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

Stød
Stød se også Bevægelsesmængde (Impuls). Uelastisk stød Kuglerne klistrer sig sammen efter støddet - René Hastighederne lægges sammen efter stødet. Altså bevæger objekterne sammen. $$m_1 \cdot \vec{v_{1_i}} + m_2 \cdot \vec{v_{2_i}} = (m_1 + m_2) \cdot \vec{v_f} = M \cdot \vec{v_f}$$ $\vec{v_{1_i/2_i}}$ : Hastighederne af objekterne *før* kollisionen. $\vec{v_{1_f/2_f}}$ : Hastighederne af objekterne *efter* kollisionen. $\vec{v_f}$ : Hastigheden af det samlede objekt efter. $M$ : Massen af det samlede objekt.
0001/01/01 · Notes
Sum of Products
Sum of Products (SOP) Create circuit from truth table For large tables use Karnaugh Map. Select all rows with outputs that are supposed to be one (red). $$y = \bar x_{1} . x_{2} . \bar x_{3} + x_{1} . \bar x_{2} . \bar x_{3} + x_{1} . \bar x_{2} . x_{3}$$ We can now simplify the equation if needed.
0001/01/01 · Notes
Surface Integrals
Surface Integrals $$\iint_{S}f(x,y,z) ,\text{dS}$$ Video Evaluation Example Parameterization $$f(x,y,z)$$ $$\vec{r} = [x(u,v),\ y(u,v),\ z(u,v)]$$ $$ \text{dS} = |\vec{r_{u}} \times \vec{r_{v}}|\ \text{du}\ \text{dv}, \ \text{where} \begin{cases} \vec{r_{u}} = \left[\frac{\partial x}{\partial u},\ \frac{\partial y}{\partial u},\ \frac{\partial z}{\partial u}\right] \ \vec{r_{v}} = \left[\frac{\partial x}{\partial v},\ \frac{\partial y}{\partial v},\ \frac{\partial z}{\partial v}\right] \ \end{cases} $$ $$ \Rightarrow \iint_{S} f(\vec{r}) \cdot |\vec{r_{u}} \times \vec{r_{v}}|\ \text{du}\ \text{dv} $$
0001/01/01 · Notes
Tangent
Tangent En tangent er en linje, der går gennem et enkelt punkt på grafen, hvor tangenten har samme hældning. En tangents hældning kan beregnes med $f'(x)$, hvor $x$ er $x$-koordinaten for tangentens berøringspunkt. Se også Linære Funktioner Sekant
0001/01/01 · Notes
Tangent Plane
Tangent Plane $$\vec n = \vec T_{1} \cdot \vec T_{2} = \left( \begin{matrix} i & j & k \ 0 & 1 & f_{2}(a,b) \ 1 & 0 & f_{1}(a,b) \end{matrix} \right) = f_{1}(a,b)i + f_{2}(a,b)j - k $$ Finding the Normal Vector Find a normal vector and equation of the tangent plane and normal line to the graph. $$z = f(x,y) = \sin (xy) \s \text{at} , P\left(\frac{\pi}{3}, -1\right)$$ Calculate $z$ $$z = \sin\left(\frac{-\pi}{3}\right) = \frac{-1}{2}$$ Partial derivatives $$f_{1} = \frac{\partial z}{\partial x} = \frac{\partial}{{\partial x}} \sin(xy) = \cos(xy) \cdot y \cdot 1$$ $$f_{2} = \frac{\partial z}{\partial y} = \frac{\partial}{{\partial y}} \sin(xy) = \cos(xy) \cdot x \cdot 1$$ Calculate $\vec n$ $$ >\vec n = f_{1}(a,b)i + f_{2}(a,b)j - k = \cos\left(\frac{\pi}{3} \cdot (-1)\right) \cdot (-1) + \cos\left(\frac{\pi}{3} \cdot (-1)\right) \cdot \frac{\pi}{3} - k >$$
0001/01/01 · Notes
Tangentplan
Tangentplan og Normaler I en funktion med to variable kan vi ikke tegne en Tangent- linje, men i stedet et tangentplan. Dette tangentplan viser alle de mulige tangenter i et givent punkt. Dette er tangentplanet til punktet $(a,b,f(a,b))$ (minder om Linarisering). $$z = f(a,b) + f'{a}(a,b)(x-a) + f'{b}(a,b)(y-b)$$ Normalvektor til tangentplanet $$\vec{n} = \vt{f'{x}(x,y)}{f'{y}(x,y)}{-1}$$ Normallinjen $$\frac{x-a}{f'{a}(a,b)} = \frac{y-b}{f'{b}(a,b)} = \frac{z-f(a,b)}{-1}$$ Begge ligheder skal være sande.
0001/01/01 · Notes
Taylorpolynomium
Taylorpolynomium $n$‘te grads taylorpolynomium udvikles om $x = a$. $$P_{n}(x) = f(a) + \frac{f’(a)}{1!}(x-a)^{1}+ \frac{f''(a)}{2!}(x-a)^{2}+ \dots + \frac{f^{(n)}(a)}{n!}(x-a)^n$$ Linarisering benytter et førstegrads taylorpolynomium. Hvis man tager nok led med kan man i nogle tilfælde finde et præcist udtryk for funktionen. Fejlvurdering (Taylors Sætning) Maksimal fejl $$|E(x)| \leq \frac{|f^{(n+1)}(s_{maks})|}{(n+1)!}(x-a)^{n+1}$$ $s_{maks}$ : Den $s$-værdi, der giver den største fejl. Fortegn for fejl Check fortegnet på $|f^{(n+1)}(s)| \cdot (x-a)^{n+1}$. Dette er fortegnet på fejlen.
0001/01/01 · Notes
TCP-IP
TCP/IP Example of the logic inside a single router $L$: Data Link Address (mac addresses), each router has a local and a public address. $N$: IP of the sender and final receiver of a packet. Transport Layer See slides. Segment = Packet, in TCP segnemt format connection Data Link Layer See slides. Sublayers Data Link Control (DLC) : this layer takes care of all the issues common to both point-to-point and broadcast links.
0001/01/01 · Notes
Terminalhastighed
Terminalhastighed $$v_t = \sqrt{\frac{mg}{\rho AC_D}},\s \text{sandt når } F_D = F_g$$ $v_t$ : Terminalhastigheden. $m$ : Massen af objektet. $g$ : Tyngdeaccelerationen. $\rho$ : Densiteten af luften. $A$ : Frontarealet af objektet. $C_D$ : Drag koefficient. $F_D$ : Luftmodstandskraften. $F_g$ : Tyngdekraften.
0001/01/01 · Notes
The C++ Preprocessor
The C++ Processor The first part of a c++-file. All commands that interact with the processor are preceded by a ‘#’. Directives Guards Control flow in the preprocessor. These are mostly used in header files. Tip If you run into errors like: “you have declared this twice”, you probably need to guard its declaration. #ifndef MY_VARIABLE #define MY_VARIABLE #include <import-one>#include <import-two>#include <import-three>#include <import-four> decrarations... #endif
0001/01/01 · Notes