Under a change of coordinates

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

u^i\to u'^i(u)

, a vector transforms as

V'^i=(\partial u'^i/\partial u^k)V^k

(from

\mathrm dx'^i=(\partial x'^i/\partial x^j)\,\mathrm dx^j

) and a covariant vector as

W'_i=(\partial u^k/\partial u'^i)W_k

(from the chain rule applied to

\partial_if

). At any point, a mixed tensor

T^i{}_j

can be written as a finite sum

T^i{}_j=\sum_a V_a^iW_{a,j}

of outer products of vectors and covariant vectors. ((a)) Use this fact, together with the separately known laws for

V^i

and

W_j

, to derive the general transformation law for

T^i{}_j

. ((b)) Show that the trace

T^i{}_i

(the sum over

i=j

) is unchanged by the coordinate change, and verify this directly for the identity tensor

T^i{}_j=\delta^i{}_j

n

dimensions.

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