Express the plane z=x in cylindrical and spherical coordinates. a) cykindrical z=r\cos(\theta) b) spherical coordinates \theta=\arcsin(\cot(\phi))

SchachtN

SchachtN

Answered question

2021-05-27

Express the plane z=x in cylindrical and spherical coordinates.
a) cykindrical
z=rcos(θ)
b) spherical coordinates
θ=arcsin(cot(ϕ))

Answer & Explanation

Clara Reese

Clara Reese

Skilled2021-05-28Added 120 answers

Step 1
The plane z=x goes through the line intersection of the planes
x=0 and z=0 and makes a π4 angle with those planes.
So, a point (x,y,z)=(x,y,x)=(x,y) is on the plane.
a) Cylindrical coordinates:
Let x=rcosθ, y=rsinθ with r0 and 0θ2π
Then. z=x
z=rcosθ
So, a point on the plane takes the form (x,y,z)=(rcosθ, rsinθ, rcosθ)
Step 2
b) Spherical coordinates:
Let x=ρsinϕcosθ, y=ρsinϕθ, z=ρcosϕ with ρ0, 0θ2π and 0ϕπ
Then, z=x
ρcosϕ=ρsinϕcosθ
ρcosϕρsinϕ=cosθ
cotϕ=cosθ
θ=arccos(cotϕ)
So, a point on the plane takes the form
(x,y,z)=(ρsinϕcosθ, ρsinϕsinθ, ρcosϕ)
=(ρsinϕcos(arccos(cotϕ)), ρsinϕsin(arccos(cotϕ)), ρcosϕ)
=(ρsinϕcos(arccos(cotϕ)), ρsinϕsin(arcsin(1cot2ϕ)), ρcosϕ)
=(ρsinϕcotϕ, ρsinϕ1cot2ϕ, ρcosϕ)
=(ρcosϕ, ρsinϕ1cot2ϕ, ρcosϕ)

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