Mathematics, Physics

Kinematics

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*Note: for the purposes of this article, variables that represent a column vector may either be denoted using bolded letters (e.g., $\mathbf{a}$) or using over-head arrow notation (e.g., $\overrightarrow{a}$)

Linear Motion

 

Variables:

  • $\mathbf{x} =$  position vector (SI unit: $\textup{m}$)

$\Delta \mathbf{x} = \mathbf{x}_f – \mathbf{x}_i$

  • $\mathbf{v} =$ velocity vector (SI unit: $\textup{m/s}$)

$\mathbf{v} = \frac{\Delta \mathbf{x}}{\Delta t} = \mathbf{\dot{x}}$

  • $\mathbf{a} =$ acceleration vector (SI unit: $\textup{m} / \textup{s}^2 $)

$\mathbf{a} = \frac{\Delta \mathbf{v}}{\Delta t} =\mathbf{\dot{v}}= \mathbf{\ddot{x}}$

Kinematic formula (1D motion with constant acceleration):

$v(t)=v_0 + at$

$\Delta x (t) = (\frac{v(t)+v_0}{2})t$

$\Delta x (t) = v_0t + \frac{1}{2}at^2$

$v^2 (t) = v_0^2 + 2a \Delta x(t)$

Rotational Motion

Variables:

  • $\mathbf{\theta} =$  angle vector (SI unit: $\textup{rad}$)

$\Delta \mathbf{\theta} = \mathbf{\theta}_f – \mathbf{\theta}_i$

  • $\mathbf{\omega} =$  angular velocity vector (SI unit: $\textup{rad/s}$)

$\mathbf{\omega} = \frac{\Delta \mathbf{\theta}}{\Delta t} = \mathbf{\dot{\theta}}$

  • $\mathbf{\alpha} =$  angular acceleration vector (SI unit: $\textup{rad} / \textup{s}^2 $)

$\mathbf{\alpha} = \frac{\Delta \mathbf{\omega}}{\Delta t} =\mathbf{\dot{\omega}}= \mathbf{\ddot{\theta}}$

Kinematic formula (uniform circular motion)

$\omega (t)=\omega_0 + \alpha t$

$\Delta \theta (t)= (\frac{\omega (t)+\omega_0}{2})t$

$\Delta \theta (t) = \omega_0t + \frac{1}{2} \alpha t^2$

$\omega^2(t) = v_0^2 + 2\alpha \Delta \theta (t)$

Relations between linear and rotational variables

Let $\mathbf{x} = \mathbf{r} = x\mathbf{\hat{\imath}}+ y\mathbf{\hat{\jmath}}$, $r = \left|| \mathbf{r} \right||$, and $\omega = \left|| \overrightarrow{\omega} \right||$

$x = r \cos{\theta}$

$y = r \sin{\theta}$

$r^2 = x^2 + y^2$

$\theta = \tan^{-1}{(y/x)}$

$\mathbf{v} = \overrightarrow{\omega} \times \mathbf{r}$

$\mathbf{a} = \mathbf{\overrightarrow{\alpha}} \times \mathbf{r} – \omega^2 \mathbf{r} $

Sources

Mathematics

Vortex Based Mathematics

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Note: I have no idea whether this is a load of crap or not, but at the very least its a fun mathematical adventure for nerds. Just remember to take some of the more philosophical stuff with a grain of salt and do your own research before coming to any major conclusions.

