Math Academy

The course begins with foundational topics that support later mathematical modeling in physics. Students study vector geometry in three-dimensional space, including equations of lines and planes and geometric relationships between them. These tools appear naturally in the description of trajectories, rays, and intersections of physical surfaces. Introductory set theory provides a precise language for describing mathematical structures and regions in space, supporting later work in multivariable analysis and probability.

Students then develop computational tools from linear algebra. Methods such as Gaussian elimination, matrix inverses, determinants, and LU factorization are used to analyze and solve systems of linear equations. These techniques arise when solving coupled equations in physical models, such as electrical circuits, mechanical systems with multiple interacting components, and discretized approximations of physical fields.

The course next develops the structure of vector spaces and linear transformations. Students study spans, bases, dimension, rank, and null spaces, and extend these ideas to abstract vector spaces. Linear maps and their matrix representations are examined, establishing the connection between geometric transformations and matrix algebra. These ideas culminate in the study of eigenvalues, eigenvectors, and matrix diagonalization, which are central tools for solving differential equations describing oscillations, normal modes of vibration, and stability of dynamical systems.

The course then turns to multivariable calculus and vector analysis. Students analyze functions of several variables, quadric surfaces, and coordinate transformations. Partial derivatives, gradients, and the multivariable chain rule are developed along with linear approximations and higher-order derivatives. These concepts describe how physical quantities such as temperature, pressure, or electric potential vary in space. Vector-valued functions and vector fields are studied together with curvature, divergence, and curl, providing the mathematical language used to describe velocity fields in fluid flow and electric and magnetic fields in electromagnetism.

Students then study multiple integration, focusing on double integrals and coordinate transformations such as polar coordinates. These techniques allow the computation of physical quantities distributed over regions, such as mass, charge, or probability density.

The final portion of the course introduces differential equations as models of dynamical systems. Students study first-order equations and qualitative behavior of solutions through phase-line analysis. Applications include motion under drag, gravitational attraction, electrical circuits, and population dynamics. The course concludes with second-order linear differential equations and oscillatory systems, including harmonic motion, damped oscillators, and resonance—fundamental mathematical models for vibrating strings, mechanical systems, and electromagnetic circuits.

Preliminaries

25 topics

1.1. Vector Geometry

	1.1.1.	The Vector Equation of a Line
	1.1.2.	The Parametric Equations of a Line
	1.1.3.	The Cartesian Equation of a Line
	1.1.4.	The Vector Equation of a Plane
	1.1.5.	The Cartesian Equation of a Plane
	1.1.6.	The Parametric Equations of a Plane
	1.1.7.	The Intersection of Two Planes
	1.1.8.	The Shortest Distance Between a Plane and a Point
	1.1.9.	The Intersection Between a Line and a Plane

1.2. Introduction to Set Theory

	1.2.1.	Special Sets
	1.2.2.	Statements and Predicates
	1.2.3.	Equivalent Sets
	1.2.4.	The Constructive Definition of a Set
	1.2.5.	The Conditional Definition of a Set
	1.2.6.	Describing Sets Using Set-Builder Notation
	1.2.7.	Describing Planar Regions Using Set-Builder Notation
	1.2.8.	Cardinality of Finite Sets
	1.2.9.	Infinite Sets
	1.2.10.	Subsets

1.3. Set Operations

	1.3.1.	The Difference of Sets
	1.3.2.	Set Complements
	1.3.3.	The Cartesian Product
	1.3.4.	Visualizing Cartesian Products
	1.3.5.	Indexed Sets
	1.3.6.	Sets and Functions

Matrices

34 topics

2.4. Gaussian Elimination

	2.4.1.	Systems of Equations as Augmented Matrices
	2.4.2.	Row Echelon Form
	2.4.3.	Solving Systems of Equations Using Back Substitution
	2.4.4.	Elementary Row Operations
	2.4.5.	Creating Rows or Columns Containing Zeros Using Gaussian Elimination
	2.4.6.	Solving 2x2 Systems of Equations Using Gaussian Elimination
	2.4.7.	Solving 2x2 Singular Systems of Equations Using Gaussian Elimination
	2.4.8.	Solving 3x3 Systems of Equations Using Gaussian Elimination
	2.4.9.	Identifying the Pivot Columns of a Matrix
	2.4.10.	Solving 3x3 Singular Systems of Equations Using Gaussian Elimination
	2.4.11.	Reduced Row Echelon Form
	2.4.12.	Gaussian Elimination For NxM Systems of Equations

