Calculus — Intermediate

Vector Calculus and Differential Equations (University Years 3–4)

Overview

The intermediate level covers the theory of vector calculus and ordinary differential equations. It ranges from vector differential operators such as gradient, divergence, and curl through systematic solution methods for differential equations.

Learning Objectives

  • Master the concepts and computation of gradient, divergence, and curl
  • Understand vector fields and the differential geometry of curves and surfaces
  • Learn systematic solution methods for ordinary differential equations
  • Understand the theory of linear differential equations
  • Get acquainted with series solution methods and special functions
  • Understand the foundations of dynamical systems

Table of Contents

  1. Chapter 1 Basics of Vector Calculus

    Gradient, divergence, curl, the nabla operator

  2. Chapter 2 Differential Geometry of Curves and Surfaces

    Curvature, torsion, the Frenet–Serret formulas, fundamental forms of surfaces

  3. Chapter 3 Second-Order Linear ODEs

    Constant coefficients, characteristic equations, inhomogeneous equations

  4. Chapter 4 Series Solutions of ODEs

    Power-series solutions, singular points, the Frobenius method

  5. Chapter 5 Systems of ODEs and Dynamical Systems

    The matrix exponential, phase portraits, stability analysis

  6. Chapter 6 Exercises

    Comprehensive exercises for the intermediate level

Prerequisites

  • The content of Basic Calculus
  • Linear algebra (eigenvalues, the matrix exponential)
  • Multivariable integration (multiple integrals, line integrals)

Frequently Asked Questions

What does Intermediate Calculus cover?

The intermediate level covers vector calculus (gradient, divergence, curl, the divergence theorem, and Stokes' theorem), the differential geometry of curves and surfaces (curvature, the Frenet formulas, Gaussian curvature), second-order linear ordinary differential equations (characteristic equations, power-series solutions, Bessel's equation), and systems of ODEs together with phase-plane analysis.

What is the difference between a line integral and a surface integral of a vector field?

The line integral $\int_C \mathbf{F}\cdot d\mathbf{r}$ computes the work done by a vector field $\mathbf{F}$ along a curve $C$. The surface integral $\iint_S \mathbf{F}\cdot d\mathbf{S}$ computes the flux through a surface $S$. Stokes' theorem converts a line integral into a surface integral, and the divergence theorem converts a surface integral into a volume integral.

What kind of concept is the Jacobian (Jacobian matrix)?

The Jacobian matrix of a map $\mathbf{f}:\mathbb{R}^m \to \mathbb{R}^n$ is the $n\times m$ matrix of partial derivatives $J_{ij}=\partial f_i/\partial x_j$. It represents the change in area or volume ($|\det J|$) in integrals under a change of variables, and it underlies the multivariable inverse function theorem and the implicit function theorem.

Essays

  • The Many Faces of the Derivative [Essay] — gradient, Jacobian matrix, operator, backpropagation. Faces that look scattered all connect to a single core: the best linear approximation.
  • Capturing Functions as Polynomials [Essay] — sine and the exponential are, inside the machine, nothing but addition and multiplication. The story of the Taylor expansion, where higher-order derivatives become "coefficients of the future."
  • A Tale of Swirls and Sources [Essay] — why does the wind swirl and water get drawn into a drain? A relaxed tour of the "habits of flow" that field derivatives — divergence and curl — read off, from fluid intuition up to Maxwell's grand theorems.