Understanding Metric Tensor in Higher Dimensions

[rbt]

Hi,

This might sound a very basic question to most of you all. But could you kindly give me some information on what eactly is a Metric Tensor and what is its significance in the higher dimension study?
(Wiki or Google info seemed too cryptic. Thus...I ask you)

Thank You

 

[lalbatros]

It allows you to calculate (short) distances or lenghts.

If v is a (small) displacement vector,
the length of the displacement is d² = v'.g.v
where v' is the transpose of v,
and g is the metric tensor,
and "." is the multiplication.

In coordinates notations d²=vi gij vj.

Go to a textbook and look for the sphere in spherical coordinates,
you will find one of the simplest metric tensor.