Statics: Force Vectors

Statics: Force Vectors

• Statics is a branch of mechanics that deals with bodies at rest under the action of forces.

Scalars and Vectors

• A scalar is a physical quantity specified by magnitude (positive or negative).
• Length, area, volume, mass, density, temperature, time, and energy are examples of scalars.
• A vector is a physical quantity requiring both magnitude and direction.
• Position, displacement, velocity, acceleration, force, and moment are examples of vectors.

Vector Operations

Multiplication and Division of a Vector by a Scalar

• Multiplying a vector by a positive scalar increases its magnitude.
• Multiplying a vector by a negative scalar reverses its direction.

Vector Addition

• Vector quantities obey the parallelogram law of addition.
• The resultant vector R is formed by adding "component" vectors A and B.
• Join tails of component vectors A and B at a point.
• Draw a line from the head of B parallel to A, and vice versa, forming a parallelogram.
• The resultant R extends from the tails of A and B to the opposite intersection point P.

Finding a Resultant Force

Parallelogram Law

• Parallelogram law is used to find the resultant of two forces
• Solve Graphically

Trigonometry

• If two forces F1 and F2 are added, the magnitude of resultant force FR is: $F_R = \sqrt{F_1^2 + F_2^2 - 2F_1F_2\cos(\theta)}$
• The direction $\theta$ of the resultant force can be determined using the law of sines: $\frac{F_1}{\sin(\theta)} = \frac{F_2}{\sin(\theta)} = \frac{F_R}{\sin(180 - \alpha)}$

Addition of Several Forces

• Successive application of the parallelogram law is performed to find the resultant force when adding more than two forces.

Cartesian Vectors

Right-Handed Coordinate System

• A right-handed coordinate system is used for vector algebra.
• Point index finger in positive x-axis direction, curl fingers towards the positive y-axis, then the thumb points in the positive z-axis direction.

Rectangular Components of a Vector

• A vector A can have one, two, or three rectangular components (Ax, Ay, Az) along the x, y, z axes.
• Using the parallelogram law, A = A' + Az and A' = Ax + Ay.
• Combining these equations, A = Ax + Ay + Az.

Cartesian Unit Vectors

• The Cartesian unit vectors (i, j, k) are dimensionless vectors with a magnitude of 1, directed along the x, y, z axes, respectively.
• The i, j, and k vectors are orthogonal to each other.

Representation of a Cartesian Vector

• A = $A_x$\textbf{i} + $A_y$\textbf{j} + $A_z$\textbf{k}, where $A_x$, $A_y$, and $A_z$ are the magnitudes of A's components.

Magnitude of a Cartesian Vector

• The magnitude of A is $A = \sqrt{A_x^2 + A_y^2 + A_z^2}$

Direction of a Cartesian Vector

• Direction of A is defined by coordinate direction angles $\alpha$, $\beta$, and $\gamma$, measured from the tail of A to the positive x, y, and z axes.
• Cosines of $\alpha$, $\beta$, and $\gamma$ are: $cos(\alpha) = \frac{A_x}{A}, cos(\beta) = \frac{A_y}{A}, cos(\gamma) = \frac{A_z}{A}$
• Also, $cos^2(\alpha) + cos^2(\beta) + cos^2(\gamma) = 1$
• Coordinate direction angles are: $\alpha = cos^{-1}(\frac{A_x}{A}), \beta = cos^{-1}(\frac{A_y}{A}), \gamma = cos^{-1}(\frac{A_z}{A})$
• Directional Cosines are useful for finding coordinate direction angles, and defining force orientation on cables or rods.

Addition of Cartesian Vectors

• Resultant force is the vector sum of its components.
• $F_R = F_1 + F_2$
• $F_R = F_{1x}$\textbf{i} + $F_{1y}$\textbf{j} + $F_{1z}$\textbf{k} + $F_{2x}$\textbf{i} + $F_{2y}$\textbf{j} + $F_{2z}$\textbf{k}
• $F_R = (F_{1x} + F_{2x})$\textbf{i} + $(F_{1y} + F_{2y})$\textbf{j} + $(F_{1z} + F_{2z})$\textbf{k}
• In general, for n forces $F_R = \sum F_x$\textbf{i} + $\sum F_y$\textbf{j} + $\sum F_z$\textbf{k}

