Does anyone have any idea of how to explain what's actually going on here, im completely lost. Id really love to understand it at last.

 

Vector spaces; metric: Consider R3 and the orthonormal frame (0; ei), i = 1,2, 3.

Let a, b and c be three vectors of that space, with contravariant components in

the basis (ei) given by

ai = (−1,−1, 0), bi = (0, 0,−2) and ci = (0, 1, 2).

(a) Calculate the contravariant components of the vectors a, b and c in the basis e′1 = e2 + e3

e'2 = e1 + e2 + e3

e′3 = −e2.

(b) Calculate the components of the metric tensor in the new basis, as well as

g^1/2 and (g′)^1/2

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Vector spaces; metric: Let a = e1 + e2 and b = e1 + 2e2 be two vectors of R2

where (e1, e2) is an orthonormal basis (i.e. g11 = g22 = 1, g12 = g21 = 0). In the

basis (e ′1 , e ′2 ), these vectors are given by a = e ′1 and b = e ′1 + e ′2 .

Calculate the covariant components g′ij of the metric in the basis (e'1 , e'2) in twodifferent ways

(i) without using the transformation matrix α relating the two bases

(ii) by using the transformation matrix α.

 

thank you in advance

I think the below url can help you out:

http://www.plmsc.psu.edu/~www/matsc597/vectors/index.html