Question #346325

Identify the surface of the ρ = sinΦsinθ by converting them into equations in the Cartesian form. Show the complete solutions.


1
Expert's answer
2022-06-01T02:45:23-0400
x=ρsinϕcosθ,y=ρsinϕsinθ,z=ρcosϕx=\rho\sin\phi\cos\theta, y=\rho\sin\phi\sin\theta, z=\rho\cos\phi

ρ2=x2+y2+z2\rho^2=x^2+y^2+z^2

Given

ρ=sinϕsinθ\rho=\sin\phi\sin\theta

Then


ρ2=ρsinϕsinθ\rho^2=\rho\sin\phi\sin\theta

Substitute


x2+y2+z2=yx^2+y^2+z^2=y

x2+(y12)2+z2=(12)2x^2+(y-\dfrac{1}{2})^2+z^2=(\dfrac{1}{2})^2

The surface is a sphere.


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