Background

Marko Rodin = original founder of vortex-based mathematics

Randy Powell = continues the work of Rodin today

Relevant Operations

Modular arithmetic notation ( defined for the purposes of this article as follows)

  •  \oplus _k   ⇒ addition modulo k
    • Example: 6 \oplus _5 5 = (6+5)\mod{5} = 11 \mod{5} = 1
  • \ominus _k   ⇒  subtraction modulo k
    • Example: 4 \ominus _7 8 = (4-8) \mod{7}= -4 \mod{7} = 3
  • \otimes _k   ⇒  multiplication modulo k
    • Example: 3 \otimes _9 7 = (3*7) \mod{9}= 21 \mod{9} = 3
  • \oslash _k   ⇒  division modulo k
    • This operation is a little more complicated than the first three
    • Importantly, care must be taken when using modulo division – it isn’t necesarilly a unique or sufficient operation
      • e.g., 9 \oslash _9 x = 9     \forall       x
    • Steps used for calculating a \oslash _k b
      • Determine the lowest possible value m that results in a + (k \cdot m) obtaining a value which is a multiple of b
      • Plug in m into the following expression: a \oslash b = (a + (k \cdot m)) / b
    • Example: find 8 \oslash _6 5
      • Givens: a = 8, b = 5,  k = 6
      • 8 + 6(2) = 20 is the lowest multiple of 5 we can obtain using a + (k \cdot m))
        • Thus, m = 2
      • 8 \oslash _6 5 = (8 + 6(2)) / 5 = 20 / 5 = 4
  • For the purposes of this article, let’s assume we are working in modulo 9 (i.e., our normal 0,1,…,9 digit system)

Digital sum (or root) = an operation taken on a positive rational real number x \in \mathbb{Q} ^+ that returns an integer n = 1 ... k (we are assuming k = 9) which corresponds to repeatedly taking the sum of all digits in x repeatedly until obtaining a single digit

  • Examples:

x = 25679 \Rightarrow 2 + 5 + 6 + 7 + 9 = 29 \Rightarrow 2 + 9 = 11 \Rightarrow 1 + 1 = 2 \Rightarrow n = 2

x = 3.6793 \Rightarrow 3 + 6 + 7 + 9 + 3 = 28 \Rightarrow 2 + 8 = 10 \Rightarrow 1 + 0 = 1 \Rightarrow n = 1

Doubling = pick a starting number x_0, multiply it by two, take the digital sum of the result (x_1) and repeat the process until you get x_i = x_0

  • As it turns out, in mod 9, and starting at x_0 = 1, we get x_6 = x_0
    • Proof:
      • x_0 = 1 \Rightarrow 2 x_0 = 2(1) = 2 = x_1
      • x_1= 2 \Rightarrow 2 x_1 = 2(2) = 4 = x_2
      • x_2 = 4 \Rightarrow x_2 = 2(4) = 8 = x_3
      • x_3 = 8 \Rightarrow 2 x_3 = 2(8) = 16 \Rightarrow 1 + 6 = 7 = x_4
      • x_4 = 7 \Rightarrow 2 x_4 = 2(7) = 14 \Rightarrow 1 + 4 = 5 = x_5
      • x_5 = 5 \Rightarrow 2 x_4 = 2(5) = 10 \Rightarrow 1 + 0 = 1 = x_6 = x_0
    • Importantly, if we ignored the step where we take the digital sum, each x_{i} value will have a digital sum corresponding to the the value it would have obtained if we did include the digital sum step
    • Morover, this \begin{matrix} 1 & 2 & 4 & 8 & 7 & 5 & 1 & ... \end{matrix}    cycle appears as long as we choose an x_0 with a digital sum of either 1, 2, 4, 5, 7, or 8
      • Example: x_0 = 7
        • x_1 = 2(7) = 14 \Rightarrow 1 + 4 = 5
        • x_2 = 2(5) = 10 \Rightarrow 1 + 0 = 1
        • x_3 = 2(1) = 2
        • x_4 = 2(2) = 4
        • x_5 = 2(4) = 8
        • x_6 = 2(8) = 16 \Rightarrow 1 + 6 = 7
      • Here is an image depicting this pattern as a sequence of transitions between nodes/states: blue path
      • Note: we also get this pattern from taking the digital sum of sequential powers of two
        • Proof:
          • 2^0 = 1
          • 2^1 = 2
          • 2^2 = 4
          • 2^3 = 8
          • 2^4 = 16 \Rightarrow 1+6 = 7
          • 2^5 = 32 \Rightarrow 3 + 2 = 5
          • 2^6  = 64 \Rightarrow 6 + 4 = 10 \Rightarrow 1+0=1
  • However, in mod 9, and starting at x_0 = 3, 6, \mathrm{or} 9 or rational numbers whose digital sum is divisible by one of these numbers, we get different patterns
    • If we let x_0 = 3, we get…
      • x_1 = 2(3) = 6
      • x_2 = 2(6) = 12 \Rightarrow 1+2 = 3 = x_0
  • Thus, for any given x_0 \in \mathbb{Q}, every x_i = 2x_{i-1} can be mapped to a sigle digit using doubling

