Vector Calculus
Year 2 · 6 sections · 56 concepts · 129 practice questions
What's covered
free1 · Curves and surfaces10 concepts
- Overview 1 · The plan of this section
- Definition 1.1 · Regular parametrised curves
- Proposition 1.2 · Arc length and the arc-length parametrisation
- Definition 1.3 · Curvature
- Proposition 1.4 · Curvature in any parametrisation
- Proposition 1.5 · Frenet–Serret formulas
- Definition 1.6 · Parametrised surfaces
- Proposition 1.7 · Normals to a surface
- Proposition 1.8 · The area element
- Check 1.9 · Check: curves and surfaces
2 · Fields and operators9 concepts
- Overview 2 · The plan of this section
- Definition 2.1 · Grad, div, curl and the Laplacian
- Definition 2.2 · Summation convention, Kronecker delta and Levi-Civita symbol
- Proposition 2.3 · The ε–δ identity
- Proposition 2.4 · Vector identities
- Proposition 2.5 · Curl of a curl
- Definition 2.6 · Irrotational and solenoidal fields
- Proposition 2.7 · Potentials force irrotational or solenoidal fields
- Check 2.8 · Check: index-notation proofs
3 · Line integrals10 concepts
- Overview 3 · The plan of this section
- Definition 3.1 · Line integrals
- Proposition 3.2 · Line integrals and reparametrisation
- Theorem 3.3 · Fundamental theorem for line integrals
- Definition 3.4 · Conservative fields
- Theorem 3.5 · Three descriptions of conservative fields
- Definition 3.6 · Simply connected regions
- Theorem 3.7 · Curl-free fields on simply connected regions are conservative
- Example 3.8 · The vortex field
- Check 3.9 · Check: is this field conservative?
4 · Surface and volume integrals8 concepts
- Overview 4 · The plan of this section
- Definition 4.1 · Scalar surface integrals
- Definition 4.2 · Orientable surfaces
- Definition 4.3 · Flux integrals
- Proposition 4.4 · Flux through graphs and spheres
- Theorem 4.5 · Change of variables in three dimensions
- Example 4.6 · Cylindrical and spherical volume elements
- Check 4.7 · Check: surface and volume integrals
5 · Integral theorems10 concepts
- Overview 5 · The plan of this section
- Theorem 5.1 · Green's theorem
- Corollary 5.2 · Area as a line integral
- Theorem 5.3 · Divergence theorem
- Corollary 5.4 · Divergence as flux per unit volume
- Theorem 5.5 · Stokes' theorem
- Corollary 5.6 · Curl as circulation per unit area
- Example 5.7 · Gauss's law for an inverse-square field
- Proposition 5.8 · The continuity equation
- Check 5.9 · Check: using the integral theorems
6 · Curvilinear coordinates9 concepts
- Overview 6 · The plan of this section
- Definition 6.1 · Orthogonal curvilinear coordinates and scale factors
- Example 6.2 · Scale factors for cylindrical and spherical coordinates
- Proposition 6.3 · Gradient in orthogonal coordinates
- Proposition 6.4 · Divergence in orthogonal coordinates
- Proposition 6.5 · Curl in orthogonal coordinates
- Corollary 6.6 · The Laplacian in orthogonal coordinates
- Example 6.7 · Radially symmetric harmonic functions
- Check 6.8 · Check: operators in curvilinear coordinates
Try it: a free card from this module
This is exactly what studying looks like inside UniMaths.
Proposition 1.2 · Arc length and the arc-length parametrisation
, parametrisation-independent; unit-speed parametrisation by .
Proof
Learn all 56 concepts like this
Cards, step-by-step proofs, and 129 questions scheduled so you review right before you forget.