Chapter 13: Q 71. (page 1068)

Let $a$ be a constant. Prove that the equation of the plane $x = a$ isrole="math" localid="1652390612497" $ρ = a c s c ϕ s e c θ$ in spherical coordinates.

Short Answer

Expert verified

Conversion is done using $x = ρ \sin ϕ \cos θ, y = ρ \sin ϕ \sin θ, z = ρ \cos ϕ$ .

Step by step solution

Given Information

Equation of plane in rectangular coordinates is given by $x = a$ ( $a$ is constant).

Simplification

The spherical coordinates are given by following mathematical relation

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

For the transformation of given plane to spherical coordinates, use $x = ρ \sin ϕ \cos θ$ in $x = a$ .

$ρ \sin ϕ \cos θ = a$

$\Rightarrow ρ = \frac{a}{\sin ϕ \cos θ}$

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

Hence, $ρ = a \csc ϕ \sec θ$

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Let $a$ be a constant. Prove that the equation of the plane $x = a$ isrole="math" localid="1652390612497" $ρ = a c s c ϕ s e c θ$ in spherical coordinates.

Short Answer

Step by step solution

Given Information

Simplification

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Let abe a constant. Prove that the equation of the plane x=aisrole="math" localid="1652390612497" ρ=acscϕsecθ in spherical coordinates.

Short Answer

Step by step solution

Given Information

Simplification

One App. One Place for Learning.

Most popular questions from this chapter

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Theoretical and Mathematical Physics

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Probability and Statistics

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Applied Mathematics

Study anywhere. Anytime. Across all devices.

Let $a$ be a constant. Prove that the equation of the plane $x = a$ isrole="math" localid="1652390612497" $ρ = a c s c ϕ s e c θ$ in spherical coordinates.