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## Main Question or Discussion Point

In a spherical polar coordinate system if the components of a vector given be (r,θ,φ)=1,2,3 respectively. Then the component of the vector along the x-direction of a cartesian coordinate system is $$rsinθcosφ$$.

But from the transformation of contravariant vector $$A^{-i}=\frac{∂x^{-i}}{∂x^j}A^j$$ where $$A^1=1 ,A^2=2 ,A^3=3$$ from this transformation if $$x^{-1}=x$$ of the cartesian coordinate system then $$A^{-1}=sin\theta cos\phi+2rcos\theta cos\phi-3rsin\theta sin\phi$$(component along x of cartesian coordinate system)

Am I missing out something??

But from the transformation of contravariant vector $$A^{-i}=\frac{∂x^{-i}}{∂x^j}A^j$$ where $$A^1=1 ,A^2=2 ,A^3=3$$ from this transformation if $$x^{-1}=x$$ of the cartesian coordinate system then $$A^{-1}=sin\theta cos\phi+2rcos\theta cos\phi-3rsin\theta sin\phi$$(component along x of cartesian coordinate system)

Am I missing out something??