If given a position vector defined for a orthogonal curvilinear coordinate system HOW would the matrices that make up the Levi Civita 3x3x3 matrix remain the same?(adsbygoogle = window.adsbygoogle || []).push({});

"Levi Civita 3x3x3 is said to be independent of any coordinate system or metric tensor"(https://en.wikipedia.org/wiki/Levi-Civita_symbol)

Position Vector in a cartesian coordinate system[x;y;z] = x*

R =i+ y*j+ z*k= x*[1;0;0] + y*[0;1;0] + z*[0;0;1]

Cartesian Unit Vectors

i =[1;0;0],j =[0;1;0],k =[0;0;1]

Angular Velocity Vector[w1;w2;w3]

w =

Velocity Vectorcross(

V =w,R) =wxR=CPM(w)*R

Cross Product Matrix (CPM) Derivation

Matrices that make up each 'page' of the 3x3x3 alternating tensor/symbol/Levi-Civita symbol

I=j*transpose(k) -k*transpose(j) = [0 0 0; 0 0 1; 0 -1 0]

J=-(i*transpose(k) -k*transpose(i)) = [0 0 -1; 0 0 0; 1 0 0]

K=i*transpose(j) -j*transpose(i) = [0 1 0; -1 0 0; 0 0 0]

LeviCivita here is a 3x3x3 matrix

(all rows, all columns, page1) =LeviCivitaI

(all rows, all columns, page2)LeviCivita= J

(all rows, all columns, page3) =LeviCivitaK

CPM= transpose(w)*LeviCivita

= [transpose(w)*(all rows, all columns, page1) ; transpose(LeviCivitaw)*(all rows, all columns, page2); ......transpose(LeviCivitaw)*(all rows, all columns, page3) ]LeviCivita

= [transpose(w)*I; transpose(w)*J; transpose(w)*K]

[ 0 -w3 w2 ]

[ w3 0 -w1 ] =CPM(w)

[ -w2 w1 0 ]

And, as expectedV=wxR=CPM(w)*R= [w2*z - w3*y; w3*x - w1*z; w1*y - w2*x]

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Again how would the Levi Civita 3x3x3 matrix remain the same in an orthogonal curvilinear coordinate system?

For example given the position vector defined in spherical coordinates. Coordinates: r = radius, o = theta, p = phi

R= [r*cos(o)*sin(p); r*sin(o)*sin(p); r*cos(p)] = r*normalized_ris the normalized unit vector for the r direction

normalized_r

Should one assume the following to arrive at the matrices that form the Levi Civita symbol:

r=o*transpose(p) -o*transpose(p)

o=-(r*transpose(p) -p*transpose(r))

p=r*transpose(o) -o*transpose(r)

where the unit vectors are found from:

r = gradient(x) = [partial_dx/partial_dr; partial_dx/partial_do; partial_dx/partial_dp]

o = gradient(y) = [partial_dy/partial_dr; partial_dy/partial_do; partial_dy/partial_dp]

p = gradient(z) = [partial_dz/partial_dr; partial_dz/partial_do; partial_dz/partial_dp]

OR Should one use the normalized unit vectors to form the matrices instead?

r=normalized_o*transpose(normalized_p) -normalized_o*transpose(normalized_o)

o=-(normalized_r*transpose(normalized_p) -normalized_p*transpose(normalized_r))

p=normalized_r*transpose(normalized_o) -normalized_o*transpose(normalized_r)

None of these resulted in the matrices that comprised the Levi Civita 3x3x3 matrix.

This spherical coordinate system is not right handed. Should one change the order from r, o, p to r, p, o?

https://math.stackexchange.com/ques...lculating-dot-and-cross-products-in-spherical

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# A Cross Product Matrix

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