topics on vectors
I Vector algebra
- 1.
definition of vector
- 2.
vector space
- 3.
parallelogram principle, median vector, difference of vectors
- 4.
geometric applications: mid-segment theorem, common point of triangle medians, median of trapezoid
- 5.
system of coordinates
- 6.
basis
- 7.
coordinate vector
- 8.
position vector
- 9.
norm (http://planetmath.org/VectorPNorm) (through Pythagoras)
- 10.
unit vector
- 11.
direction cosines
- 12.
dot product
- 13.
http://planetmath.org/node/6178parallelism
condition
- 14.
http://planetmath.org/node/6178orthogonality condition
- 15.
vector components and scalar components
- 16.
cross product
- 17.
area (http://planetmath.org/CrossProduct) of parallelogram
- 18.
triple scalar product, volume of prism
- 19.
triple cross product
- 20.
distance of non-parallel lines (an application)
- 21.
matrices and determinants
- 22.
matrices and linear mappings
- 23.
linear systems and solution methods
II Vector calculus
- 1.
definiton of real valued vector function
- 2.
derivative of vector function
- 3.
properties of derivative of vector function
- 4.
derivative of a vector function with constant norm
- 5.
nabla
- 6.
cylindrical coordinates
- 7.
polar coordinates
- 8.
spherical coordinates
- 9.
differential geometry
(http://planetmath.org/ClassicalDifferentialGeometry)
- 10.
tangent
(http://planetmath.org/TangentSpace), normal (http://planetmath.org/NormalVector) and binormal vectors
- 11.
osculating plane
, normal plane
and binormal planes
- 12.
Frenet frame
- 13.
Frenet-Serret equations
- 14.
kinematic method for calculating the radius of curvature
(http://planetmath.org/CurvatureOfACurve)
- 15.
gradient
of a scalar function
- 16.
divergence
of a vector function
- 17.
solenoidal field
- 18.
vector potential
- 19.
curl of a vector function
- 20.
irrotational field
, lamellar field
- 21.
Helmholtz decomposition
- 22.
integration of vector functions
- 23.
line integral
- 24.
tensors and differential forms
- 25.
covariant differentiation
III Integral theorems
- 1.
Gauss theorem
- 2.
solid angle
- 3.
Green theorems
- 4.
Stokes theorem
- 5.
circulation and vorticity
- 6.
Kelvin theorem
- 7.
Helmholtz theorems
IV Vector advanced topics
- 1.
alternate characterization of curl
- 2.
tensor notation for a vector
- 3.
transformation law for a vector
- 4.
vector fields
: Lagrangian and Eulerian description
- 5.
motion of continuum
- 6.
Jacobians
connected with transformation of integration regions
- 7.
Reynolds transport theorem
- 8.
rotations
- 9.
linear transformation spaces
- 10.
linear functionals
or covectors
- 11.
bivectors
- 12.
exterior or Grassmann algebra
- 13.
Clifford algebra
- 14.
quaternions
- 15.
projective geometry
- 16.
Grassmann-Cayley algebra
- 17.
vector bundles
- 18.
connections
- 19.
spinors
- 20.
twistors
- 21.
spin structures
- 22.
linear programming and the simplex method
- 23.
representation theory
- 24.
linear extension
- 25.
K-theory
- 26.
Category
V Endomorphism decomposition
- 1.
eigenvalues
, eigenvectors
- 2.
characteristic
and minimal polynomials
- 3.
eigen-subspaces and invariant subspaces
- 4.
Hamilton-Cayley theorem (http://planetmath.org/CayleyHamiltonTheorem)
- 5.
Jordan blocks
and canonical decomposition
- 6.
singular value decomposition
VI Lie groups and Lie algebras
- 1.
the connection between Lie groups and Lie algebras
- 2.
commutators or Lie bracket
- 3.
matrix groups and algebras
- 4.
Pauli matrices