teori dasar. (1)

Well,, ternyata teteup masih lebih enakeun blogger daripada multiply.. Asa rariweuh multiply maah.. Tapi kelebihannya di multiply bisa nge-attach file.. Termasuk .mp3 n .pdf. Itu penting..

Hah sudah2.. Mulai coba bikin isi tesis. Tgl 21 nilai dah harus masuk. Gawat. Gw rada pesimis apa bisa sidang 1 sebelum tgl 21. Hiks..

Where do we start? Firstly, it will be good if we start with explaining covariant and contra-variant vectors..

To construct a real scalar function from several vectors, these vectors must be operated (dioperasikan?) by a tensor. One ways for operating a vector with tensor is to multiplied each vector components with its covariant. Contravariant vector (usually simply called 'vectors') is a type of vector which transform exactly the same with a coordinate transformation. Suppose a coordinate transformation from O to O' below:

(ada persamaan..)

For transformation in differential form, equation (diatas) become:

(persamaan lagi)

Thus, a contravariant vectors transform as:

(persamaan)

A gradient of a scalar function is obtained from:

(persamaan)

Then, a different kind of vectors which transform using the form of equation (diatas) is called a covariant vector.
A second order covariant tensor (with 16 components) transform as:

(persamaan)

while its covariant transforms as:

(persamaan)

Thus, a combined-second order tensor will transform as:
(persamaan).

An inner-product of 2 vector is defined as a product of its covariant and contravariant:

(persamaan)

It can be shown that the inner-product of 2 vector is invariant by coordinate transformation.

(persamaan)

The relation between contravariant and covariant vector are:

(persamaan)

with g describe the metric of the space (manifold).
As an example, in Minkowski spacetime, its metric will have a form of:

(persamaan)

and the metric inverse is:

(persamaan)

haaaah.. istirahat duluu.. :D