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f(rcosx,rsinx)=(brcos2x,brsin2x,abr) where b>0. Show f is a local isometry iff $a=3^{0.5}$ and $b=0.5$.

My thoughts: I can use the fact that f is a local isometry iff for each p in the domain, there is a basis $e_{1}$, $e_{2}$ for the tangent space at p such that $df_{p}(e_{i}).df_{p}(e_{j})=e_{i}.e_{j}$ fo i,j=1,2, where $df_{p}$ is the differential of f at p. I thought of getting the basis vectors from a co-ordinate chart.

Thanks