Math Notes

Lesson Objectives ▼ Understand the concept of vector functions and their derivatives. Learn about partial derivatives and the gradient of a function. Compute divergence and curl of a vector field. Evaluate multiple integrals (double and triple integrals). Apply line and surface integrals. Understand the fundamental theorems of vector calculus. Lesson Outline ▼ Definition of Vector Functions Partial Derivatives and Gradient Divergence and Curl Multiple Integrals: Double and Triple Integrals Line and Surface Integrals Fundamental Theorems of Vector Calculus Examples Definition of Vector Functions A **vector function** assigns a vector to each point in space. It is written as: \[ \mathbf{r}(t) = x(t) \mathbf{i} + y(t) \mathbf{j} + z(t) \mathbf{k} \] where \( x(t), y(t), z(t) \) are functions of \( t \). Partial Derivatives and Gradient If ...

Lesson Objectives ▼ Understand the concept of parametric equations and their applications. Convert between parametric and Cartesian equations. Learn the fundamentals of polar coordinates. Convert between polar and Cartesian coordinates. Perform calculus operations on parametric and polar equations. Lesson Outline ▼ Definition of Parametric Equations Eliminating the Parameter Introduction to Polar Coordinates Conversion between Polar and Cartesian Coordinates Calculus with Parametric and Polar Equations Examples Definition of Parametric Equations A curve in the plane can be represented by a set of **parametric equations**: \[ x = f(t), \quad y = g(t), \quad t \text{ is the parameter} \] For example, the parametric equations: \[ x = \cos t, \quad y = \sin t, \quad 0 \leq t \leq 2\pi \] represent a unit circle. Eliminating the Parameter To convert from ...

Lesson Objectives ▼ Understand the concept of sequences and their limits. Learn the definition of infinite series and when they converge. Apply common convergence tests for series. Explore power series and Taylor series. Lesson Outline ▼ Definition of Sequences Limits of Sequences Definition of Series Convergence Tests Power Series and Taylor Series Examples Definition of Sequences A **sequence** is an ordered list of numbers generated by a rule. A sequence is usually written as: \[ a_1, a_2, a_3, \dots, a_n, \dots \] For example, the sequence \( a_n = \frac{1}{n} \) is: \[ 1, \frac{1}{2}, \frac{1}{3}, \frac{1}{4}, \dots \] Limits of Sequences The **limit** of a sequence \( a_n \) is defined as: \[ \lim_{n \to \infty} a_n = L \] If this limit exists, the sequence **converges** to \( L \); otherwise, it diverges. Definition of Series A **series** is the su...

Lesson Objectives ▼ Understand the least squares method. Learn how to solve overdetermined systems. Derive the least squares solution mathematically. Apply the method to regression and optimization problems. Lesson Outline ▼ Definition of Least Squares Derivation of Least Squares Solution Solving Overdetermined Systems Applications of Least Squares Examples Definition of Least Squares The **least squares method** is a technique used to find the best approximation to an overdetermined system (more equations than unknowns). For a system \( A\mathbf{x} = \mathbf{b} \), where \( A \) is \( m \times n \) with \( m > n \), an exact solution may not exist. The least squares method finds \( \mathbf{x} \) that minimizes: \[ \| A\mathbf{x} - \mathbf{b} \| \] Derivation of Least Squares Solution To minimize \( \| A\mathbf{x} - \mathbf{b} \|^2 \), differentiate with respect ...

Lesson Objectives ▼ Understand the concept of Singular Value Decomposition (SVD). Decompose a matrix into orthogonal matrices and singular values. Learn how to compute SVD step by step. Explore applications of SVD in data compression and machine learning. Lesson Outline ▼ Definition of SVD Computing SVD Step by Step Applications of SVD Examples Definition of Singular Value Decomposition The **Singular Value Decomposition (SVD)** of an \( m \times n \) matrix \( A \) is a factorization of the form: \[ A = U \Sigma V^T \] where: \( U \) is an \( m \times m \) orthogonal matrix (left singular vectors). \( \Sigma \) is an \( m \times n \) diagonal matrix with **singular values** on the diagonal. \( V \) is an \( n \times n \) orthogonal matrix (right singular vectors). Computing SVD Step by Step To compute \( A = U \Sigma V^T \): Find the eigenvalues and eige...

Lesson Objectives ▼ Understand the definition of eigenvalues and eigenvectors. Learn how to compute eigenvalues and eigenvectors. Explore the characteristic equation. Interpret eigenvalues and eigenvectors in transformations. Understand diagonalization and its applications. Lesson Outline ▼ Definition of Eigenvalues and Eigenvectors Characteristic Equation Diagonalization Examples Definition of Eigenvalues and Eigenvectors In a linear transformation, an eigenvector is a nonzero vector that only changes by a scalar factor when the transformation is applied. For a square matrix \( A \), an eigenvector \( \mathbf{v} \) and its corresponding eigenvalue \( \lambda \) satisfy: \[ A\mathbf{v} = \lambda \mathbf{v} \] \( A \) is an \( n \times n \) matrix. \( \mathbf{v} \neq 0 \) is an eigenvector. \( \lambda \) is a scalar eigenvalue. Characteristic Equation ...

Lesson Objectives ▼ Understand the definition of inner product spaces. Compute norms, angles, and distances in inner product spaces. Explore the concept of orthogonality. Learn the Gram-Schmidt orthogonalization process. Use inner products for projections and least squares approximation. Lesson Outline ▼ Definition of Inner Product Spaces Norms, Angles, and Orthogonality Gram-Schmidt Orthogonalization Projections and Least Squares Examples Definition of Inner Product Spaces An inner product space is a vector space with an operation called the **inner product**, which measures similarity between vectors. For a real vector space, the inner product of two vectors \( \mathbf{u}, \mathbf{v} \) is: \[ \langle \mathbf{u}, \mathbf{v} \rangle = u_1 v_1 + u_2 v_2 + \dots + u_n v_n \] For a complex vector space, the inner product is: \[ \langle \mathbf{u}, \mathbf{v} \ran...