Position Vectors

Position Vector

• A position vector r locates a point in space relative to another.
• r = x\textbf{i} + y\textbf{j} + z\textbf{k}

Force Vector Directed Along a Line

• The direction of a 3D force is specified by two points on its line of action.
• F can be formulated as a Cartesian vector with the same direction as the position vector r.
• Direction is the unit vector u = $\frac{\textbf{r}}{r}$.
• F = F\textbf{u} = $F\frac{\textbf{r}}{r}$

Dot Product

Dot Product

• The dot product of vectors A and B is: A $\cdot$ B = A B cos($\theta$)
• A and B represent magnitudes of A and B
• $\theta$ is the angle between the tails of A and B

Laws of Operation

• Commutative law: A $\cdot$ B = B $\cdot$ A
• Multiplication by a scalar: a (A $\cdot$ B) = (aA) $\cdot$ B = A $\cdot$ (aB) = (A $\cdot$ B) a
• Distributive law: A $\cdot$ (B + C) = (A $\cdot$ B) + (A $\cdot$ C)

Cartesian Vector Formulation

• If A = $A_x$\textbf{i} + $A_y$\textbf{j} + $A_z$\textbf{k} and B = $B_x$\textbf{i} + $B_y$\textbf{j} + $B_z$\textbf{k}, then A $\cdot$ B = $A_x$$B_x$ + $A_y$$B_y$ + $A_z$$B_z$

Applications

• Angle between two vectors: $\theta$ = $cos^{-1}$ $\frac{\textbf{A} \cdot \textbf{B}}{AB}$
• Component of a vector in a specified direction: $A_{a}$ = A cos($\theta$) = A $\cdot$ $\textbf{u}$, where $A_{a}$ is the magnitude of the component of A in the direction of a

Introduction to Probability

The probability of an event indicates its likelihood of occurring, and is quantified between 0 and 1.

• "0" represents impossibility, while "1" means certainty.

Definition

• The probability of event E, denoted P(E), is calculated as: $P(E) = \frac{M}{N} = \frac{\text{Number of favorable outcomes}}{\text{Total number of possible outcomes}}$, where N is the total number of equally likely outcomes and M is the number of outcomes in which E occurs.

Example

• Tossing a fair coin provides two possible outcomes: heads (H) or tails (T), each with equal likelihood.
• $P(H) = \frac{1}{2}$

Basic Concepts

Sample Space

• The sample space is the set of all possible outcomes of an experiment, denoted by $S$.
• A six-sided die $S = {1, 2, 3, 4, 5, 6}$

Event

• An event is a subset of the sample space.
• Rolling an even number on a die is the event $E = {2, 4, 6}$.

Types of Events

• Simple Event: An event consisting of only one outcome.
• Compound Event: An event consisting of more than one outcome.

Basic Rules of Probability

Rule 1: Probability Values

• The probability of any event must be between 0 and 1: $0 \leq P(E) \leq 1$

Rule 2: Sum of Probabilities

• The sum of probabilities must equal 1: $\sum P(E_i) = 1$

Rule 3: Complement Rule

• The complement of an event $E'$, consists of all outcomes in sample space that are not in E: $P(E') = 1 - P(E)$

Rule 4: Addition Rule

• For any two events Probability is: $P(A \cup B) = P(A) + P(B) - P(A \cap B)$
• If A and B are mutually exclusive (disjoint), then $P(A \cap B) = 0$
• The rule simplifies to: $P(A \cup B) = P(A) + P(B)$

Rule 5: Multiplication Rule

• For any events the probability is: $P(A \cap B) = P(A) \cdot P(B|A)$ where $P(B|A)$ is the conditional probability of B given that A has occurred.
• If A and B independent the equation is: $P(A \cap B) = P(A) \cdot P(B)$

Conditional Probability

• $P(A|B) = \frac{P(A \cap B)}{P(B)}$, provided $P(B) > 0$

Independence

• Two events $A$ and $B$ are independent if the occurrence of one does not affect the probability of the other mathematically.
• $P(A|B) = P(A)$ or $P(B|A) = P(B)$
• $P(A \cap B) = P(A) \cdot P(B)$