2.5. Elementary Matrices

	2.5.1.	Elementary 2x2 Matrices
	2.5.2.	Row Operations on 2x2 Matrices as Products of Elementary Matrices
	2.5.3.	Elementary 3x3 Matrices
	2.5.4.	Row Operations on 3x3 Matrices as Products of Elementary Matrices

2.6. The Inverse of a Matrix

	2.6.1.	The Invertible Matrix Theorem in Terms of 2x2 Systems of Equations
	2.6.2.	Finding the Inverse of a 2x2 Matrix Using Row Operations
	2.6.3.	Finding the Inverse of a 3x3 Matrix Using Row Operations
	2.6.4.	Finding the Inverse of an NxN Square Matrix Using Row Operations
	2.6.5.	Matrices With Easy-to-Find Inverses

2.7. Determinants

	2.7.1.	The Determinant of an NxN Matrix
	2.7.2.	Finding Determinants Using Laplace Expansions
	2.7.3.	Basic Properties of Determinants
	2.7.4.	Further Properties of Determinants
	2.7.5.	Row and Column Operations on Determinants
	2.7.6.	Conditions When a Determinant Equals Zero
	2.7.7.	Finding Determinants Using Row and Column Operations
	2.7.8.	Cramer's Rule for 2x2 Systems of Linear Equations

2.8. LU Factorization

	2.8.1.	Triangular Matrices
	2.8.2.	LU Factorization of 2x2 Matrices
	2.8.3.	LU Factorization of 3x3 Matrices
	2.8.4.	LU Factorization of NxN Matrices
	2.8.5.	Solving Systems of Equations Using LU Factorization

Vector Spaces

27 topics

3.9. Vectors in N-Dimensional Space

	3.9.1.	Vectors in N-Dimensional Euclidean Space
	3.9.2.	Linear Combinations of Vectors in N-Dimensional Euclidean Space
	3.9.3.	Linear Span of Vectors in N-Dimensional Euclidean Space
	3.9.4.	Linear Dependence and Independence

3.10. Subspaces of N-Dimensional Space

	3.10.1.	Subspaces of N-Dimensional Space
	3.10.2.	Subspaces of N-Dimensional Space: Geometric Interpretation
	3.10.3.	The Column Space of a Matrix
	3.10.4.	The Null Space of a Matrix

3.11. Bases of N-Dimensional Space

	3.11.1.	Finding a Basis of a Span
	3.11.2.	Finding a Basis of the Column Space of a Matrix
	3.11.3.	Finding a Basis of the Null Space of a Matrix
	3.11.4.	Expressing the Coordinates of a Vector in a Given Basis
	3.11.5.	Writing Vectors in Different Bases
	3.11.6.	The Change-of-Coordinates Matrix
	3.11.7.	Changing a Basis Using the Change-of-Coordinates Matrix

3.12. Dimension and Rank in N-Dimensional Space

	3.12.1.	The Dimension of a Span
	3.12.2.	The Rank of a Matrix
	3.12.3.	The Dimension of the Null Space of a Matrix
	3.12.4.	The Invertible Matrix Theorem in Terms of Dimension, Rank and Nullity
	3.12.5.	The Rank-Nullity Theorem

3.13. Abstract Vector Spaces

	3.13.1.	Introduction to Abstract Vector Spaces
	3.13.2.	Defining Abstract Vector Spaces
	3.13.3.	Linear Independence in Abstract Vector Spaces
	3.13.4.	Subspaces of Abstract Vector Spaces
	3.13.5.	Bases in Abstract Vector Spaces
	3.13.6.	The Coordinates of a Vector Relative to a Basis in Abstract Vector Spaces
	3.13.7.	Dimension in Abstract Vector Spaces

Linear Transformations

10 topics

4.14. Linear Transformations

	4.14.1.	The Standard Matrix of a Linear Transformation in Terms of the Standard Basis
	4.14.2.	The Kernel of a Linear Transformation
	4.14.3.	The Image and Rank of a Linear Transformation
	4.14.4.	The Image of a Linear Transformation in the Cartesian Plane
	4.14.5.	The Invertible Matrix Theorem in Terms of Linear Transformations
	4.14.6.	The Rank-Nullity Theorem in Terms of Linear Transformations

4.15. Linear Maps and Their Matrix Representations

	4.15.1.	The Matrix of a Linear Transformation Relative to a Basis
	4.15.2.	Connections Between Matrix Representations of a Linear Transformation
	4.15.3.	Linear Maps Between Two Different Vector Spaces
	4.15.4.	The Matrix of a Linear Map Relative to Two Bases

