Solutions To Differential Geometry Problems (Part 1)

 1) Consider two vectors A, B spanning a subspace of   R ⁴   where:

A = (1, 2, 1, 0) and B = (1, 2, 3, 1)  

Find:  A/ ‖A ‖    and:  B/ ‖A‖

Solution: For A we have the orthonormal basis:

A/ ‖A ‖ = (1, 2, 1, 0)/ Ö{1 ²  +  2 ²  + 1 ²  +  0 ² } =

(1, 2, 1, 0)/ Ö 6

For B we have:

B/ ‖A‖ = (1, 2, 3, 1)/ Ö {1 ² + 2 ² +  3 ² + 1 ²} =

 

(1, 2, 3, 1)/ Ö 15

2)  Given  x'_{i =    x i  +   b i}

And:  x _i   =

(2….1…..0)

(3….-5.…6)

(-7….0… 4)

Find the new coordinates x'_i   if :

  _b i  =

(1….2…..1)

(2….1.….3)

(3….2… .1)

Write out the matrix elements for  a _ik .

Solutions:

The key to the solution is noting - from the original post text: 

A(= a _ik ) equals the Kronecker delta (unit matrix) so then the new Cartesian coordinate system is given by:

x'_{i =    x i  +   b i}             i=  1, 2, 3.....

Then we know immediately the matrix for a _ik is:

(1….0…..0)

(0….1…..0)

(0….0… .1)

And the solution follows from simple Mathcad matrix operations summarized below:

For which the final - new - coordinates after transformation become:

(3….3…..1)

(3….-4.…9)

(-4….2… 5)