topics on vectors

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/6178

parallelism

condition

14.

http://planetmath.org/node/6178

orthogonality 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

Vect

ℝ

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

Title

topics on vectors

Canonical name

TopicsOnVectors

Date of creation

2013-03-22 15:49:56

Last modified on

2013-03-22 15:49:56

Owner

perucho (2192)

Last modified by

perucho (2192)

Numerical id

32

Author

perucho (2192)

Entry type

Topic

Classification

msc 53A45

Synonym

topic entry on vectors

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