# topics on vectors

## I Vector algebra

1. 1.

definition of vector

2. 2.
3. 3.
4. 4.
5. 5.

system of coordinates

6. 6.

basis

7. 7.

coordinate vector

8. 8.
9. 9.

norm (http://planetmath.org/VectorPNorm) (through Pythagoras)

10. 10.
11. 11.
12. 12.
13. 13.

http://planetmath.org/node/6178parallelism condition

14. 14.

http://planetmath.org/node/6178orthogonality condition

15. 15.

vector components and scalar components

16. 16.
17. 17.

area (http://planetmath.org/CrossProduct) of parallelogram

18. 18.

triple scalar product, volume of prism

19. 19.
20. 20.
21. 21.

matrices and determinants

22. 22.

matrices and linear mappings

23. 23.

linear systems and solution methods

## II Vector calculus

1. 1.

definiton of real valued vector function

2. 2.

derivative of vector function

3. 3.

properties of derivative of vector function

4. 4.

derivative of a vector function with constant norm

5. 5.

nabla

6. 6.
7. 7.
8. 8.
9. 9.

differential geometry (http://planetmath.org/ClassicalDifferentialGeometry)

10. 10.

tangent (http://planetmath.org/TangentSpace), normal (http://planetmath.org/NormalVector) and binormal vectors

11. 11.

osculating plane, normal plane and binormal planes

12. 12.

Frenet frame

13. 13.
14. 14.

kinematic method for calculating the radius of curvature (http://planetmath.org/CurvatureOfACurve)

15. 15.

gradient of a scalar function

16. 16.

divergence of a vector function

17. 17.
18. 18.
19. 19.

curl of a vector function

20. 20.
21. 21.
22. 22.

integration of vector functions

23. 23.
24. 24.

tensors and differential forms

25. 25.

covariant differentiation

## III Integral theorems

1. 1.

Gauss theorem

2. 2.
3. 3.
4. 4.

Stokes theorem

5. 5.
6. 6.

Kelvin theorem

7. 7.

## IV Vector advanced topics

1. 1.
2. 2.

tensor notation for a vector

3. 3.

transformation law for a vector

4. 4.

vector fields: Lagrangian and Eulerian description

5. 5.
6. 6.

Jacobians connected with transformation of integration regions

7. 7.
8. 8.
9. 9.

linear transformation spaces

10. 10.
11. 11.
12. 12.

exterior or Grassmann algebra

13. 13.
14. 14.

quaternions

15. 15.
16. 16.

Grassmann-Cayley algebra

17. 17.

vector bundles

18. 18.
19. 19.

spinors

20. 20.

twistors

21. 21.

spin structures

22. 22.

linear programming and the simplex method

23. 23.
24. 24.
25. 25.

K-theory

26. 26.

Category ${\rm Vect}_{\mathbb{R}}$

## V Endomorphism decomposition

1. 1.
2. 2.
3. 3.

eigen-subspaces and invariant subspaces

4. 4.

Hamilton-Cayley theorem (http://planetmath.org/CayleyHamiltonTheorem)

5. 5.
6. 6.

## VI Lie groups and Lie algebras

1. 1.
2. 2.
3. 3.
4. 4.
Title topics on vectors TopicsOnVectors 2013-03-22 15:49:56 2013-03-22 15:49:56 perucho (2192) perucho (2192) 32 perucho (2192) Topic msc 53A45 topic entry on vectors