Micky natural science study notes

Eigenvalues and Eigenvectors: Motivation

• $A$ represents an $n \times n$ matrix.
• Linear transformation $T_A : \mathbb{R}^n \rightarrow \mathbb{R}^n$ is given by $T_A(x) = Ax$.
• An area of interest is if there are any special vectors $x$ where $Ax$ is a scalar multiple of $x$.
• Need to identify if there is nonzero $x$ where $Ax = \lambda x$ for a scalar $\lambda$.
• If such $x$ exists, $A$'s engagement on $x$ is scalar multiplication, making $x$ special.

Eigenvalues and Eigenvectors: Definitions

• An eigenvector of $A$ is a nonzero vector $x$ where $Ax = \lambda x$ for scalar $\lambda$.
• Scalar $\lambda$ is the eigenvalue of $A$ if there's a nontrivial solution $x$ of $Ax = \lambda x$.
• The nontrivial solution $x$ is an eigenvector corresponding to $\lambda$.
• $x = 0$ can never be an eigenvector.
• $\lambda = 0$ is allowed to be an eigenvalue.
• If $x$ is an eigenvector corresponding to $\lambda$, then $Ax = \lambda x$.

Example of Eigenvectors

• For $A = \begin{bmatrix} 1 & 6 \ 5 & 2 \end{bmatrix}$, it is to be determined if $u = \begin{bmatrix} 6 \ -5 \end{bmatrix}$ and $v = \begin{bmatrix} 1 \ 1 \end{bmatrix}$ are eigenvectors of $A$.
  • $Au = \begin{bmatrix} 1 & 6 \ 5 & 2 \end{bmatrix} \begin{bmatrix} 6 \ -5 \end{bmatrix} = \begin{bmatrix} -24 \ 20 \end{bmatrix} = -4 \begin{bmatrix} 6 \ -5 \end{bmatrix} = -4u$, thus $u$ is an eigenvector of $A$ with eigenvalue $-4$.
  • $Av = \begin{bmatrix} 1 & 6 \ 5 & 2 \end{bmatrix} \begin{bmatrix} 1 \ 1 \end{bmatrix} = \begin{bmatrix} 7 \ 7 \end{bmatrix} = 7 \begin{bmatrix} 1 \ 1 \end{bmatrix} = 7v$, thus $v$ is an eigenvector of $A$ with eigenvalue $7$.

Geometric Interpretation

• With $x$ as the eigenvector of $A$, $Ax$ is a scalar multiple of $x$.
• $Ax$ is on the same line as $x$.
• A maps $x$ to a vector on the same line as $x$.

Summary Guidelines

• A summary should convey the core of a text concisely.
• Summaries help readers grasp content quickly.

Considerations for Summaries

• Adhere to the original text.
• Include the main ideas.
• Restate in your own words.
• Be brief and succinct.
• Do not introduce new information, interpretations, or judgments.

Summary Structure

• Begin with an introduction stating the topic of the text.
• Include a main section highlighting principal points.
• Finish with a conclusion from the text.

Effective Language

• Maintain objectivity.
• Use short, simple sentences and precise wording.
• Write in the present tense and use connectors for smooth transitions.

Air Pollution Example: Original Text

• Air pollution: Is a serious environmental issue
• Caused by: vehicle emissions, industrial facilities, and heating systems.
• Negative effects: human health and the environment.
• Leads to: respiratory diseases and contributes to climate change and ozone depletion.
• Reducing Emissions: reduce vehicle consumption, add filters to industrial facilities, and improve heating systems.
• Renewable Energies: These must be promoted to lessen fossil fuel dependency.

Air Pollution Example: Summary

• The text addresses air pollution, caused by emissions from vehicles, industrial sites, and heating.
• Air pollution negatively impacts health and the environment by respiratory diseases, climate change, and ozone depletion.
• To combat this, emissions must be reduced through fuel-efficient vehicles, filters, and improved heating.
• Renewable energy sources should be advanced.

Graph Definition

• Denoted as $G = (V, E)$.
• $V$ is the set of vertices or nodes.
• $E$ is the set of edges or links connecting vertex pairs.
• Edges can have direction (directed) or lack direction (undirected).

Graph Example

• Vertices: A, B, C, D.
• Edges: {(A, B), (A, C), (A, D), (B, D)}.

