How do you convert x^2+y^2=zx2+y2=z into spherical and cylindrical form?

1 Answer
A. S. Adikesavan
Aug 8, 2016

Spherical form+ r=cos phi csc^2 thetar=cosϕcsc2θ.
Cylindrical form: r=z csc^2thetar=zcsc2θ

Explanation:

The conversion formulas,

Cartesian to→ spherical::

(x, y, z)=r(sin phi cos theta, sin phi sin theta, cos phi), r=sqrt(x^2+y^2+z^2)(x,y,z)=r(sinϕcosθ,sinϕsinθ,cosϕ),r=√x2+y2+z2

Cartesian to→ cylindrical:

(x, y, z)=(rho cos theta, rho sin theta, z), rho=sqrt(x^2+y^2)(x,y,z)=(ρcosθ,ρsinθ,z),ρ=√x2+y2

Substitutions in x^2+y^2=zx2+y2=z lead to the forms in the answer.

Note the nuances at the origin:

r = 0 is Cartesian (x, y, z) = (0, 0, 0). This is given by

(r, theta, phi) = (0, theta, phi)(r,θ,ϕ)=(0,θ,ϕ), in spherical form, and

(rho, theta, z)=(0, theta, 0)(ρ,θ,z)=(0,θ,0), in cylindrical form...
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