David Tong: Lectures on Vector Calculus

These lectures are aimed at first year undergraduates. They describe the basics of div, grad and curl and various integral theorems. The lecture notes are around 120 pages. Please do email me if you find any typos or mistakes.

PDF

Content

• 1. Curves:   PDF
  Tangent vectors and arc lengths, curvature and torsion; Line integrals. Conservative fields and the gradient.
• 2. Surfaces (and Volumes):   PDF
  Area integrals and volume integrals, Jacobians, spherical and cylindrical polar coordinates. Flux. The Gauss-Bonnet theorem.
• 3. Grad, Div and Curl:   PDF
  The gradient, div, curl; conservative, irrotational and solenoidal fields; the Laplacian. Orthogonal curvilinear coordinates, spherical polar coordinats, cylindrical polar coordinates.
• 4. The Integral Theorems:   PDF
  The divergence theorem, conservation laws. Green's theorem in the plane. Stokes' theorem.
• 5. Some Vector Calculus Equations:   PDF
  Gravity and electrostatics, Gauss' law and potentials. The Poisson equation and the Laplace equation. Special solutions and the Green's function.
• 6. Tensors:   PDF
  Transformation law, maps, and invariant tensors. Tensor fields. A unification of integral theorems.

Problem Sheets

• Problem Sheet 1:   PDF    Curves and Surfaces. Line and Area Integrals.


• Problem Sheet 2:   PDF    Grad, Div and Curl. The Divergence Theorem.


• Problem Sheet 3:   PDF    Green's Theorem, Stokes' Theorem. Vector Calculus Equations.   


• Problem Sheet 4:   PDF    Feeling tenser.

Other Lecture Notes on the Web

• Vector Calculus   previous lecture notes by Ben Allanach and Jonathan Evans   


• Vector Calculus   yet earlier lecture notes by Stephen Cowley. Be prepared to draw your own figures!


• Vector Calculus   by Matthias Dorrzapf.

Some History

• Remarks on the Mathematical Classification of Physical Quantities   James Clerk Maxwell proposing the names "Slope", "Convergence" and "Curl"

"I shall conclude by proposing for the consideration of mathematicians certain phrases... I should be greatly obliged to anyone who can give me suggestions on this subject, as I feel that the onomastic power is very faint in me."