Chapter 13: Q 72. (page 1068)

Let $b$ be a constant. Prove that the equation of the plane $y = b$ is $r = b c s c φ c s c θ$ in spherical coordinates.

Short Answer

Expert verified

This is proved using relation $x = r \sin ϕ \cos θ, y = r \sin ϕ \sin θ, z = r \cos ϕ$ .

Step by step solution

Given Information

Equation of rectangular coordinates is given as $y = b$ ( $b$ is constant).

Simpification

We know that

$x = r \sin ϕ \cos θ, y = r \sin ϕ \sin θ, z = r \cos ϕ$

For conversion of given equation of plane to spherical coordinates, use $y = r \sin ϕ \sin θ$ in given equation

$r \sin ϕ \sin θ = b$

$\Rightarrow r = \frac{b}{\sin ϕ \sin θ}$

$= b (\frac{1}{\sin ϕ}) (\frac{1}{\sin θ})$

role="math" localid="1652391295106" $r = b \csc ϕ \csc θ$

Hence proved.

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Let $b$ be a constant. Prove that the equation of the plane $y = b$ is $r = b c s c φ c s c θ$ in spherical coordinates.

Short Answer

Step by step solution

Given Information

Simpification

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Let bbe a constant. Prove that the equation of the plane y=bisr=bcscφcscθin spherical coordinates.

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Let $b$ be a constant. Prove that the equation of the plane $y = b$ is $r = b c s c φ c s c θ$ in spherical coordinates.