vectors and tensors

Recommended Posts

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

-------------------------------------------

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.

Create an account

Register a new account