Diagonalization of Matrices

20 topics

5.16. Eigenvectors and Eigenvalues

	5.16.1.	The Eigenvalues and Eigenvectors of a 2x2 Matrix
	5.16.2.	Calculating the Eigenvalues of a 2x2 Matrix
	5.16.3.	Calculating the Eigenvectors of a 2x2 Matrix
	5.16.4.	The Characteristic Equation of a Matrix
	5.16.5.	The Cayley-Hamilton Theorem
	5.16.6.	Calculating the Eigenvectors of a 3x3 Matrix With Distinct Eigenvalues
	5.16.7.	Calculating the Eigenvectors of a 3x3 Matrix in the General Case
	5.16.8.	The Invertible Matrix Theorem in Terms of Eigenvalues

5.17. Diagonalization

	5.17.1.	Diagonalizing a 2x2 Matrix
	5.17.2.	Properties of Diagonalization
	5.17.3.	Diagonalizing a 3x3 Matrix With Distinct Eigenvalues
	5.17.4.	Diagonalizing a 3x3 Matrix in the General Case

5.18. Real Matrices With Complex Eigenvalues

	5.18.1.	Vectors Over the Complex Numbers
	5.18.2.	Matrices Over the Complex Numbers
	5.18.3.	Finding Complex Eigenvalues of Real 2x2 Matrices
	5.18.4.	Finding Complex Eigenvectors of Real 2x2 Matrices

5.19. Generalized Eigenvectors

	5.19.1.	Nilpotent and Idempotent Matrices
	5.19.2.	Generalized Eigenvectors
	5.19.3.	Ranks of Generalized Eigenvectors
	5.19.4.	Finding Generalized Eigenvectors of Specific Ranks

Multivariable Functions

45 topics

6.20. Introduction to Multivariable Functions

	6.20.1.	Interior and Boundary Points
	6.20.2.	Interiors and Boundaries of Sets
	6.20.3.	Open and Closed Sets
	6.20.4.	The Domain of a Multivariable Function
	6.20.5.	Level Curves
	6.20.6.	Level Surfaces
	6.20.7.	Limits and Continuity of Multivariable Functions

6.21. Quadric Surfaces and Cylinders

	6.21.1.	Ellipsoids
	6.21.2.	Hyperboloids
	6.21.3.	Paraboloids
	6.21.4.	Elliptic Cones
	6.21.5.	Cylinders
	6.21.6.	Intersections of Lines and Planes With Surfaces
	6.21.7.	Identifying Quadric Surfaces

6.22. Partial Derivatives

	6.22.1.	Introduction to Partial Derivatives
	6.22.2.	Computing Partial Derivatives Using the Rules of Differentiation
	6.22.3.	Geometric Interpretations of Partial Derivatives
	6.22.4.	Partial Differentiability of Multivariable Functions
	6.22.5.	Higher-Order Partial Derivatives
	6.22.6.	Equality of Mixed Partial Derivatives
	6.22.7.	Tangent Planes to Surfaces
	6.22.8.	Linearization of Multivariable Functions
	6.22.9.	Differentiating Under the Integral Sign

6.23. The Gradient Vector

	6.23.1.	The Gradient Vector
	6.23.2.	The Gradient as a Normal Vector
	6.23.3.	Tangent Lines to Level Curves
	6.23.4.	Tangent Planes to Level Surfaces
	6.23.5.	Directional Derivatives
	6.23.6.	The Multivariable Mean-Value Theorem

6.24. Plane Transformations

	6.24.1.	Affine Transformations
	6.24.2.	The Image of an Affine Transformation
	6.24.3.	The Inverse of an Affine Transformation
	6.24.4.	Nonlinear Transformations of Plane Regions
	6.24.5.	Polar Coordinate Transformations
	6.24.6.	Transformations of Regions Between Curves

6.25. Differentiation

	6.25.1.	The Jacobian
	6.25.2.	The Inverse Function Theorem
	6.25.3.	The Jacobian of a Three-Dimensional Transformation
	6.25.4.	The Derivative of a Multivariable Function
	6.25.5.	Symmetric Matrices
	6.25.6.	The Second Derivative of a Multivariable Function
	6.25.7.	Second-Degree Taylor Polynomials of Multivariable Functions

6.26. The Multivariable Chain Rule

	6.26.1.	The Multivariable Chain Rule
	6.26.2.	The Multivariable Chain Rule in Vector Form
	6.26.3.	Differentials

Vector Functions and Vector Fields

26 topics

7.27. Vector-Valued Functions

	7.27.1.	The Domain of a Vector-Valued Function
	7.27.2.	Differentiation Rules for Vector-Valued Functions
	7.27.3.	Integration Rules for Vector-Valued Functions
	7.27.4.	The Arc Length of a Vector-Valued Function
	7.27.5.	Tangent Vectors and Tangent Lines to Curves
	7.27.6.	Unit Tangent Vectors
	7.27.7.	Principal Normal Vectors
	7.27.8.	Binormal Vectors
	7.27.9.	The Osculating Plane
	7.27.10.	Parameterization by Arc Length

