Past exam of the mathematics course of the University of Cambridge / 2020 / ii / Paper 3 / 37D / a

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- Solution a

Solution

In coordinates

x^{μ}

, write the type-

(1, 1)

T = T^{μ}_{ν} \partial_{μ} \otimes d x^{ν} .

η = η_{μ} d x^{μ}

V = V^{ν} \partial_{ν}

,

T (η, V) = T^{μ}_{ν} η_{μ} V^{ν} .

Since this must equal

η (V) = η_{ν} V^{ν}

for every

η

and

V

, the components are

T^{μ}_{ν} = δ^{μ}_{ν} .

Thus

T

is the identity tensor.

The tensor component transformation law gives

T^{μ^{'}}_{ν^{'}} = \frac{\partial x ^{μ^{'}}}{\partial x ^{α}} \frac{\partial x ^{β}}{\partial x ^{ν^{'}}} T^{α}_{β} = \frac{\partial x ^{μ^{'}}}{\partial x ^{α}} \frac{\partial x ^{α}}{\partial x ^{ν^{'}}} = δ^{μ^{'}}_{ν^{'}} .

The components are therefore the same Kronecker delta in every coordinate system.

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