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

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