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Wyatt's Notes — University

Multivariable Calculus

Multivariable Calculus

Multivariable calculus extends the differential and integral calculus to functions of several real variables. The partial derivative generalises the ordinary derivative; the multiple integral generalises the definite integral; and the three integral theorems of vector calculus (Green, Stokes, and the Divergence Theorem) connect the local, differential behaviour of a field to its global, integral behaviour.

The central objects are scalar and vector fields on subsets of Euclidean space, and the central tools are the gradient, divergence, curl, Jacobian, and Hessian. Each is defined precisely, its coordinate expression is derived, and its geometric meaning is stated. The treatment is coordinate-aware: where a result holds independent of the choice of coordinates, that invariance is proved rather than assumed.

Contents

  1. Partial Derivatives
  2. Multiple Integrals
  3. Vector Calculus
  4. Optimization
  5. Curves and Surfaces
  6. Problem Set

Overview

University-level multivariable calculus notes covering partial derivatives, vector calculus, and optimisation.

Topics Covered

  • Partial Derivatives: Chain rule, gradient, directional derivatives
  • Multiple Integrals: Double and triple integrals, change of variables
  • Vector Calculus: Green”s theorem, Stokes’ theorem, divergence theorem
  • Optimisation: Local extrema, Lagrange multipliers, Hessian

Prerequisites

  • Single-variable calculus (differentiation, integration)
  • Linear algebra (vectors, matrices)
  • Basic topology (open sets, continuity)
  • Mathematical proofs and logic

How to Use These Notes

Start with partial derivatives to build foundational knowledge, then progress to vector calculus and optimisation. Each section includes worked examples and practice problems.

Use the sidebar to browse topics, or start with the introductory pages linked from the sidebar.

Additional Resources

Each section includes:

  • Detailed explanations of key concepts
  • Worked examples with step-by-step solutions
  • Practice problems with answers
  • Common pitfalls and how to avoid them
  • Connections to other areas of mathematics

Study Tips

  1. Master the definitions: Multivariable calculus requires precise understanding of derivatives and integrals
  2. Practise proofs: Learn to write clear, rigorous proofs
  3. Draw diagrams: Visualise surfaces, curves, and vector fields
  4. Learn standard examples: Know the properties of common surfaces (planes, spheres, paraboloids)
  5. Connect to physics: Relate multivariable calculus to electromagnetism and fluid dynamics