Examples

Example 1: Drawing Cards

• A card is drawn from a standard deck of 52 cards. What is the probability that the card is a heart?
• $P(\text{Heart}) = \frac{\text{Number of hearts}}{\text{Total number of cards}} = \frac{13}{52} = \frac{1}{4}$

Example 2: Rolling a Die

• What is the probability of rolling a 3 or a 4 on a six-sided die?
• $P(3 \cup 4) = P(3) + P(4) = \frac{1}{6} + \frac{1}{6} = \frac{2}{6} = \frac{1}{3}$

Example 3: Tossing Two Coins

• Two fair coins are tossed. What is the probability of getting at least one head?
• Event $E$ (at least one head) = ${HH, HT, TH}$
• $P(E) = \frac{3}{4}$

Lecture 24: Introduction to Gradient Descent

Motivation

• The goal is to minimize a differentiable function, $\min_x f(x)$, where $f: \mathbb{R}^d \rightarrow \mathbb{R}$.

Idea

• Start with an initial guess $x_0$, and iteratively improve the guess: $x_{t+1} = x_t + \Delta x_t$

Key Question

• How to choose $\Delta x_t$? Want $f(x_{t+1}) < f(x_t)$

Taylor's Theorem

• Taylor's Theorem provides insight; $f(x + \Delta x) = f(x) + \langle \nabla f(x), \Delta x \rangle + O(||\Delta x||^2)$
• If $||\Delta x||$ is small, then $f(x + \Delta x) \approx f(x) + \langle \nabla f(x), \Delta x \rangle$
• So, $f(x + \Delta x) < f(x)$ if $\langle \nabla f(x), \Delta x \rangle < 0$

Gradient Descent

• Gradient Descent approach: Choose $\Delta x_t = -\eta \nabla f(x_t)$ for some $\eta > 0$ (called the learning rate or step size)
• Update rule: $x_{t+1} = x_t - \eta \nabla f(x_t)$

A Simple Example

• $f(x) = x^2$, $x \in \mathbb{R}$, and $\nabla f(x) = 2x$.
• $x_{t+1} = x_t - \eta \cdot 2x_t = (1 - 2\eta)x_t$

Cases

• If $0 < \eta < 1$, then $|1 - 2\eta| < 1$, so $x_t \rightarrow 0$ as $t \rightarrow \infty$.
• If $\eta = 1$, then $x_{t+1} = -x_t$, so oscillates between $x_0$ and $-x_0$.
• If $\eta > 1$, then $|1 - 2\eta| > 1$, so $|x_t| \rightarrow \infty$ as $t \rightarrow \infty$.

Takeaway

• If the learning rate $\eta$ is too large, gradient descent can diverge.
• If $\eta$ is too small, gradient descent can be slow.

Convergence Analysis

• Set up: $f: \mathbb{R}^d \rightarrow \mathbb{R}$ is a convex function, $| \nabla f(x) | \le G$ for all $x$ (G is a constant), $x_t = x_{t-1} - \eta \nabla f(x_{t-1})$
• If $f$ is convex after $T$ steps of gradient descent we have: and $| \nabla f(x) | \le G$ then the convergence rate is: $f(\bar{x}_T) - \min_x f(x) \le \frac{\eta G^2}{2} + \frac{|x_0 - x|^2}{2\eta T}$ where $\bar{x}T = \frac{1}{T} \sum{t=1}^T x_t$

Choosing $\eta$

• The optimal learning rate is $\eta = \frac{R}{G\sqrt{T}}$; plugging this in results in $f(\bar{x}_T) - \min_x f(x) \le \frac{RG}{\sqrt{T}}$.

Strongly Convex Case

• Assumption : is is $\mu$-strongly convex if $f(y) \ge f(x) + \langle \nabla f(x), y-x \rangle + \frac{\mu}{2} |y-x|^2$ for all $x, y$. If $f$ is $\mu$-strongly convex and we choose $\eta = \frac{1}{\mu}$, convergence rate:
• $|x_t - x^|^2 \le \left(1 - \frac{\mu^2}{L}\right)^t |x_0 - x^|^2$, where $L$ is the smoothness parameter of $f$.

Takeaway

• Strongly convex functions converge faster than convex functions, and the convergence rate depends on the condition number $\frac{L}{\mu}$.