7.28. Curvature

	7.28.1.	Introduction to Curvature
	7.28.2.	Finding Curvature Using the Cross Product
	7.28.3.	Radius of Curvature
	7.28.4.	The Curvature of a Plane Curve

7.29. Vector Fields

	7.29.1.	Vector Fields
	7.29.2.	Visualizing Vector Fields
	7.29.3.	Gradient Vector Fields
	7.29.4.	Conservative Vector Fields in the Cartesian Plane
	7.29.5.	Calculating Potential Functions
	7.29.6.	Simple, Closed, and Oriented Curves
	7.29.7.	Connected and Simply-Connected Regions
	7.29.8.	Stream Functions

7.30. Divergence and Curl

	7.30.1.	The Divergence of a Vector Field
	7.30.2.	Properties of the Divergence Operator
	7.30.3.	The Curl of a Vector Field
	7.30.4.	Properties of the Curl Operator

Multiple Integrals

12 topics

8.31. Double Integrals

	8.31.1.	Double Integrals Over Rectangular Domains
	8.31.2.	Double Integrals Over Non-Rectangular Domains
	8.31.3.	Properties of Double Integrals
	8.31.4.	Type I and II Regions in Two-Dimensional Space
	8.31.5.	Double Integrals Over Type I Regions
	8.31.6.	Double Integrals Over Type II Regions
	8.31.7.	Double Integrals Over Partitioned Regions
	8.31.8.	Changing the Order of Integration in Double Integrals

8.32. Change of Variables for Double Integrals

	8.32.1.	Double Integrals in Plane Polar Coordinates
	8.32.2.	Double Integrals Between Polar Curves
	8.32.3.	Computing Areas Using a Change of Variables
	8.32.4.	Computing Double Integrals Using a Change of Variables

Probability & Random Variables

28 topics

9.33. Probability

	9.33.1.	The Law of Total Probability
	9.33.2.	Extending the Law of Total Probability
	9.33.3.	Bayes' Theorem
	9.33.4.	Extending Bayes' Theorem

9.34. Random Variables

	9.34.1.	Probability Density Functions of Continuous Random Variables
	9.34.2.	Calculating Probabilities With Continuous Random Variables
	9.34.3.	Continuous Random Variables Over Infinite Domains
	9.34.4.	Cumulative Distribution Functions for Continuous Random Variables
	9.34.5.	Median, Quartiles and Percentiles of Continuous Random Variables
	9.34.6.	Finding the Mode of a Continuous Random Variable
	9.34.7.	Approximating Discrete Random Variables as Continuous
	9.34.8.	One-to-One Transformations of Discrete Random Variables
	9.34.9.	Many-to-One Transformations of Discrete Random Variables

9.35. Distributions of Two Discrete Random Variables

	9.35.1.	Double Summations
	9.35.2.	Joint Distributions for Discrete Random Variables
	9.35.3.	The Joint CDF of Two Discrete Random Variables
	9.35.4.	Marginal Distributions for Discrete Random Variables
	9.35.5.	Conditional Distributions for Discrete Random Variables
	9.35.6.	Independence of Discrete Random Variables

9.36. Expectation of Random Variables

	9.36.1.	Expected Values of Discrete Random Variables
	9.36.2.	Properties of Expectation for Discrete Random Variables
	9.36.3.	Variance of Discrete Random Variables
	9.36.4.	Moments of Discrete Random Variables
	9.36.5.	Properties of Variance for Discrete Random Variables
	9.36.6.	Moments of Continuous Random Variables
	9.36.7.	Expected Values of Continuous Random Variables
	9.36.8.	Variance of Continuous Random Variables
	9.36.9.	The Rule of the Lazy Statistician

10.

Continuous Probability Distributions

10 topics

10.37. The Normal Distribution

	10.37.1.	The Z-Score
	10.37.2.	The Standard Normal Distribution
	10.37.3.	Symmetry Properties of the Standard Normal Distribution
	10.37.4.	The Normal Distribution
	10.37.5.	Mean and Variance of the Normal Distribution
	10.37.6.	Percentage Points of the Standard Normal Distribution
	10.37.7.	Modeling With the Normal Distribution

10.38. The Exponential Distribution

	10.38.1.	The Exponential Distribution
	10.38.2.	Modeling With the Exponential Distribution
	10.38.3.	Mean and Variance of the Exponential Distribution

11.

