Curvilinear Coordinates - Generalization To n Dimensions

Generalization To n Dimensions

The formalism extends to any finite dimension as follows.

Consider the real Euclidean n-dimensional space, that is Rn = R × R × ... × R (n times) where R is the set of real numbers and × denotes the Cartesian product, which is a vector space.

The coordinates of this space can be denoted by: x = (x₁, x₂,...,x_n). Since this is a vector (an element of the vector space), it can be written as:

where e1 = (1,0,0...,0), e2 = (0,1,0...,0), e3 = (0,0,1...,0),...,en = (0,0,0...,1) is the standard basis set of vectors for the space Rn, and i = 1, 2,...n is an index labelling components. Each vector has exactly one component in each dimension (or "axis") and they are mutually orthogonal (perpendicular) and normalized (has unit magnitude).

More generally, we can define basis vectors b_i so that they depend on q = (q₁, q₂,...,q_n), i.e. they change from point to point: b_i = b_i(q). In which case to define the same point x in terms of this alternative basis: the coordinates with respect to this basis v_i also necessarily depend on x also, that is v_i = v_i(x). Then a vector v in this space, with respect to these alternative coordinates and basis vectors, can be expanded as a linear combination this basis (which simply means to multiply each basis vector e_i by a number v_i – scalar multiplication):

The vector sum that describes v in the new basis is composed of different vectors, although the sum itself remains the same.