Diagram

vortex_mathematics

  • Blue nodes corresponds with “physical” energy states
  • Red nodes corresponds with “metaphysical” energy states

Sources

Mathematics

Multivariable Calculus

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Visualizing multivariable functions

Parametric surface = surfaces in 3D space that are actually “maps” of somje 2D space

  • Start with a 2D input that somehow “moves” into 3D
  • e.g., nraveling a cylinder into a rectangular plane

Vector fields = every point in the input space is associated with a vector in the output

  • Used for visualizing functions that have the same number of dimensions in their input as their output
  • Note: often, output vectors are scaled down to get a much better (and cleaner) feel for the output corresponding with each input point
    • Importantly, this is kind of like lying, so we should always indicate when we are scaling down our vectors
  • Useful for “fluid flow” interpretations

Contour maps = allows us to visualize 3D graphs in a 2D setting (associate each input point with a color that indicates the magnitude of the output)

  • Contour lines = lines that are used to denote constant output values on a contour map
    • Can “squish” contour lines together along an xy-plane to get a 2D representation of some (but not all) of our outputs
    • Each line represents a constant output of our function
  • Usually, “warmer” colors are used to represent higher output values and “cooler” colors are used to represent lower output values

Operators

Gradient = a way of packing all the partial derivative information of a function into a vector

  • Mathematically defined as an arbitrary n-dimensional column vector of all the first-ordered partial derivatives (n is equal to the number of inputs an arbitrary function takes)
  • Note: for any arbitrary multivariable function, the gradient is always the direction of steepest ascent
  • General definition of the gradient of a multivariable function:

\nabla f(x_1 , ..., x_n) = \begin{bmatrix} \frac { \partial f}{ \partial x_{ 1 } } \\ \vdots \\ \frac { \partial f}{ \partial x_{ n } } \end{bmatrix}

  • We can think of \nabla as an operator full of partial derivative operations:

\nabla = \begin{bmatrix} \frac { \partial }{ \partial x_{ 1 } } \\ \vdots \\ \frac { \partial }{ \partial x_{ n } } \end{bmatrix}

Directional derivative = tells us how a “nudge” in the direction of a particular vector \overrightarrow{v} changes the output of a multivariable function

  • General equation in 2D:
    • For \overrightarrow{v} = \begin{bmatrix} a \\ b \end{bmatrix},

\nabla _{\overrightarrow{v}} f = \overrightarrow{v} \cdot \nabla f = a \frac{\partial f}{\partial x} + b \frac{\partial f}{\partial y}

    • Normalized formula for the directional gradient:

\nabla _{\overrightarrow{v}} f = \frac{ \overrightarrow{v} \cdot \nabla f}{\left\| \overrightarrow{v} \right\| }

Laplacian = the divergence of the gradient of a MISO (multi-input, single-output) system

  • General equation: \Delta f(\overrightarrow{x}) = \nabla \cdot \nabla f(\overrightarrow{x})
  • Note: harmonic functions are defined as MISO systems that have \Delta f(\overrightarrow{x}) = 0 for all input vectors \overrightarrow{x}

Multivariable Chain Rule

Used to determine the derivative of a multivariable function composed of single-valued functions (that have the same input variable) with respect to that variable

