An oblique basis has

PHYSICS · Multi-part · Fermi University Physics · Mathematical Methods for Physics · Tensors and differential geometry · Tensors and differential geometry

Invariance, Ch General bases, the metric and tensors in spacetime

Problem

|\mathbf{e}_1| = 3.0\,\text{\AA}

|\mathbf{e}_2| = 4.0\,\text{\AA}

and an angle of

75^\circ

between them, so that

g_{11} = 9.0\,\text{\AA}^2

g_{22} = 16.0\,\text{\AA}^2

and

g_{12} = g_{21} = |\mathbf{e}_1||\mathbf{e}_2|\cos75^\circ

. A vector has (dimensionless) contravariant components

V^i = (2, 1)

. ((a)) Find the covariant components

V_i = g_{ij}V^j

. ((b)) Find the length

|\mathbf{V}| = \sqrt{V^iV_i}

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