First-Order Differential Equations

10 topics

11.39. First-Order ODEs

	11.39.1.	Introduction to First-Order Linear ODEs
	11.39.2.	General Solutions of First-Order Linear ODEs
	11.39.3.	Solving First-Order Linear ODEs With Exponential Forcing
	11.39.4.	Solving First-Order Linear ODEs With Sinusoidal Forcing
	11.39.5.	Solving First-Order Linear ODEs Using Integrating Factors
	11.39.6.	Solving First-Order ODEs by Substitution
	11.39.7.	Further Solving First-Order ODEs by Substitution
	11.39.8.	Reducing ODEs to First-Order Linear by Substitution
	11.39.9.	Phase Lines of First-Order ODEs
	11.39.10.	Classifying Equilibrium Solutions of First-Order ODEs

12.

Modeling With First-Order Differential Equations

21 topics

12.40. Applications of First-Order ODEs

	12.40.1.	Modeling With First-Order ODEs
	12.40.2.	Further Modeling With First-Order ODEs
	12.40.3.	Modeling Mixture Problems With First-Order Separable ODEs
	12.40.4.	Modeling Mixture Problems With First-Order Linear ODEs
	12.40.5.	Orthogonal Trajectories

12.41. Growth and Decay Models With First-Order ODEs

	12.41.1.	Exponential Growth and Decay Models With First-Order ODEs
	12.41.2.	Exponential Growth and Decay Models With First-Order ODEs: Calculating Unknown Times and Initial Values
	12.41.3.	Exponential Growth and Decay Models With First-Order ODEs: Half-Life Problems
	12.41.4.	Inhibited Growth Models With First-Order ODEs
	12.41.5.	Inhibited Decay Models With First-Order ODEs
	12.41.6.	Logistic Growth Models With First-Order ODEs
	12.41.7.	Qualitative Analysis of the Logistic Growth Equation
	12.41.8.	Solving the Logistic Growth Equation

12.42. Physical Applications of First-Order ODEs

	12.42.1.	Velocity and Acceleration as Functions of Displacement
	12.42.2.	Determining Properties of Objects Described as Functions of Displacement
	12.42.3.	Falling Body Problems With Linear Drag
	12.42.4.	Falling Body Problems With Quadratic Drag
	12.42.5.	Newton's Law of Universal Gravitation
	12.42.6.	Modeling Escape Velocity With First-Order ODEs
	12.42.7.	Modeling RL Circuits With First-Order ODEs
	12.42.8.	Modeling RC Circuits With First-Order ODEs

13.

Linear Differential Equations

24 topics

13.43. Theory of Solutions to Linear ODEs

	13.43.1.	Differential Operators
	13.43.2.	Linear Differential Operators
	13.43.3.	Introduction to Second-Order Linear ODEs
	13.43.4.	The Superposition Principle
	13.43.5.	Reduction of Order
	13.43.6.	Linear Independence of Solutions to Homogeneous ODEs
	13.43.7.	General Solutions of Linear ODEs
	13.43.8.	Abel's Identity

13.44. Second-Order Linear ODEs With Constant Coefficients

	13.44.1.	Second-Order Homogeneous ODEs: Characteristic Equations With Distinct Real Roots
	13.44.2.	Second-Order Homogeneous ODEs: Characteristic Equations With Repeated Roots
	13.44.3.	Second-Order Homogeneous ODEs: Characteristic Equations With Complex Roots
	13.44.4.	Second-Order Homogeneous Initial Value Problems
	13.44.5.	Second-Order Inhomogeneous ODEs With Polynomial Forcing
	13.44.6.	Second-Order Inhomogeneous ODEs With Exponential Forcing
	13.44.7.	Second-Order Inhomogeneous ODEs With Sinusoidal Forcing

13.45. Modeling With Second-Order ODEs

	13.45.1.	Simple Harmonic Oscillators
	13.45.2.	Damped Oscillators
	13.45.3.	Forced Oscillators
	13.45.4.	Resonance in Vibrating Systems
	13.45.5.	Modeling Capacitor Charge in RCL Circuits With Second-Order ODEs
	13.45.6.	Modeling Current in RCL Circuits With Second-Order ODEs

13.46. Variation of Parameters

	13.46.1.	Variation of Parameters for First-Order Linear ODEs
	13.46.2.	Variation of Parameters for Second-Order ODEs
	13.46.3.	Solving Second-Order ODEs Using Variation of Parameters

Mathematical Methods for the Physical Sciences I

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