A two-dimensional oblique lattice has basis vectors and (Cartesian components in an…

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

A two-dimensional oblique lattice has basis vectors

\mathbf{e}_1 = (2, 0)\,\mathrm{m}

and

\mathbf{e}_2 = (1, 3)\,\mathrm{m}

(Cartesian components in an auxiliary orthonormal frame). A vector has contravariant components

V^i = (3, 1)

in this basis. Someone argues: "To lower the index I just write

V_i = \delta_{ij}V^j

, so

V_1 = 3

and

V_2 = 1

: lowering an index never changes the numbers, it only changes where you write them." ((a)) Explain what is wrong with this reasoning. ((b)) Compute the correct covariant components

V_i

. ((c)) Use

V^iV_i

to find the length

|\mathbf{V}|

, and check it is not

\sqrt{3^2+1^2}\,\mathrm{m} = \sqrt{10}\,\mathrm{m}

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