Guía de inicio rápido de Geometry Pad (Geometry Pad Quick Start Guide)

This is a quick start guide for the Geometry Pad application. Geometry Pad is an application for dynamic geometry on tablets. It allows users to create geometric constructs and explore the properties by moving elements with a finger.

Creating a point

There are three different ways to create a point:

• Touch an empty place in the canvas.
• Touch a line to create a point on the line.
• Touch the intersecion of two lines, circles, etc.

Selecting Objects

• Before acting on an object, it must first be selected. To select an object, tap it. The selected object will then be highlighted. To select multiple objects, tap them one after the other.

Moving an Object

• To move an object it must first be selected. It can be moeve by dragging it to the new placement.

Creación de otros objetos (Creating other objects)

• To create other objects, a tool must first be selected from the toolbar. Then, follow the instructions that are displayed on the screen. For example, to create a line you must select the line tool, and then tap two points.

Cambiando el color de un objeto (Changing the color of an object)

• To change an object's color, it must first be selected. Next, tap the color button on the toolbar. From there, a color menu will appear allowing the selecting of colors.

Eliminando un objeto (Eliminating an Object)

• To eliminate an object, it must first be selected. Next tap the eliminator tool on the toolbar.

Deshacer/Rehacer (Undo/Redo)

• To undo the last action, tap the deshacer button on the toolbar. T redo. To redo the last action, tap the rehacer button on the toolbar.

Guardar/Abrir (Save/Open)

• To save a construction, tap the guarder button on the tool bar. To open a saved construction tap the open on the toolbar.

Analytische Geometrie (Analytical Geometry)

This document covers Vectors, Lines, Planes, and Distance Calculations in Analytical Geometry.

1. Vektoren (Vectors)

Definition

• A Vector $\vec{v}$ is an object with direction and length, and is represented as an arrow.

Darstellung (Representation)

• Koordinatendarstellung(Coordinate Representation): $\vec{v} = \begin{pmatrix} v_1 \ v_2 \ v_3 \end{pmatrix}$
• Ortsvektor (Location Vector): Vector from the origin to a point P. $\vec{OP} = \vec{p} = \begin{pmatrix} p_1 \ p_2 \ p_3 \end{pmatrix}$
• Verbindungsvektor (Connecting Vector): Vector between two points A and B. $\vec{AB} = \vec{b} - \vec{a} = \begin{pmatrix} b_1 - a_1 \ b_2 - a_2 \ b_3 - a_3 \end{pmatrix}$

Rechnen mit Vektoren (Calculating with Vectors)

• Addition/Subtraktion (Addition/Subtraction): $\vec{a} \pm \vec{b} = \begin{pmatrix} a_1 \pm b_1 \ a_2 \pm b_2 \ a_3 \pm b_3 \end{pmatrix}$
• Skalarmultiplikation (Scalar multiplication): $\lambda \cdot \vec{a} = \begin{pmatrix} \lambda a_1 \ \lambda a_2 \ \lambda a_3 \end{pmatrix}$
• Betrag (Länge) (Magnitude/Length): $|\vec{a}| = \sqrt{a_1^2 + a_2^2 + a_3^2}$
• Skalarprodukt (Scalar Product) : $\vec{a} \cdot \vec{b} = a_1b_1 + a_2b_2 + a_3b_3 = |\vec{a}| \cdot |\vec{b}| \cdot \cos(\alpha)$
• $\vec{a} \cdot \vec{b} = 0 \Leftrightarrow \vec{a} \perp \vec{b}$
• Kreuzprodukt (Vektorprodukt) (Cross Product/Vector Product): $\vec{a} \times \vec{b} = \begin{pmatrix} a_2b_3 - a_3b_2 \ a_3b_1 - a_1b_3 \ a_1b_2 - a_2b_1 \end{pmatrix}$
• $|\vec{a} \times \vec{b}|$ = Area of the parallelogram spanned by $\vec{a}$ and $\vec{b}$
• $\vec{a} \times \vec{b} \perp \vec{a}, \vec{b}$

2. Geraden (Lines)

Parameterform (Parametric Form)

• $\vec{x} = \vec{p} + \lambda \cdot \vec{v}$, where:
• $\vec{x}$: Location vector of an arbitrary point on the line
• $\vec{p}$: Location vector of a known point on the line
• $\vec{v}$: direction vector of the line
• $\lambda$: paramter ($\lambda \in \mathbb{R}$)

Lagebeziehungen (Positional Relationships)

• Identisch (Identical): Direction vectors are collinear and a point on one line lies on the other
• Parallel: Direction vectors are collinear, but not point on one lines lie on the other
• Schneidend (Intersecting): The lines have an intersection point.
• Windschief (Skew): The lines are neither parallel nor do they intersect.