Undirected Graphs

• Have edges that lack direction.
• Edges are two-way.
• Social networks depicting friendships are an example.
• If $(A, B) \in E$, then $(B, A) \in E$

Directed Graphs

• Edges have a direction.
• Edges are one-way.
• Twitter follower relationships are an example.
• If $(A, B) \in E$, it doesn't necessitate that $(B, A) \in E$

Adjacency Matrix

• This is a 2-dimensional array where rows and columns represent vertices.
• $A[i][j] = 1$ indicates an edge from vertex i to vertex j, while $A[i][j] = 0$ means there isn't.
• The space complexity is $O(|V|^2)$.
• This is suitable for dense graphs.

Adjacency List

• This incorporates an array of lists, with each representing a vertex's neighbors.
• The space complexity is $O(|V| + |E|)$.
• This is suitable for sparse graphs.

Vertex Degree

• The count of edges linked to a vertex.
• In directed graphs, it distinguishes between in-degree (incoming edges) and out-degree (outgoing edges).

Path Definition

• This is a sequence of vertices that are connected by edges.
• A simple path doesn't revisit vertices.

Cycle Definition

• This is a path that starts and stops at the same vertex.
• A simple cycle doesn't exhibit vertex or edge repetition (excluding the first and last vertex).

Connected Graph

• Every vertex pair has connecting path.
• In directed graphs, it can be strongly connected (pathways in both directions) or weakly connected (pathways in just one direction).

Breadth-First Search (BFS)

• Explore at level-by-level vertices.
• A queue to keep track of vertices that need to be visited.
• Identifies the shortest route within unweighted graphs.

Depth-First Search (DFS)

• Explores vertices along each branch as far as possible before going backward.
• A stack is often utilized for tracking which vertices need a visit..
• Utilized to uncover cycles.

Derivatives: Basic Formulas

• $(cf)' = cf'$, where $c$ is a constant and $f$ is a differentiable function
• $(f \pm g)' = f' \pm g'$, where $f$ and $g$ are differentiable functions
• $(fg)' = f'g + fg'$ (Product Rule)
• $(\frac{f}{g})' = \frac{f'g - fg'}{g^2}$ (Quotient Rule)
• $\frac{d}{dx}(c) = 0$
• $\frac{d}{dx}(x^n) = nx^{n-1}$ (Power Rule, $n$ is a constant)
• $\frac{d}{dx}(f(g(x))) = f'(g(x))g'(x)$ (Chain Rule)
• $\frac{d}{dx}(e^x) = e^x$
• $\frac{d}{dx}(a^x) = a^x\ln(a)$
• $\frac{d}{dx}\ln(x) = \frac{1}{x}, x>0$
• $\frac{d}{dx}\log_a(x) = \frac{1}{x\ln(a)}$

Derivatives: Trig Formulas

• $\frac{d}{dx}(\sin x) = \cos x$
• $\frac{d}{dx}(\cos x) = -\sin x$
• $\frac{d}{dx}(\tan x) = \sec^2 x$
• $\frac{d}{dx}(\sec x) = \sec x \tan x$
• $\frac{d}{dx}(\csc x) = -\csc x \cot x$
• $\frac{d}{dx}(\cot x) = -\csc^2 x$

Derivatives: Inverse Trig Formulas

• $\frac{d}{dx}(\sin^{-1} x) = \frac{1}{\sqrt{1-x^2}}$
• $\frac{d}{dx}(\cos^{-1} x) = -\frac{1}{\sqrt{1-x^2}}$
• $\frac{d}{dx}(\tan^{-1} x) = \frac{1}{1+x^2}$
• $\frac{d}{dx}(\sec^{-1} x) = \frac{1}{|x|\sqrt{x^2-1}}$
• $\frac{d}{dx}(\csc^{-1} x) = -\frac{1}{|x|\sqrt{x^2-1}}$
• $\frac{d}{dx}(\cot^{-1} x) = -\frac{1}{1+x^2}$

Integrals: Basic Formulas

• $\int cf(x) dx = c\int f(x) dx$, where $c$ is a constant.
• $\int f(x) \pm g(x) dx = \int f(x) dx \pm \int g(x) dx$
• $\int_a^b f(x) dx = F(b) - F(a)$ where $F'(x) = f(x)$
• $\int x^n dx = \frac{x^{n+1}}{n+1} + c, n \neq -1$
• $\int \frac{1}{x} dx = \ln |x| + c$
• $\int e^x dx = e^x + c$
• $\int a^x dx = \frac{a^x}{\ln(a)} + c$
• $\int f'(g(x))g'(x) dx = f(g(x)) + c$

Integrals: Trig Formulas

• $\int \cos x dx = \sin x + c$
• $\int \sin x dx = -\cos x + c$
• $\int \sec^2 x dx = \tan x + c$
• $\int \sec x \tan x dx = \sec x + c$
• $\int \csc x \cot x dx = -\csc x + c$
• $\int \csc^2 x dx = -\cot x + c$

Integrals: Inverse Trig Formulas

• $\int \frac{1}{\sqrt{a^2 - x^2}} dx = \sin^{-1}(\frac{x}{a}) + c$
• $\int \frac{1}{a^2 + x^2} dx = \frac{1}{a}\tan^{-1}(\frac{x}{a}) + c$
• $\int \frac{1}{x\sqrt{x^2 - a^2}} dx = \frac{1}{a}\sec^{-1}(\frac{x}{a}) + c$

Standard Integration Techniques

• $\int f(g(x))g'(x) dx = \int f(u) du$
• $\int u dv = uv - \int v du$
• $\int_a^b f(x) dx = -\int_b^a f(x) dx$
• $\int_a^b f(x) dx = \int_a^c f(x) dx + \int_c^b f(x) dx$

Useful Formulas: Series

• $\sum_{i=0}^n i = \frac{n(n+1)}{2}$
• $\sum_{i=0}^n i^2 = \frac{n(n+1)(2n+1)}{6}$
• $\sum_{i=0}^n i^3 = (\frac{n(n+1)}{2})^2$
• $\sin(2x) = 2\sin(x)\cos(x)$
• $\cos(2x) = \cos^2(x) - \sin^2(x)$
• $\cos^2(x) = \frac{1}{2}(1 + \cos(2x))$
• $\sin^2(x) = \frac{1}{2}(1 - \cos(2x))$
• $e^x = \sum_{n=0}^{\infty} \frac{x^n}{n!}$
• $\sin(x) = \sum_{n=0}^{\infty} (-1)^n \frac{x^{2n+1}}{(2n+1)!}$
• $\cos(x) = \sum_{n=0}^{\infty} (-1)^n \frac{x^{2n}}{(2n)!}$
• $\left| x \right| < 1, \frac{1}{1-x} = \sum_{n=0}^{\infty} x^n$

Useful Formulas: Average Function Value

• $f_{avg} = \frac{1}{b-a} \int_a^b f(x) dx$

Useful Formulas: Arc Length

• $L = \int_a^b \sqrt{1 + [f'(x)]^2} dx$

Surface Area

• Rotate about x-axis: $A = \int_{a}^{b} 2\pi y \sqrt{1 + (\frac{dy}{dx})^2} dx$
• Rotate about y-axis: $A = \int_{a}^{b} 2\pi x \sqrt{1 + (\frac{dy}{dx})^2} dx$

Volume

• $V = \int_{a}^{b} A(x) dx$
• Rotation about the x-axis: $V = \int_{a}^{b} \pi [f(x)]^2 dx$
• Rotation about the x-axis: $V = \int_{a}^{b} \pi ([f(x)]^2 - [g(x)]^2) dx$
• Rotation about the y-axis: $V = \int_{a}^{b} 2\pi x f(x) dx$

Work

• $W = \int_{a}^{b} F(x) dx$

Center of Mass

• $\overline{x} = \frac{\int x \delta dA}{\int \delta dA}$
• $\overline{y} = \frac{\int y \delta dA}{\int \delta dA}$

Area

• $A = \int_{a}^{b} [f(x) - g(x)] dx$
• $A = \int_{a}^{b} r dr d\theta$

Polar Coordinates

• $x = r\cos(\theta)$
• $y = r\sin(\theta)$
• $\tan(\theta) = \frac{y}{x}$
• $x^2 + y^2 = r^2$

Explanation of Eigenvalues and Eigenvectors in Linear Algebra. Includes the motivation and definitions of eigenvalues and eigenvectors. Also provides an example to better understand the concept.