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View Contravariant and Covariant components of a 3-vector with Mathematica

In the following notebook I will show two transformations of some vector V (RED; you control its coordinates with the x,y,z sliders) which is acted upon by some matrix (it is defined with 3 free parameters to be set by the sliders). Suppose that we apply the matrix to the canonical 3 base vectors (BLACK), obtaining a transformed reference system (PURPLE). What kind of "vector" do we want to deal with?
We have two cases.
1.The vector is contravariant. Its components in the new reference system are equal to those of a vector obtained by taking the product of the Inverse matrix with V .

2. The vector is covariant, which is, its component get transformed in the new reference system by taking the product of the matrix and V.

So contravariant vectors transform with the inverse of the base changing matrix; they are **observer-dependent** vectors. (ORANGE vector) Example: position.
Covariant vectors transform as the base changing matrix; they are **observer-independent **vectors. (BLUE vector) Example: gradient.
In the notebook the sliders are referred to: V=(x,y,z), M=(((a)(0)(0)) , ((0)(b)(0)) , ((0)(0)(c))) and l is a scaling factor.
Feel free to experiment by modifying the matrix! Example:
M=(((a)(a+c)(0)) , ((0)(b-c)(b+c)) , ((0)(0)(c)))

Google Drive folder with downloadable file: https://drive.google.com/folderview?id=0B36YNo_GwIhxb1F4TkVFdEtsY0U&usp=sharingFilename: **covar_contra.nb**

N.B. Always evaluate before using.

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