Transformation from Cartesian to spherical polar coordinates

[andrey21]

Transformation from Cartesian to spherical polar coordinates

In dimensions:

x=r sinθ cos [itex]\varphi[/itex] and y= r sin θ sin [itex]\varphi[/itex] z=r cos θ

Show one example of:

∂z^{[itex]\alpha[/itex]}/ ∂x^μ . ∂x^μ/ ∂z^{[itex]\alpha[/itex]} = δ^{[itex]\alpha[/itex]}_{[itex]\beta[/itex]}

Now here is my answer:

δ^y_x=(∂y/∂r . ∂r/∂x) + (∂y/∂θ . ∂θ/∂x) + (∂y/∂[itex]\varphi[/itex] . ∂[itex]\varphi[/itex]/∂x)



Is this correct? If not where have I made an error... Thank you

 

[andrey21]

Just like to pick up in this old thread, still having trouble with the question.

Using what I have already done:

δ^r_θ=(∂r/∂x . ∂x/∂θ) + (∂r/∂y . ∂y/∂θ) + (∂r/∂z . ∂z/∂θ) (1)

Where:

x=r sin θ cos φ and y= r sin θ sin φ z= r cos θ

Would (1) then become:

δyx= = ((sin θ cos φ) . ( r cos θ cos φ)) + ((sin θ sin φ) . (cos θ sin φ)) + ((cos θ) . (-r sin θ))

Then multiply out the brackets and simplify