Prototypical example: let $x(t)$ and $y(t)$ be inputs to a 2D multivariable function $f(x(t),y(t))$

  • The multivariable chain rule says that in general, the derivative of $f(x(t),y(t))$ (i.e., the composition of multiple single-valued functions) with respect to $t$ (i.e., the input variable for our single-valued functions) will be the sum of the products between the partial derivative of our MVF with respect to one of its inputs and the derivative of that input with respect to $t$:

$\frac{df}{dt}(x(t),y(t)) = \frac{\partial f}{\partial x} \frac{dx}{dt}+ \frac{\partial f}{\partial y} \frac{dy}{dt}$

Sources

  • Khan Academy – Multivariable Calculus
Mathematics

Math Laws, Identities, and Definitions (Pre-Calculus)

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Sets and Types of Numbers

  • Prime numbers = postive integer that are only divisible by themselves and 1
    • The lowest prime number is 2 (this is also the only even prime number)
    • Note: by convention, 1 is NOT considered a prime number
    • Cannot be factorized ⇒ prime numbers are considered the fundamental building blocks of positive integers
      • Important concept for coding
    • The Riemann hypothesis involves the distribution of prime numbers among the positive integers
  • Composite numbers = positive integers that are neither 1 nor a prime
    • Every composite number can be written as a unique product of prime factors
      • Examples:
        • 12 = 2^2 \times 3
        • 21 = 3 \times 7
        • 270 = 2 \times 3^3 \times 5
    • Currently, there is no genral algorithm that can determine the uniqe combination of prime factors for any given positive integer
      • This process is an ideal basis fo encryption systems

Trigonometry

Tangent Identities

  • \tan \theta = \frac{\sin \theta}{\cos \theta}
  • \cot \theta = \frac{\cos \theta}{\sin \theta}

Pythagorean Identities

  • \sin ^2 \theta  + \cos ^2 \theta = 1
  • \tan ^2 \theta + 1 = \sec ^2 \theta
  • \cot ^2 \theta + 1 = \csc ^2 \theta

Periodic Identities

  • \sin (\theta + 2 \pi n) = \sin \theta
  • \cos (\theta + 2 \pi n) = \cos \theta
  • \tan (\theta + \pi n) = \tan \theta
  • \csc (\theta + 2 \pi n) = \csc \theta
  • \sec (\theta + 2 \pi n) = \sec \theta
  • \cot (\theta + \pi n) = \cot \theta

Sources:

  • Glendinning, P. (2013). Math in minutes. New York, NY: Quercus.
Geomancy Fun, Mathematics

Cones and Conic Sections

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Relevant terminology

Generator = a linear line (green line) that intersects a vertical line called the axis at a fixed point (purple dot) called the vertex 

  • Note: the angle between the generator and the axis is called the vertex angle 
  • Rotating the generator 360° around the vertex and tracing out the rotation into a 3D surface gives a conecone generator

Double right circular cone = a cone whose bases (the top and bottom faces of the cone) are perpendicular to the vertical axis and form a perfect circle cone

  •  Directrix = 2D outline of the base
    • The directrix of a right circular cone is always a perfect circle
  • Nappe = lateral surface of a single circular cone
    • For the diagram above,
      • The upper nappe is outlined in pink
      • The lower nappe is outlined in blue

Overview of Conic Sections

Conic Sections 3D
Source Link
Picture
Source Link

conic section is the 2D curve corresponing to a particular 2D “slice” of a 3D cone

  • The angle at which the 2D plane slices the 3D cone determines the type of conic section we get

Parabolas

General equation:

y - k = a(x - h)^2

x - h = a(y - k)^2

Circles

General equation:

(x-h)^2 + (y-k)^2 = R^2

  • The center of the circle is defined as C:(h,k)
  • The radius of the cirlce is defined as R

Ellipses

General equation:

\frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2} = 1

Hyperbolas

General equation:

\frac{(x-h)^2}{a^2} - \frac{(y-k)^2}{b^2} = 1

Sources: