Basic Engineering Mathematics (Term-Work)

flowchart TD
A[Start: What is a Determinant?] --> B{Key Idea: A Special Scalar Value<br>Calculated from a Square Matrix};
B --> C{Primary Interpretations};
C --> C1["Geometric: Scaling Factor<br>Area/Volume change by<br>linear transformation"];
C --> C2["Algebraic: Invertibility Test<br>detA = 0 → Matrix is singular<br>detA ≠ 0 → Matrix is invertible"];
C1 & C2 --> D[Methods of Computation];
subgraph D[Computation Methods by Matrix Size]
    D1["1x1 Matrix<br>det[a] = a"]
    D2["2x2 Matrix<br>det⎡a b⎤ = ad - bc<br>    ⎣c d⎦"]
    D3["3x3 Matrix (Sarrus' Rule)<br>+aei + bfg + cdh - ceg - bdi - afh"]
    D4["nxn Matrix (Laplace Expansion)<br>Choose row/column, sum of<br>a_ij * -1ⁱ⁺ʲ * detM_ij<br>M_ij = Minor matrix"]
end
D --> E[Important Properties];
subgraph E[Key Properties & Rules]
    E1["detI = 1"]
    E2["detAᵀ = detA"]
    E3["detAB = detA * detB"]
    E4["Swapping a row → flips sign"]
    E5["Proportional rows → det = 0"]
end
E --> F{Applications & Uses};
F --> G1["Solving Systems of Equations<br>Cramer's Rule"];
F --> G2["Finding Matrix Inverse<br>A⁻¹ = adjA / detA"];
F --> G3["Calculus: Jacobian Determinant<br>For change of variables in<br>multiple integrals"];
F --> G4["Geometry: Calculating Area/<br>Volume of parallelograms/<br>parallelepipeds"];
F --> G5["Eigenvalue Problems<br>Characteristic equation: detA - λI = 0"];
G1 & G2 & G3 & G4 & G5 --> H[End: Determinant as a Fundamental<br>Tool in Linear Algebra];

flowchart TD
A[Start: Properties of Determinants] --> B["detA is a scalar value<br>calculated from a square matrix A"];
B --> C{Core Properties};
subgraph C[Fundamental Rules]
    C1["detI = 1<br>Determinant of Identity matrix"]
    C2["detAᵀ = detA<br>Transpose doesn't change determinant"]
    C3["Triangular Matrix Diagonal Included<br>detA = product of diagonal elements"]
end
C --> D{Effect of Row/Column Operations};
subgraph D[Elementary Operations]
    D1["Row/Column Switching<br>Swaps sign of determinant: det → -det"]
    D2["Row/Column Scaling by k<br>Scales determinant by k: det → k⋅det"]
    D3["Row/Column Addition<br>Adding a multiple of one row to another<br>Does NOT change determinant: det unchanged"]
end
D --> E{Multiplicative Property & Invertibility};
subgraph E[Product and Inversion]
    E1["detAB = detA ⋅ detB<br>Determinant of product equals product of determinants"]
    E2["detA⁻¹ = 1 / detA<br>if A is invertible"]
    E3["detAⁿ = detAⁿ"]
end
E1 --> F{"If detA = 0?"};
F -- Yes --> G["Matrix A is Singular<br>Not Invertible"];
F -- No --> H["Matrix A is Nonsingular<br>Invertible"];
G --> I["Rows/Columns are Linearly Dependent"];
H --> J["Rows/Columns are Linearly Independent"];
D & E --> K{Special Cases & Other Properties};
subgraph K[Additional Important Properties]
    K1["Proportional Rows/Columns<br>If two rows/columns are proportional, then detA = 0"]
    K2["Zero Row/Column<br>If a row or column is all zeros, then detA = 0"]
    K3["Linearity Property<br>det is linear in each row/column separately"]
    K4["Block Triangular Matrix<br>det = detblock₁₁ ⋅ detblock₂₂ ⋅ ... ⋅ detblockₙₙ"]
    K5["Scalar Multiplication of Matrix<br>detkA = kⁿ ⋅ detA<br>for an n×n matrix A"]
end
K --> L{Applications of Properties};
subgraph L[Uses of These Properties]
    L1["Simplify Determinant Calculation<br>Use row operations to create zeros<br> or triangular form"]
    L2["Proving Matrix Invertibility"]
    L3["Solving Systems of Equations Cramer's Rule"]
    L4["Finding Eigenvalues Solve detA - λI = 0"]
    L5["Calculating Area/Volume Scale Factor<br> Absolute value of det gives area/volume change"]
end
L --> M[End: Properties Simplify Computation and Theory];

flowchart TD
A[Start: Cramer's Rule] --> B["A method to solve a system of<br>n linear equations with n variables"];
B --> C{Prerequisites};
subgraph C[Conditions for Application]
    C1["System must be Square<br>Number of equations = Number of unknowns"]
    C2["Coefficient Matrix A must be Invertible<br>i.e., detA ≠ 0"]
end
C --> D["Given System: A⋅X = B<br>Where:
<br>A = n×n Coefficient Matrix
<br>X = n×1 Variable Matrix x₁, x₂,...,xₙ
<br>B = n×1 Constant Matrix"];
D --> E["Calculate Determinant of A, denoted detA"];
E --> F{Is detA = 0?};
F -- No, detA ≠ 0 --> G["Proceed with Cramer's Rule"];
F -- Yes, detA = 0 --> H["Stop. Cramer's Rule fails.<br>The system has either no solution or infinitely many solutions."];
G --> I["For each variable xᵢ"];
I --> J["Form Matrix Aᵢ by replacing<br>the i-th column of A with the constant matrix B"];
J --> K["Calculate determinant of Aᵢ, denoted detAᵢ"];
K --> L["Solution for xᵢ = detAᵢ / detA"];
L --> M["Repeat for i = 1 to n<br>to find all variables"];
M --> N["Solution Vector X = x₁, x₂, ..., xₙ"];
N --> O{Advantages & Disadvantages};
subgraph O[Analysis]
    O1["Advantages:<br>- Straightforward formula<br>- Useful for theoretical purposes<br>- Good for small systems n=2,3"]
    O2["Disadvantages:<br>- Computationally inefficient for n>3<br>- Requires detA ≠ 0<br>- Not practical for large systems"]
end
O --> P[End: A Elegant but Limited Tool];

flowchart TD
A[Start: What is a Matrix?] --> B["A rectangular array of numbers<br> arranged in rows and columns."]
B --> C{Primary Classification:<br> By Dimensions & Structure}
subgraph C[Types of Matrices]
    direction TB
    C1["Row Matrix<br>1 x n"]
    C2["Column Matrix<br>m x 1"]
    C3["Square Matrix<br>n x n"]
    C4["Rectangular Matrix<br>m x n, m≠n"]
    C5["Diagonal Matrix"]
    C6["Identity Matrix I"]
    C7["Zero / Null Matrix"]
    C8["Symmetric Matrix<br>A = Aᵀ"]
    C9["Singular Matrix<br>det = 0"]
end
C --> D[Matrix Algebra & Operations]
subgraph D[Key Operations]
    D1[Addition & Subtraction<br>Element-wise, requires same dimension]
    D2[Scalar Multiplication<br>Multiply every element by a scalar]
    D3[Matrix Multiplication<br>Rows of A × Columns of B<br>Requires: colsA = rowsB]
    D4[Transpose Aᵀ<br>Flip rows and columns]
    D5["Inverse A⁻¹<br>Exists only for square,<br>non-singular detA ≠ 0 matrices"]
    D6["Determinant detA or A<br>Scalar value for square matrices only"]
end
D --> E{Key Properties of Operations}
subgraph E[Properties]
    E1["Associative: A+B+C = A+B+C<br>AB C = A BC"]
    E2["Commutative: A+B = B+A<br>But AB ≠ BA generally"]
    E3["Distributive: A B+C = AB + AC"]
    E4["Transpose of Product: ABᵀ = BᵀAᵀ"]
end
E --> F{Applications & Real-World Uses}
F --> G1["Computer Graphics:<br>3D Transformations Rotation, Scaling, Translation"]
F --> G2["Data Science:<br>Dataset Representation, Linear Regression, PCA"]
F --> G3["Physics & Engineering:<br>Solving Systems of Linear Equations, Circuit Analysis"]
F --> G4["Economics & Operations Research:<br>Input-Output Models, Linear Programming"]
F --> G5["Computer Science:<br>Graph Theory Adjacency Matrix, Search Algorithms"]
F --> G6["Cryptography:<br>Encoding and Decoding Messages"]
G1 & G2 & G3 & G4 & G5 & G6 --> H[End: Matrices as the Language of Linear Algebra]

flowchart TD
A[Start: Matrix] --> B["A rectangular array of numbers,<br> symbols, or expressions arranged in rows and columns"];
B --> C{Classification of Matrices};
subgraph C[Based on Dimensions & Structure]
    C1["Row Matrix 1 x n"]
    C2["Column Matrix m x 1"]
    C3["Square Matrix n x n"]
    C4["Rectangular Matrix m x n m ≠ n"]
    C5["Zero Matrix All elements are 0"]
    C6["Diagonal Matrix<br>Square matrix with<br> non-zero elements only on main diagonal"]
    C7["Identity Matrix I<br>Diagonal matrix with 1's on diagonal"]
    C8["Symmetric Matrix A = Aᵀ"]
    C9["Skew-Symmetric Matrix A = -Aᵀ"]
    C10["Triangular Matrix Upper & Lower"]
    C11["Orthogonal Matrix Aᵀ = A⁻¹"]
    C12["Singular Matrix detA = 0"]
    C13["Non-Singular Matrix detA ≠ 0"]
end
C --> D{Matrix Algebra};
subgraph D[Operations on Matrices]
    D1["Addition & Subtraction<br>Element-wise, requires same dimensions"]
    D2["Scalar Multiplication<br>Multiply every element by a scalar"]
    D3["Matrix Multiplication<br>#Columns in A = #Rows in B<br> to form AB"]
    D4["Transpose Aᵀ or A'<br>Flip rows and columns"]
    D5["Inverse A⁻¹<br>Exists only for square, non-singular matrices<br> A A⁻¹ = I"]
    D6["Determinant detA or A<br>Scalar value for square matrices only"]
    D7["Trace Sum of diagonal elements"]
end
D --> E{Properties of Operations};
subgraph E[Key Algebraic Properties]
    E1["Addition: Commutative A+B=B+A, Associative"]
    E2["Multiplication: Associative, Distributive, NOT Commutative"]
    E3["Transpose: ABᵀ = BᵀAᵀ, A+Bᵀ = Aᵀ+Bᵀ"]
    E4["Inverse: AB⁻¹ = B⁻¹A⁻¹, Aᵀ⁻¹ = A⁻¹ᵀ"]
    E5["Determinant: detAB = detA detB, detAᵀ = detA"]
end
E --> F{Special Matrix Operations};
subgraph F[Advanced Concepts]
    F1["Matrix Decomposition<br>LU, QR, Eigen, SVD decompositions"]
    F2["Rank of Matrix Maximum number of<br> linearly independent rows/columns"]
    F3["Eigenvalues and Eigenvectors<br>Ax = λx"]
    F4["Matrix Calculus<br>Derivatives with respect to vectors/matrices"]
end
F --> G{Applications of Matrices};
subgraph G[Real-World Applications]
    G1["Computer Graphics<br>Transformations rotation, scaling, translation"]
    G2["Physics & Engineering<br>Solving systems of linear equations, quantum mechanics"]
    G3["Economics & Operations Research<br>Input-output models, linear programming"]
    G4["Data Science & Statistics<br>Datasets, covariance matrices, machine learning"]
    G5["Computer Science<br>Graph theory adjacency matrices, search algorithms"]
    G6["Cryptography<br>Encoding and decoding messages"]
end
G --> H[End: Matrices as Fundamental Building Blocks];

flowchart TD
A[Start: Matrix Properties] --> B["A matrix is a rectangular array<br>of numbers, symbols, or expressions"];
B --> C{Classification by Dimensions & Structure};
subgraph C[Types of Matrices]
    C1["Row & Column Matrix"]
    C2["Square & Rectangular Matrix"]
    C3["Diagonal & Identity Matrix I"]
    C4["Zero or Null Matrix"]
    C5["Symmetric A = Aᵀ & Skew-Symmetric A = -Aᵀ"]
    C6["Triangular Upper & Lower"]
    C7["Orthogonal AᵀA = I"]
    C8["Singular det=0 & Nonsingular det≠0"]
end
C --> D{Matrix Algebra Properties};
subgraph D[Operation-Specific Properties]
    D1["Addition/Subtraction<br>- Commutative: A+B = B+A<br>- Associative: A+B+C = A+B+C<br>- Additive Identity: A + 0 = A<br>- Additive Inverse: A + -A = 0"]
    D2["Scalar Multiplication<br>- Associative: c d A = c d A<br>- Distributive: c A+B = cA + cB<br>-
c+d A = cA + dA"]
    D3["Matrix Multiplication<br>- NOT Commutative: AB ≠ BA general<br>- Associative: AB C = A BC<br>- Distributive: A B+C = AB + AC<br>-
B+C A = BA + CA<br>- Multiplicative Identity: AI = IA = A"]
    D4["Transpose<br>- Aᵀᵀ = A<br>- A+Bᵀ = Aᵀ + Bᵀ<br>- cAᵀ = cAᵀ<br>- ABᵀ = BᵀAᵀ crucial"]
    D5["Inverse exists if detA ≠ 0<br>- A A⁻¹ = A⁻¹A = I<br>- AB⁻¹ = B⁻¹A⁻¹<br>- Aᵀ⁻¹ = A⁻¹ᵀ<br>- Aⁿ⁻¹ = A⁻¹ⁿ"]
    D6["Trace Sum of diagonal elements<br>- trA+B = trA + trB<br>- trcA = c trA<br>- trAB = trBA"]
end
D --> E{Special Properties & Theorems};
subgraph E[Key Results]
    E1["Determinant Properties<br>- detI = 1<br>- detAᵀ = detA<br>- detAB = detA detB<br>- detA⁻¹ = 1/detA<br>- detcA = cⁿ detA for n×n matrix"]
    E2["Rank Properties<br>- ρA = ρAᵀ<br>- ρAB ≤ min ρA, ρB<br>- ρA+B ≤ ρA + ρB"]
    E3["Eigenvalue & Eigenvector Properties<br>- Sum of eigenvalues = trA<br>- Product of eigenvalues = detA"]
end
E --> F{Applications of Properties};
subgraph F[Real-World Significance]
    F1["Solving Linear Systems<br>Consistency, solutions"]
    F2["Matrix Decompositions<br>LU, QR, SVD for numerical stability"]
    F3["Vector Spaces & Linear Transformations<br>Basis, dimension, rank-nullity theorem"]
    F4["Computer Graphics & Physics<br>Transformations, quantum mechanics"]
    F5["Data Science & Optimization<br>PCA, least squares solutions"]
end
F --> G[End: Properties Define Matrix Behavior];

flowchart TD
A[Start: Matrix Inversion] --> B["Find A⁻¹ for a square matrix A<br>such that A × A⁻¹ = I"];
B --> C{Prerequisite Check};
subgraph C[Is Inversion Possible?]
    C1["Matrix must be Square n×n"]
    C2["Matrix must be Nonsingular<br>i.e., detA ≠ 0"]
end
C1 & C2 --> D{Choose an Inversion Method};
subgraph D[Methods for Finding A⁻¹]
    D1["Elementary Row Operations<br>Gauss-Jordan Elimination"]
    D2["Adjugate Matrix Method<br>A⁻¹ = adjA / detA"]
    D3["Decomposition Methods<br>LU Decomposition, Cholesky"]
    D4["Special Case Methods<br>For 2x2, 3x3, orthogonal, etc."]
end
D1 --> E["Augment matrix A with Identity Matrix: A | I"];
E --> F["Perform Gauss-Jordan Elimination<br>Apply row ops to reduce A to I"];
F --> G["The right block becomes A⁻¹<br>Result: I | A⁻¹"];
D2 --> H["Calculate the Determinant of A, detA"];
H --> I["Find the Matrix of Minors"];
I --> J["Create Matrix of Cofactors<br>Apply checkerboard pattern of signs +/ -"];
J --> K["Transpose to get the Adjugate Matrix adjA"];
K --> L["A⁻¹ = 1 / detA × adjA"];
D3 --> M["Factorize A into simpler matrices<br>e.g., A = L × U<br>Where L is lower triangular, U is upper triangular"];
M --> N["Solve for A⁻¹ by solving systems<br>using forward/back substitution"];
D4 --> O["For 2x2 Matrix:<br>a b⁻¹ = 1/ad-bc × d -b<br>c d    -c  a"];
G & L & N & O --> P["Verify Result: A × A⁻¹ = I<br>to avoid calculation errors"];
P --> Q{Applications of Matrix Inverse};
subgraph Q[Why Invert a Matrix?]
    Q1["Solve Systems of Linear Equations<br>A⋅x = b → x = A⁻¹⋅b"]
    Q2["Compute Division in Matrix Algebra"]
    Q3["Find Basis for Vector Space"]
    Q4["Coordinate Changes & Transformations"]
end
Q --> R[End: Inverse is a Fundamental Tool];

flowchart TD
A[Start: What is a Vector?] --> B{Vector Fundamental Concepts};
subgraph B[Core Properties]
    direction LR
    B1[Magnitude<br>Length of the arrow]
    B2[Direction<br>Angle or bearing]
end
B --> C{Representation};
C --> C1["Cartesian Coordinates<br>(x, y, z)"];
C --> C2["Polar Coordinates<br>(r, θ)"];
C --> C3["Unit Vectors<br>î, ĵ, k̂"];
C1 & C2 & C3 --> D[Key Vector Operations];
subgraph D[Operations]
    D1[Addition & Subtraction<br>Tip-to-Tail or Parallelogram Method]
    D2[Scalar Multiplication<br>Scaling magnitude,<br>flipping direction]
    D3[Dot Product<br>Result: Scalar<br>Measures Projection]
    D4[Cross Product<br>Result: Vector<br>Perpendicular to both inputs]
end
D --> E{Applications in Physics & Engineering};
E --> F1["Mechanics<br>Force, Velocity, Acceleration"];
E --> F2["Electromagnetism<br>Electric & Magnetic Fields"];
E --> F3["Computer Graphics<br>3D Modeling, Lighting"];
E --> F4["Navigation<br>Heading, Wind Correction"];
F1 & F2 & F3 & F4 --> G[End: Vector as a Fundamental Tool];

flowchart TD
A[Start: What is a Function?] --> B["A relation where each input (x)<br>has exactly one output (y)<br>f: X → Y"];
B --> C{Ways to Represent a Function};
subgraph C[Representation]
    direction LR
    C1["Algebraically<br>f(x) = x² + 1"]
    C2["Graphically<br>Plot on Coordinate Plane"]
    C3["Numerically<br>Table of Values"]
    C4["Verbally<br>Description in Words"]
end
C --> D{Classify by Behavior & Properties};
subgraph D1[Key Properties]
    D11["Injective (One-to-One)<br>f(a) = f(b) ⇒ a = b"]
    D12["Surjective (Onto)<br>Range = Codomain"]
    D13["Bijective<br>Both 1-1 and Onto<br>(Has an inverse)"]
end
subgraph D2[Symmetry]
    D21["Even Function<br>f(-x) = f(x)<br>Symmetric about y-axis"]
    D22["Odd Function<br>f(-x) = -f(x)<br>Symmetric about origin"]
end
subgraph D3[Rates of Change]
    D31["Increasing<br>x₁ < x₂ ⇒ f(x₁) < f(x₂)"]
    D32["Decreasing<br>x₁ < x₂ ⇒ f(x₁) > f(x₂)"]
    D33["Constant<br>f(x) = c"]
end
D --> E{Common Types of Functions};
subgraph E[Function Types]
    E1["Polynomial<br>f(x) = aₙxⁿ + ... + a₁x + a₀"]
    E2["Rational<br>f(x) = P(x)/Q(x)"]
    E3["Exponential<br>f(x) = a ⋅ bˣ"]
    E4["Logarithmic<br>f(x) = logₐ(x)"]
    E5["Trigonometric<br>sin(x), cos(x), tan(x)"]
    E6["Piecewise<br>Defined by different rules<br>on different intervals"]
end
E --> F[Operations on Functions];
subgraph F[Operations]
    F1["Arithmetic<br>f+g, f-g, f⋅g, f/g"]
    F2["Composition<br>(f∘g)(x) = f(g(x))"]
    F3["Inversion<br>f⁻¹(x) exists only if f is bijective<br>f(f⁻¹(x)) = x"]
end
F --> G{Applications of Functions};
G --> H1["Physics: Describe motion<br>Position, velocity, acceleration<br>as functions of time"]
G --> H2["Economics & Finance:<br>Cost, revenue, profit functions,<br>Compound interest"]
G --> H3["Computer Science:<br>Functions as subroutines,<br>Input-Output processing"]
G --> H4["Engineering: Signal processing,<br>Electrical waveforms"]
G --> H5["Everyday Life: Modeling trends,<br>Converting units, Temperature scales"]
H1 & H2 & H3 & H4 & H5 --> I[End: Functions as a Fundamental Concept];

flowchart TD
A[Start: What is a Limit?] --> B["The value a function f(x)<br> approaches as x approaches<br> some value c.<br>limₓ→c f(x) = L"];
B --> C{Approach to c can be from};
C --> C1["Both Sides limₓ→c"]
C --> C2["Left-Hand Limit limₓ→c⁻"]
C --> C3["Right-Hand Limit limₓ→c⁺"]
C1 & C2 & C3 --> D{The Limit Exists IF};
D --> D1["limₓ→c⁻ f(x) = L"]
D --> D2["limₓ→c⁺ f(x) = L"]
D --> D3["AND L is a finite number"]
D1 & D2 & D3 --> D4["limₓ→c f(x) = L exists"]
D4 --> E{How to Find the Limit?<br>Evaluation Methods};
subgraph E[Methods of Evaluation]
    E1["Direct Substitution<br>Plug in x = c"]
    E2["Algebraic Manipulation<br>Factor & Cancel, Rationalize"]
    E3["Special Limits<br>limₓ→0 sinx/x = 1"]
end
E1 --> F1{Get an Answer?};
E2 --> F1;
E3 --> F1;
F1 -->|Yes, a finite number L| G["Success!<br>limₓ→c f(x) = L"];
F1 -->|No, get Indeterminate Form 0/0, ∞/∞, etc.| H["Use Advanced Techniques"];
subgraph H[Advanced Techniques for Indeterminate Forms]
    H1["More Algebra<br>Factorization, Rationalization"]
    H2["L'Hôpital's Rule<br>Take derivative of numerator and denominator"]
end
H --> I["Re-evaluate the limit"];
I --> F1;
E --> J["Limits at Infinity<br>limₓ→±∞ f(x)"];
J --> J1["Compare degrees of numerator<br>and denominator in rational functions"]
J --> J2["Use rules of growth rates<br>e.g., Exponential > Polynomial > Log"]
G --> K{Applications of Limits};
subgraph K[Where Limits Are Used]
    K1["Define Continuity<br>f is continuous at c if<br>limₓ→c f(x) = f(c)"]
    K2["Define the Derivative<br>f'(x) = limₕ→₀ f(x+h)-f(x)/h"]
    K3["Define Improper Integrals<br>∫ₐ∞ f(x dx = limₜ→∞ ∫ₐᵗ f(x dx"]
    K4["Analyze Asymptotes & Behavior<br>Horizontal & Vertical Asymptotes"]
end
K --> L[End: Limits are the Foundation of Calculus];

flowchart TD
    A[Start: Is fx continuous at x = c?] --> B{Three Conditions Must Hold}
    
    subgraph B[Continuity Checklist]
        direction TB
        B1["1. fc is defined<br>c is in the domain of f"]
        B2["2. limₓ→c fx exists<br>LHL = RHL = L"]
        B3["3. limₓ→c fx = fc<br>The limit equals the function value"]
    end
    
    B1 -->|Yes| B2
    B1 -->|No| C1["Condition FAILED<br>Discontinuity at x=c"]
    
    B2 -->|Yes, L exists| B3
    B2 -->|No, LHL ≠ RHL| C2["Condition FAILED<br>Limit does not exist"]
    
    B3 -->|Yes| D["All Conditions MET<br>f is CONTINUOUS at x = c"]
    B3 -->|No, L ≠ fc| C3["Condition FAILED<br>Removable Discontinuity"]
    
    C1 & C2 & C3 --> E["f is DISCONTINUOUS at x = c"]
    
    E --> F{Classify the Type of Discontinuity}
    
    subgraph F[Types of Discontinuities]
        F1["Removable Hole<br>lim exists but ≠ fc<br>or fc is undefined"]
        F2["Jump<br>LHL and RHL exist but are not equal"]
        F3["Infinite Asymptotic<br>Limit is ±∞"]
        F4["Oscillating<br>Limit does not exist due to<br>wild oscillation e.g., sin1/x"]
    end
    
    D --> G[Extended Analysis]
    
    subgraph G[Beyond a Single Point]
        G1["Continuity on an Interval<br>Continuous at every point in a, b"]
        G2["Types of Continuous Functions<br>Polynomial, Rational, Root, Trig,<br>Exp, Log functions are continuous<br>on their domains"]
        G3["The Intermediate Value Theorem IVT<br>If f is continuous on a, b,<br>it takes on every value<br>between fa and fb"]
    end
    
    G --> H[End: Continuity Ensures Predictable, Connected Behavior]

flowchart TD
    A[Start: Application Problem] --> B{What type of problem?}
    B --> C[Find Rate of Change]
    B --> D[Analyze Function Graph<br>Increasing/Decreasing]
    B --> E[Find Local Extrema<br>Max/Min]
    B --> F[Find Absolute Extrema<br>on an Interval]
    B --> G[Solve Optimization Problem]
    B --> H[Related Rates Problem]
    C --> C1[Compute derivative f'x<br>This IS the rate of change]
    C1 --> C2[Evaluate f'a for<br>instantaneous rate at x=a]
    D --> D1[Find critical points<br>Solve f'x = 0 or undefined]
    D1 --> D2[Test intervals around<br>critical points]
    D2 --> D3[Conclusion:<br>f'x > 0 → Increasing<br>f'x < 0 → Decreasing]
    E --> E1[Find critical points]
    E1 --> E2[Use First or Second<br>Derivative Test]
    E2 --> E3[First Derivative Test:<br>Sign change + to - → max<br>- to + → min]
    E2 --> E4[Second Derivative Test:<br>f''c > 0 → local min<br>f''c < 0 → local max]
    F --> F1[Find critical points<br>in interval a,b]
    F1 --> F2[Evaluate fa, fb, and<br>fc at critical points]
    F2 --> F3[Largest value = abs max<br>Smallest value = abs min]
    G --> G1[Understand problem<br>Draw diagram if needed]
    G1 --> G2[Write objective function<br>Q = formula to max/min]
    G2 --> G3[Reduce to one variable<br>using constraints]
    G3 --> G4[Find critical points<br>on domain]
    G4 --> G5[Use First/Second Deriv Test<br>or Closed Interval Method]
    G5 --> G6[Verify answer makes sense]
    H --> H1[Identify given rate<br>and needed rate]
    H1 --> H2[Find relating equation]
    H2 --> H3[Differentiate both sides<br>with respect to time t]
    H3 --> H4[Substitute known values<br>and solve for unknown rate]
    C2 --> Z[State Final Answer<br>with units]
    D3 --> Z
    E3 --> Z
    E4 --> Z
    F3 --> Z
    G6 --> Z
    H4 --> Z
    style A fill:#e0f7fa
    style Z fill:#c8e6c9,stroke:green
    style C fill:#ffebee
    style D fill:#e8eaf6
    style E fill:#e3f2fd
    style F fill:#f3e5f5
    style G fill:#fff8e1
    style H fill:#e8f5e8

flowchart TD
    A[Start: Choose a Coordinate System] --> B{What type of problem?}
    B --> C[2D Problems]
    B --> D[3D Problems]
    B --> E[Specialized Applications]
    %% Subgraph for 2D Systems
    subgraph C [2D Coordinate Systems]
        direction TB
        C1{Most suitable 2D system?}
        
        C1 --> C2[Cartesian Rectangular]
        C2 --> C3[Coordinates: x, y<br>Base vectors: i, j]
        C3 --> C4[Best for: Grid-based problems,<br>polygons, most algebra]
        
        C1 --> C5[Polar]
        C5 --> C6[Coordinates: r, θ<br>r = radius, θ = angle]
        C6 --> C7[Best for: Circular motion,<br>spiral patterns, rotation]
        
        C8{Conversion between systems?}
        C8 --> C9[Polar to Cartesian<br>x = r cosθ, y = r sinθ]
        C8 --> C10[Cartesian to Polar<br>r = √x² + y², θ = tan⁻¹y/x]
    end
    %% Subgraph for 3D Systems
    subgraph D [3D Coordinate Systems]
        direction TB
        D1{Most suitable 3D system?}
        
        D1 --> D2[Cartesian 3D]
        D2 --> D3[Coordinates: x, y, z<br>Base vectors: i, j, k]
        D3 --> D4[Best for: 3D shapes,<br>volumes, spatial analysis]
        
        D1 --> D5[Cylindrical]
        D5 --> D6[Coordinates: r, θ, z<br>Polar + height]
        D6 --> D7[Best for: Cylinders,<br>pipes, rotational symmetry about z-axis]
        
        D1 --> D8[Spherical]
        D8 --> D9[Coordinates: ρ, θ, φ<br>ρ = radius, θ = azimuth, φ = polar angle]
        D9 --> D10[Best for: Spheres,<br>global coordinates, radial fields]
        
        D11{3D Conversions?}
        D11 --> D12[Between all systems<br>using trigonometric relations]
    end
    %% Subgraph for Specialized Systems
    subgraph E [Specialized Systems]
        direction TB
        E1{Special application?}
        
        E1 --> E2[Log-Polar]
        E2 --> E3[Used in: Computer vision,<br>image recognition]
        
        E1 --> E4[Barycentric]
        E4 --> E5[Used in: Computer graphics,<br>triangle interpolation]
        
        E1 --> E6[Curvilinear]
        E6 --> E7[Used in: Advanced physics,<br>general relativity]
        
        E1 --> E8[Homogeneous]
        E8 --> E9[Used in: Computer graphics,<br>projective geometry]
    end
    C --> F{Need to solve problem?}
    D --> F
    E --> F
    
    F --> G[Apply appropriate coordinate<br>transformations if needed]
    G --> H[Use system-specific formulas<br>and equations]
    H --> I[Final Solution]
    style A fill:#e0f7fa
    style I fill:#c8e6c9,stroke:green
    style C fill:#fff8e1
    style D fill:#e8f5e8
    style E fill:#e3f2fd
    style F fill:#f3e5f5

flowchart TD
    A[Start: Straight Line Problem] --> B{What information is given?}
    B --> C[Given Slope & Information]
    B --> D[Given Two Points]
    B --> E[Given Intercepts]
    B --> F[Given Two Lines<br>Find Relationship]
    %% Subgraph for Slope-based Equations
    subgraph C [Given Slope & Information]
        direction TB
        C1{What specific information?}
        C1 --> C2[Slope m & Y-intercept c]
        C2 --> C3[Slope-Intercept Form:<br>y = mx + c]
        
        C1 --> C4[Slope m & Point x₁,y₁]
        C4 --> C5[Point-Slope Form:<br>y - y₁ = mx - x₁]
        
        C1 --> C6[Angle θ with x-axis]
        C6 --> C7[Slope m = tanθ]
    end
    %% Subgraph for Two Points
    subgraph D [Given Two Points]
        direction TB
        D1[Points: x₁,y₁ & x₂,y₂]
        D1 --> D2[Calculate Slope<br>m = y₂-y₁ / x₂-x₁]
        D2 --> D3[Use Point-Slope Form<br>y - y₁ = mx - x₁]
        D2 --> D4[Two-Point Form<br>y-y₁ / y₂-y₁ = x-x₁ / x₂-x₁]
    end
    %% Subgraph for Intercepts
    subgraph E [Given Intercepts]
        direction TB
        E1{Which intercepts?}
        E1 --> E2[x-intercept a & y-intercept b]
        E2 --> E3[Intercept Form:<br>x/a + y/b = 1]
        
        E1 --> E4[Given both intercepts equal]
        E4 --> E5[x/a + y/a = 1 or x + y = a]
    end
    %% Subgraph for Line Relationships
    subgraph F [Given Two Lines]
        direction TB
        F1{Find what relationship?}
        
        F1 --> F2[Angle between lines]
        F2 --> F3[tanθ = │m₁-m₂ / 1+m₁m₂│]
        
        F1 --> F4[Check if parallel]
        F4 --> F5[Parallel if: m₁ = m₂]
        
        F1 --> F6[Check if perpendicular]
        F6 --> F7[Perpendicular if: m₁ × m₂ = -1]
        
        F1 --> F8[Find intersection point]
        F8 --> F9[Solve equations simultaneously]
        
        F1 --> F10[Distance between parallel lines]
        F10 --> F11[d = │c₁-c₂│ / √a²+b²]
        
        F1 --> F12[Distance from point to line]
        F12 --> F13[d = │Ax₀+By₀+C│ / √A²+B²]
    end
    C --> G[Convert to required form]
    D --> G
    E --> G
    F --> H[Apply specific formula]
    G --> I[General Form: Ax + By + C = 0]
    H --> I
    
    I --> J[Final Answer]
    style A fill:#e0f7fa
    style J fill:#c8e6c9,stroke:green
    style C fill:#fff8e1
    style D fill:#e8f5e8
    style E fill:#e3f2fd
    style F fill:#f3e5f5

flowchart TD
    A[Start: Need Equation of a Line] --> B{What information is given?}
    B --> C[Slope & Y-Intercept]
    B --> D[Slope & A Point]
    B --> E[Two Points]
    B --> F[X & Y Intercepts]
    B --> G[Normal Form]
    B --> H[Parallel/Perpendicular to<br>another line + Point]
    C --> C1[<b>Slope-Intercept Form</b><br>y = mx + c]
    C1 --> C2[<b>m</b> = slope<br><b>c</b> = y-intercept]
    D --> D1[<b>Point-Slope Form</b><br>y - y₁ = mx - x₁]
    D1 --> D2[<b>m</b> = slope<br><b>x₁, y₁</b> = given point]
    E --> E1[Calculate Slope<br>m = y₂ - y₁ / x₂ - x₁]
    E1 --> E2[<b>Two-Point Form</b><br>y - y₁ / y₂ - y₁ = x - x₁ / x₂ - x₁]
    E2 --> E3[<b>x₁, y₁, x₂, y₂</b> = given points]
    F --> F1[<b>Intercept Form</b><br>x/a + y/b = 1]
    F1 --> F2[<b>a</b> = x-intercept<br><b>b</b> = y-intercept]
    G --> G1[<b>Normal Form</b><br>x cosθ + y sinθ = p]
    G1 --> G2[<b>p</b> = perpendicular distance<br>from origin<br><b>θ</b> = angle with normal]
    H --> H1[Find slope relationship]
    H1 --> H2{Which relationship?}
    H2 --> H3[Parallel: m₁ = m₂]
    H2 --> H4[Perpendicular: m₁ × m₂ = -1]
    H3 --> H5[Use Point-Slope Form with new slope]
    H4 --> H5
    C2 --> Z[Convert to required form if needed]
    D2 --> Z
    E3 --> Z
    F2 --> Z
    G2 --> Z
    H5 --> Z
    Z --> FINAL[Final Line Equation]
    style A fill:#e0f7fa
    style FINAL fill:#c8e6c9,stroke:green
    style C1 fill:#fff8e1
    style D1 fill:#e8f5e8
    style E2 fill:#e3f2fd
    style F1 fill:#f3e5f5
    style G1 fill:#ffebee
    style H1 fill:#e8f5e8

flowchart TD
    A[Start: Identify Conic Section] --> B{General Second Degree Equation:<br>Ax² + Bxy + Cy² + Dx + Ey + F = 0}
    B --> C{Calculate Discriminant:<br>Δ = B² - 4AC}
    C --> D[Δ < 0]
    C --> E[Δ = 0]
    C --> F[Δ > 0]
    D --> G{Ellipse Type}
    G --> H[A = C and B = 0]
    H --> I[<b>Circle</b><br>Special case of ellipse]
    G --> J[Otherwise]
    J --> K[<b>Ellipse</b>]
    E --> L[<b>Parabola</b>]
    F --> M{Check A and C}
    M --> N[A = -C]
    N --> O[<b>Rectangular Hyperbola</b>]
    M --> P[Otherwise]
    P --> Q[<b>Hyperbola</b>]
    %% Detailed Analysis of Each Conic
    K --> R[Standard Form of Ellipse<br>x²/a² + y²/b² = 1 or<br>x²/b² + y²/a² = 1]
    R --> S[Identify: Center, Vertices,<br>Foci, Major & Minor Axes]
    I --> T[Standard Form of Circle<br>x-h² + y-k² = r²]
    T --> U[Identify: Center h,k, Radius r]
    L --> V[Standard Forms of Parabola<br>y² = 4ax or x² = 4ay or<br>y-k² = 4ax-h or x-h² = 4ay-k]
    V --> W[Identify: Vertex, Focus,<br>Directrix, Axis]
    Q --> X[Standard Form of Hyperbola<br>x²/a² - y²/b² = 1 or<br>y²/a² - x²/b² = 1]
    X --> Y[Identify: Center, Vertices,<br>Foci, Asymptotes]
    O --> Z[Standard Form<br>xy = c² or x² - y² = a²]
    Z --> AA[Identify: Asymptotes,<br>Rectangular nature]
    S --> FINAL[Graph & Analyze Conic]
    U --> FINAL
    W --> FINAL
    Y --> FINAL
    AA --> FINAL
    style A fill:#e0f7fa
    style FINAL fill:#c8e6c9,stroke:green
    style D fill:#fff8e1
    style E fill:#e8f5e8
    style F fill:#e3f2fd
    style H fill:#f3e5f5
    style N fill:#ffebee

flowchart TD
    A[Start: Statistical Analysis] --> B{What is your goal?}
    B --> C[Describe & Summarize Data]
    B --> D[Make Inferences & Predictions]
    B --> E[Explore Relationships]
    %% DESCRIPTIVE STATISTICS BRANCH
    subgraph C [Descriptive Statistics]
        direction TB
        C1{Data Type?}
        C1 --> C2[Categorical/Nominal]
        C2 --> C3[Frequency Tables,<br>Mode, Bar Charts, Pie Charts]
        
        C1 --> C4[Numerical]
        C4 --> C5{Central Tendency &<br>Dispersion Measures}
        C5 --> C6[Mean, Median, Mode,<br>Range, Variance, Std Dev, IQR]
        
        C4 --> C7{Data Shape &<br>Distribution}
        C7 --> C8[Skewness, Kurtosis,<br>Histograms, Box Plots]
    end
    %% INFERENTIAL STATISTICS BRANCH
    subgraph D [Inferential Statistics]
        direction TB
        D1{What to infer?}
        
        D1 --> D2[Estimate Parameters]
        D2 --> D3{Parameter Type?}
        D3 --> D4[Population Mean]
        D4 --> D5[Confidence Intervals<br>t-interval or z-interval]
        D3 --> D6[Population Proportion]
        D6 --> D7[Proportion CI<br>z-interval for p]
        
        D1 --> D8[Test Hypotheses]
        D8 --> D9{Comparison Type?}
        D9 --> D10[One Sample]
        D10 --> D11[One-sample t-test<br>or z-test]
        
        D9 --> D12[Two Samples]
        D12 --> D13{Data Type?}
        D13 --> D14[Independent]
        D14 --> D15[Independent t-test]
        D13 --> D16[Paired/Matched]
        D16 --> D17[Paired t-test]
        
        D9 --> D18[Three+ Samples]
        D18 --> D19[ANOVA]
    end
    %% RELATIONSHIPS BRANCH
    subgraph E [Explore Relationships]
        direction TB
        E1{Variable Types?}
        
        E1 --> E2[Two Numerical]
        E2 --> E3[Scatter Plot]
        E3 --> E4[Correlation Coefficient r]
        E4 --> E5[Regression Analysis<br>y = a + bx]
        
        E1 --> E6[Two Categorical]
        E6 --> E7[Contingency Table]
        E7 --> E8[Chi-square Test<br>of Independence]
        
        E1 --> E9[One Numerical, One Categorical]
        E9 --> E10[Group Comparison<br>t-test or ANOVA]
    end
    C --> F[Check Data Quality &<br>Assumptions]
    D --> F
    E --> F
    F --> G{Normality Check?}
    G --> H[Data Normal?]
    H --> I[Yes -> Use Parametric Tests]
    H --> J[No -> Use Non-Parametric Tests<br>Mann-Whitney, Wilcoxon, Kruskal-Wallis]
    
    I --> K[Proceed with Analysis]
    J --> K
    K --> L[Interpret Results &<br>Draw Conclusions]
    L --> M{Results Significant?}
    M --> N[Yes -> Report Effect Size &<br>Practical Significance]
    M --> O[No -> Consider Power Analysis &<br>Sample Size]
    N --> FINAL[Final Conclusion &<br>Recommendations]
    O --> FINAL
    style A fill:#e0f7fa
    style FINAL fill:#c8e6c9,stroke:green
    style C fill:#fff8e1
    style D fill:#e8f5e8
    style E fill:#e3f2fd
    style F fill:#f3e5f5