Schnittpunkt (Intersection Point)

• Equate the line equations and solve the linear equation system.

3. Ebenen (Planes)

Parameterform (Parametric Form)

• $\vec{x} = \vec{p} + \lambda \cdot \vec{v} + \mu \cdot \vec{w}$, where:
• $\vec{x}$: Location vector of an arbitrary point on the plane
• $\vec{p}$: Location vector of a known point on the plane
• $\vec{v}, \vec{w}$: Direction vectors of the plane) linearly independant
• $\lambda, \mu$: paramenters ($\lambda, \mu \in \mathbb{R}$)

Koordinatenform (Normalenform) Coordinate Form (Normal Form)

• $ax_1 + bx_2 + cx_3 = d$ or $\vec{n} \cdot \vec{x} = d$, where:
• $\vec{n} = \begin{pmatrix} a \ b \ c \end{pmatrix}$: Normal vector of the plane.
• $d$: Distance of the plane from the origin when $|\vec{n}|$=1

Umwandlung Parameterform $\rightarrow$ Koordinatenform: Conversion of Parameter to Coordinate form.

1. Calculate the normal vector: $\vec{n} = \vec{v} \times \vec{w}$
2. Insert a point of the plane int the coordinate form to determine d, : $d=\vec{n} \vec{p}$

Lagebeziehungen zwischen Ebenen (Positional Relationships between Planes)

• Identisch (Identical): Normal vectors are collinear and the point of one plane lies on the other.
• Parallel: Normal vectors are collinear, but no point of one plane lies on the other.
• Schneidend (intersecting): The planes intersect in a straght line.

Lagebeziehungen zwischen Gerade und Ebene (positional Relationships between Lines and Planes)

• Liegt in der Ebene (Lies in the plane: All the points of the line lie in the plane.
• Parallel: Direction vector of the line is orthogonal to the normal vector of the plane, and the line has no point in the plane.
• Schneidend (intersecting): The line intersects the plane at one point.

Schnittpunkt Gerade und Ebene (Intersection Point Line and Plane)

• Substitute the line equation into the plane equation and solve for the parameter.

4. Abstandsberechnungen (Distance Calculations)

• Punkt zu Punkt (point to point) $d(A, B) = |\vec{AB}| = |\vec{b} - \vec{a}|$
• Punkt zu Gerade (point to Line) $d(P, g) = \frac{|\vec{AP} \times \vec{v}|}{|\vec{v}|}$, where A is a point on the line.
• Punkt zu Ebene (point to Plane) = $d(P, E) = \frac{|\vec{n} \cdot (\vec{p} - \vec{a})|}{|\vec{n}|}$, where A is a point on the plane.
• Gerade zu Gerade (windschief) (Line to Line Skew) : = $d(g, h) = \frac{|(\vec{a} - \vec{b}) \cdot (\vec{v} \times \vec{w})|}{|\vec{v} \times \vec{w}|}$, where A is on g and B is on h.
• Gerade zu Ebene (parallel)(Line to Plane Parallel) = $d(g, E) = d(P, E)$, where P is a point on the line g.
• Ebene zu Ebene (paralel)(Plane to Plane Parallel) = $d(E_1, E_2) = d(P, E_2)$ , where $P$ is a point on $E_1$.

Chemical Principles

Properties of Gases

• Gases fill any container, are highly compressible, and form homogeneous mixtures.

Pressure

• Pressure is the force per unit area exerted by gas molecules striking surfaces.
• $Pressure = \frac{Force}{Area}$
• SI unit: pascal (Pa), $1 Pa = 1 \frac{N}{m^2}$
• A barometer measures atmospheric pressure, while a manometer measures the pressure of a gas other than the atmosphere.

Standard pressure

Normal atmospheric pressure at sea level is defined as:

• $1 atm = 760 mm Hg = 760 torr = 1.01325 \times 10^5 Pa = 101.325 kPa$

The Gas Laws

Boyle's Law

• At constant temperature, the volume of a fixed quantity of gas is inversely proportional to the pressure.
• $V \propto \frac{1}{P}$
• $P_1V_1 = P_2V_2$

Charles's Law

• At constant pressure, the volume of a fixed quantity of gas is directly proportional to the absolute temperature.
• $V \propto T$
• $\frac{V_1}{T_1} = \frac{V_2}{T_2}$

Avogadro's Law

• At a given temperature and pressure, the volume of a gas is directly proportional to the number of moles of gas.
• $V \propto n$
• $\frac{V_1}{n_1} = \frac{V_2}{n_2}$

The Ideal Gas Equation

• $PV = nRT$, where R is the ideal gas constant $R = 0.08206 \frac{L \cdot atm}{mol \cdot K} = 8.314 \frac{J}{mol \cdot K}$

Gas Densities and Molar Mass

• $d = \frac{m}{V}$
• $d = \frac{PM}{RT}$
• Where:
  • d is the density
  • M is the molar mass
• $M = \frac{dRT}{P}$

Dalton's Law of Partial Pressures

• The total pressure of a gas mixture is the sum of each gas's individual pressure.
• $P_T = P_1 + P_2 + P_3 +...$
• $P_1 = X_1P_T$
• Where:
  • $P_1$ is the partial pressure of component 1, $X_1$ is its mole fraction, and $P_T$ is the total pressure.

The Kinetic-Molecular Theory of Gases

• Gases consist of many molecules in constant, random motion.
• The combined volume of molecules is negligible compared to the container volume.
• Attractive and repulsive forces between gas molecules are negligible.
• Energy can transfer between molecules during collisions, but the average kinetic energy stays constant at constant temperature.
• Average kinetic energy of molecules is proportional to absolute temperature.

Molecular Speed

KE = $\frac{1}{2}mv^2$, $u_{rms} = \sqrt{\frac{3RT}{M}}$

• Where : is the root-mean-square speed and M is the molar mass.

Diffusion and Effusion

• Diffusion is the spread of one substance throughout a space or throughout a second substance.

• Effusion is the escape of gas molecules through a tiny hole into an evacuated space.

• $\frac{r_1}{r_2} = \sqrt{\frac{M_2}{M_1}}$

Where:

• $r_1$ and $r_2$ are the rates of effusion of two different gases
• $M_1$ and $M_2$ are the molar masses of the gases.

Real Gases: Deviations from Ideal Behavior

Real gases deviate from ideal behavior at high pressure and low temperature.

The van der Waals Equation

• $(P + \frac{an^2}{V^2})(V - nb) = nRT$
• a and b are van der Waals constants, unique for each gas, quantifying molecular attraction and volume.

Anatomy and Neurobiology

Introduction

Covers the anatomy of the brain and nervous system, an overview of neurobiology.

Brain Anatomy

Major Structures

• Cerebrum: The largest part of the brain. Responsible for higher-level cognitive functions such as language, memory, and reasoning.
• Cerebellum: Located at the back of the brain. Coordinates movement and balance.
• Brainstem: Connects the brain to the spinal cord. Controls basic life functions such as breathing and heart rate.

Lobes of the Cerebrum

• Frontal Lobe: Involved in planning, decision-making, and motor control.
• Parietal Lobe: Processes sensory information such as touch, temperature, and pain.
• Temporal Lobe: Involved in auditory processing, memory, and language.
• Occipital Lobe: Processes visual information.

Neurons

• Structure of a Neuron:
  • Cell Body (Soma): Contains the nucleus and other organelles.
  • Dendrites: Branch-like extensions that receive signals from other neurons.
  • Axon: A long, slender projection that transmits signals to other neurons, muscles, or glands.
  • Myelin Sheath: A fatty substance that insulates the axon and speeds up signal transmission.
  • Nodes of Ranvier: Gaps in the myelin sheath where the axon is exposed.
  • Axon Terminals: The end of the axon, where signals are transmitted to other cells.

Types of Neurons

• Sensory Neurons: Carry information from sensory receptors to the central nervous system.
• Motor Neurons: Carry information from the central nervous system to muscles and glands.
• Interneurons: Connect sensory and motor neurons within the central nervous system.

