of Dirac Delta functions. $$ y(t) = \int_{-\infty}^{\infty}u(\tau) h(t, \tau), d\tau $$ For Linear Systems this also applies: $$ y(t) = \int_{-\infty}^{\infty}u(t - \tau) h(\tau), d\tau $$
0001/01/01 · Notes
Diskret Fourier Transformation
Diskret Fourier Transformation See slides. Skalleringen ($1/N$) kan være i begge formler, bare den kun er i én af dem. $$X(m) := \frac{1}{N} \sum_{n=0}^{N-1}x(nT) W_{N}^{mn}$$ $$x(n) = \sum_{m=0}^{N-1}X(m) W_{n}^{-mn}$$ $$W_{N}= e^{-j2\pi /N}$$ $N$: Samples pr. period ($\frac{1}{N} = FT$) $T$: Time between samples **The distance between samples in the frequency spectrum is $f_{s}/N$.** Therefore we can get a higher resolution by zero-padding, which just means adding a bunch of zeros to the end of the signal in the time domain before transforming it.
0001/01/01 · Notes
Divergence
Divergence Denotes the ratio of input flow and output flow at point in a vector field. $$\mathbf{div}\ \vec{F}(x,y) = \nabla \bullet \vec{F}(x,y) = \frac{\partial \vec{F}_1}{\partial x} + \frac{\partial \vec{F}_2}{\partial y}$$ $$ \begin{cases} \mathbf{div}\ \vec{F}(x_{0}, y_{0}) > 0 &\Rightarrow &\text{more outflow} \ \mathbf{div}\ \vec{F}(x_{0}, y_{0}) = 0 &\Rightarrow &\text{equal infow and outflow} \ \mathbf{div}\ \vec{F}(x_{0}, y_{0}) < 0 &\Rightarrow &\text{more inflow} \end{cases} $$ Overview Nabla notation
0001/01/01 · Notes
Dæmpede Svingninger - Differentialligning
Dæmpede Svingninger $$a \cdot y'' + b \cdot y' + c \cdot y = 0 \arrows (a \cdot r^2 + br + c) \cdot e^{rt} = 0$$ Udledning Udledning $$a \cdot y'' + b \cdot y' + c \cdot y = 0$$ Gæt på løsning $$\begin{align} >y(t) & = e^{rt} \ >y'(t) & = r \cdot e^{rt} \ >y''(t) & = r^2 \cdot e^{rt} >\end{align}$$ Sætter ind $$a \cdot r^2 \cdot e ^{rt} + b \cdot r \cdot e^{rt} + c \cdot e^{rt} = 0$$
0001/01/01 · Notes