Editing Curvilinear coordinates (section)

==Generalization to ''n'' dimensions==

The formalism extends to any finite dimension as follows.

Consider the [[real number|real]] [[Euclidean space|Euclidean]] ''n''-dimensional space, that is '''R'''<sup>''n''</sup> = '''R''' × '''R''' × ... × '''R''' (''n'' times) where '''R''' is the [[set (mathematics)|set]] of [[real numbers]] and × denotes the [[Cartesian product]], which is a [[vector space]].

The [[coordinates]] of this space can be denoted by: '''x''' = (''x''<sub>1</sub>, ''x''<sub>2</sub>,...,''x<sub>n</sub>''). Since this is a vector (an element of the vector space), it can be written as:

:<math> \mathbf{x} = \sum_{i=1}^n x_i\mathbf{e}^i </math>

where '''e'''<sup>1</sup> = (1,0,0...,0), '''e'''<sup>2</sup> = (0,1,0...,0), '''e'''<sup>3</sup> = (0,0,1...,0),...,'''e'''<sup>''n''</sup> = (0,0,0...,1) is the ''[[standard basis]] set of vectors'' for the space '''R'''<sup>''n''</sup>, 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 vector|orthogonal]] ([[perpendicular]]) and normalized (has [[unit vector|unit magnitude]]).

More generally, we can define basis vectors '''b'''<sub>''i''</sub> so that they depend on '''q''' = (''q''<sub>1</sub>, ''q''<sub>2</sub>,...,''q<sub>n</sub>''), i.e. they change from point to point: '''b'''<sub>''i''</sub> = '''b'''<sub>''i''</sub>('''q'''). In which case to define the same point '''x''' in terms of this alternative basis: the ''[[coordinate vector|coordinates]]'' with respect to this basis ''v<sub>i</sub>'' also necessarily depend on '''x''' also, that is ''v<sub>i</sub>'' = ''v<sub>i</sub>''('''x'''). Then a vector '''v''' in this space, with respect to these alternative coordinates and basis vectors, can be expanded as a [[linear combination]] in this basis (which simply means to multiply each basis [[Coordinate vector|vector]] '''e'''<sub>''i''</sub> by a number ''v''<sub>''i''</sub> – [[scalar multiplication]]):

:<math> \mathbf{v} = \sum_{j=1}^n \bar{v}^j\mathbf{b}_j = \sum_{j=1}^n \bar{v}^j(\mathbf{q})\mathbf{b}_j(\mathbf{q}) </math>

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