Question

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

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

Answers (1)

2021-05-28
Step 1
The plane \(z=x\) goes through the line intersection of the planes
\(x=0\) and \(z=0\) and makes a \(\frac{\pi}{4}\) angle with those planes.
So, a point \((x,y,z)=(x,y,x)=(x,y)\) is on the plane.
a) Cylindrical coordinates:
Let \(x=r\cos\theta,\ y=r\sin\theta\) with \(r\geq0\) and \(0\leq\theta\leq2\pi\)
Then. \(z=x\)
\(\Rightarrow z=r\cos\theta\)
So, a point on the plane takes the form \((x,y,z)=(r\cos\theta,\ r\sin\theta,\ r\cos\theta)\)
Step 2
b) Spherical coordinates:
Let \(x=\rho\sin\phi\cos\theta,\ y=\rho\sin\phi\theta,\ z=\rho\cos\phi\) with \(\rho\geq0,\ 0\leq\theta\leq2\pi\) and \(0\leq\phi\leq\pi\)
Then, \(z=x\)
\(\Rightarrow\rho\cos\phi=\rho\sin\phi\cos\theta\)
\(\Rightarrow\frac{\rho\cos\phi}{\rho\sin\phi}=\cos\theta\)
\(\Rightarrow\cot\phi=\cos\theta\)
\(\Rightarrow\theta=\arccos(\cot\phi)\)
So, a point on the plane takes the form
\((x,y,z)=(\rho\sin\phi\cos\theta,\ \rho\sin\phi\sin\theta,\ \rho\cos\phi)\)
\(=(\rho\sin\phi\cos(\arccos(\cot\phi)),\ \rho\sin\phi\sin(\arccos(\cot\phi)),\ \rho\cos\phi)\)
\(=(\rho\sin\phi\cos(\arccos(\cot\phi)),\ \rho\sin\phi\sin(\arcsin(\sqrt{1-\cot^{2}\phi})),\ \rho\cos\phi)\)
\(=(\rho\sin\phi\cot\phi,\ \rho\sin\phi\sqrt{1-\cot^{2}\phi},\ \rho\cos\phi)\)
\(=(\rho\cos\phi,\ \rho\sin\phi\sqrt{1-\cot^{2}\phi},\ \rho\cos\phi)\)
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