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.
A version of these notes appeared as a series of appendices in a textbook on electromagnetism.
Curves
Tangent vectors and arc lengths, curvature and torsion; Line integrals. Conservative fields and the gradient.
Surfaces (and Volumes)
Area integrals and volume integrals, Jacobians, spherical and cylindrical polar coordinates. Flux. The Gauss-Bonnet theorem.
Grad, Div and Curl
The gradient, div, curl; conservative, irrotational and solenoidal fields; the Laplacian. Orthogonal curvilinear coordinates, spherical polar coordinats, cylindrical polar coordinates.
The Integral Theorems
The divergence theorem, conservation laws. Green's theorem in the plane. Stokes' theorem.
Some Vector Calculus Equations
Gravity and electrostatics, Gauss' law and potentials. The Poisson equation and the Laplace equation. Special solutions and the Green's function.
Tensors
Tensor transformation law, maps, and invariant tensors. Tensor fields. A unification of integral theorems.
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