Right-handed, left-handed systems for different coordinate systems, et al

[spaghetti3451]

I know the orientations of the x-, y- and z- axes for a right-handed and a left-handed system. But that's for the cartesian coordinate system. How are the orientations of the coordinate axes for other coordinate systems defined?

Also, i X j = k, j X k = i and k X i = j. How does this apply for the unit vectors along the coordinate axes for cylindrical and spherical coordinate systems?

 

[jedishrfu]

For cylindrical they are orthogonal, ie one is along the radial vector another is tangent to the circle and the third is in the z direction.

Similarly for spherical coordinates, one is along the radial direction, the second is tangential to the sphere and third is also tangential and perpendicular to the second.

 

[chiro]

Hey failexam.

What typically happens for generalized co-ordinate systems is that we have a transformation from Euclidean systems to the actual system itself in the form of what is known as tensor calculus.

Tensor calculus formulates the generalization of geometry for general co-ordinate systems.

Like the euclidean system we have inner products and we have a metric just like we do in the normal euclidean system. We also have a way to go from one system to another in terms of a vector using an operator (i.e. think of it as a matrix) where we can take one vector and then get the same vector in another system.

It's the same kind of thing intuitively as if you have x and you wish to apply f(x) to get a value under the transformation of f. If an inverse for f exists you can then apply f_inverse(f(x)) = x to get back x (assuming f inverse exists for some x).

If you recall in linear algebra, we use the idea of inverses to do the above and not surprisingly inverses tell us when we can do the above kind of transformations between co-ordinate systems.

 

[spaghetti3451]

Thanks for th replies. But I guess the replies do not answer the question.

All I am asking for is a picture of right- and left- handedness for cylindrical and spherical coordinate systems and the cross products of orthogonal unit vectors in those coordinate systems?

For example, cylindrical right-handed system: r points away from the origin along the xy plane, phi points counterclockwise and z points upwards. ?

For example, cylindrical left-handed system: r points away from the origin along the xy plane, phi points clockwise and z points upwards. ?

For example, r X phi = z ? If so, why?

I don't know if these are the right answers but I am just trying to work out this type of answer to the above questions.

 

[spaghetti3451]

Actually, i think i figured out the answer.

The formula for right-handedness is here: http://en.wikipedia.org/wiki/Unit_vector#Curvilinear_coordinates

and the unit vectors are orthogonal, so the usual rules for multiplying orthogonal vectors in the cross product apply.

 

[chiro]

Actually, i think i figured out the answer.

The formula for right-handedness is here: http://en.wikipedia.org/wiki/Unit_vector#Curvilinear_coordinates

and the unit vectors are orthogonal, so the usual rules for multiplying orthogonal vectors in the cross product apply.

Look at the sign of the determinant of the matrix for its orientation: +1 for positive oriented and -1 for negatively oriented.