Synaptic Transmission

• Synapse: The junction between two neurons, where signals are transmitted.
• Neurotransmitters: Chemical messengers that transmit signals across the synapse.
• Receptors: Proteins on the receiving neuron that bind to neurotransmitters.

Steps of Synaptic Transmission

• An action potential reaches the axon terminal of the presynaptic neuron. Calcium channels open, and calcium ions enter the axon terminal. Calcium entry triggers the release of neurotransmitters from vesicles. Neurotransmitters diffuse across the synaptic cleft and bind to receptors on the postsynaptic neuron. The binding of neurotransmitters to receptors causes a change in the postsynaptic neuron, such as an opening of ion channels. The neurotransmitter is either degraded by enzymes, taken back up into the presynaptic neuron, or diffuses away from the synapse.

Glial Cells

• Astrocytes: Provide support and nutrients to neurons, regulate the chemical environment, and form the blood-brain barrier.
• Oligodendrocytes: Form the myelin sheath in the central nervous system.
• Schwann Cells: Form the myelin sheath in the peripheral nervous system.
• Microglia: Act as immune cells in the brain, removing debris and pathogens.

Neuroplasticity

• Definition: The ability of the brain to change and reorganize itself in response to experience or injury.
  • Mechanisms:
    • Synaptic Plasticity: Changes in the strength of synaptic connections.
    • Neurogenesis: The formation of new neurons.

Neurotransmitters and Their Functions

• Acetylcholine - muscle contraction, memory, and attention
• Dopamine - pleasure, motivation, motor control
• Serotonin - mood regulation, sleep, appetite
• Norepinephrine - Alertness, arousal, stress response GABA - Inhibitory neurotransmitter, reduces neural excitability Glutamate - Excitatory neurotransmitter, involved in learning and memory Endorphins - Pain relief, pleasure

Common Neurological Disorders

Disorders and descriptions

Alzheimer's Disease - Progressive neurodegenerative disorder that leads to memory loss, cognitive decline, and impaired daily functioning Parkinson's Disease - Neurodegenerative disorder that affects motor control, leading to tremors, rigidity, and slow movement Multiple Sclerosis - Autoimmune disorder that affects the brain and spinal cord, causing a range of symptoms including muscle weakness Epilepsy - Neurological disorder characterized by recurrent seizures Stroke - Occurs when blood supply to the brain is interrupted, leading to brain damage

Research Methods in Neurobiology

• fMRI (functional Magnetic Resonance Imaging): Measures brain activity by detecting changes in blood flow. EEG (Electroencephalography): Measures electrical activity in the brain using electrodes placed on the scalp.
• TMS (Transcranial Magnetic Stimulation): Uses magnetic fields to stimulate or inhibit brain activity.
• Lesion Studies: Examine the effects of damage to specific brain regions on behavior and cognition.
• Genetic Studies: Investigate the role of genes in neurological disorders and brain function.

Quantum Physics

• Quantum physics studies atoms and subatomic particles that dictate universe function.

History

• 1900: Max Planck's quantization theory says energy comes in packets.
• 1905: Einstein's photoelectric effect shows light as photons (packets).
• 1924: Louis de Broglie: Electrons can act like waves.
• 1927: Heisenberg's Uncertainty Principle formulated.
• 1927: Solvay Conference gathers scientists to understand findings.
• 1935: Schrödinger's cat thought experiment shows problems.

Heisenberg's Uncertainty Principle

• States more accurately a particle’s position is known, less accurately its momentum can be known, and vice versa.
• It is impossible to perfectly know both properties.
• Uncertainty in position * Uncertainty in momentum >= $\frac{h}{4\pi}$, ($h$ is Planck's Constant)
• Quantum effects aren’t noticed in daily life because the uncertainty is so small.

Wave-particle Duality

• Shows light and matter exhibit wave and particle features.
• The Double-Slit Experiment is used to demonstrate this.
• Observed patterns depend on the experiment performed.
• Electrons behave as particles; they go through one slit/another.
• Interference pattern from electrons behaving like waves.
• This suggests each electron interacts with itself during observation.

Quantum Entanglement

• Links two or more particles to share the same fate no matter how far apart.
• Particles lack definite properties until measured.
• Measuring the spin on one particle instantly reveals the opposite for its pair, regardless of distance.
• This principle is being adopted in Quantum Computing and Cryptography Einstein called this "spooky action at a distance".