Question about relating basis of curvilinear coordinate systems.

[Group_Complex]

Wikipedia gives the relationship between a cartesian and curvlinear coordinate system as
gi=(partial)x1/(partial)zi +(partial)x2/(partial)zi
http://en.wikipedia.org/wiki/Curvilinear_coordinates

Where gi is the i'th basis in the curvlinear coordinate system, x1 and x2 are the cartesian coordinates, zi, is the i'th curvlinear coordinate. Taking i as 1 or 2 in the two dimensional case.

I am having trouble deriving a proof for why this is true. Despite being a first year mathematics student i still feel that this is something within my grasp (I am currently studying vector calculus). Could someone please provide me with some geometric intuition for this identity?

 

[pat_connell]

The easiest way to think about this relationship is by visualizing it graphically. Consider a two-dimensional Cartesian coordinate system (x1, x2). In this system, a point P can be represented by its coordinates (x1, x2). Now imagine that you want to represent the same point using a curvilinear coordinate system (z1, z2). To do this, you first need to define the basis vectors g1 and g2 of the curvilinear coordinate system. These basis vectors are the unit vectors along the two axes of the curvilinear coordinate system. Now to represent point P in the curvilinear coordinate system, you need to determine how much of each basis vector g1 and g2 is required to reach the point P. To do this, you can draw lines from the origin of the curvilinear coordinate system to the point P, such that each line is parallel to one of the basis vectors. The lengths of these lines represent the amount of each basis vector required to reach the point P. These lengths can then be calculated using the relationship given in the question, where the partial derivatives of x1 and x2 with respect to z1 and z2 respectively give the amounts of g1 and g2 required to reach point P. This is why the relationship between the cartesian and curvilinear coordinate systems is as stated in the question.