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## Homework Statement

Given a tensor M

^{a}

_{b}, Prove that its trace is a scalar.

## Homework Equations

## The Attempt at a Solution

To prove the trace is a scalar, I know I have to prove it doesn't transform under coordinate transformations.

Now, we can transform M^a_b as follows.

M'

^{c}

_{d}=(dx'

^{c}/dx

^{a})(dx

^{b}/dx'

^{d})M

^{a}

_{b}

For the trace M

^{a}

_{a}, the transformation is as follows. I've used the Kronecker delta for changing M

^{a}

_{b}to M

^{a}

_{a}. ( Am I along the right lines in doing so?).

M'

^{c}

_{d}=(dx'

^{c}/dx

^{a})(dx

^{b}/dx'

^{d})M

^{a}

_{b}(Kronecker delta

^{a}

_{b})

Then I thought using the delta I change dx

^{b}to dx

^{a}and cancel the two dx

^{a}. However that would then the delta away from M

^{a}

_{b}, leaving it as M

^{a}

_{b}as before rather than M

^{a}

_{a}.

That's the best I can do, please help.

Last edited: