Chapter 13: Q 9 (page 1065)

. To convert from rectangular to spherical coordinates
$p =, θ =, ϕ =$

Short Answer

Expert verified

$p = \sqrt{x^{2} + y^{2} + z^{2}} θ = \tan^{- 1} (\frac{y}{x}) ϕ = \cos^{- 1} (\frac{z}{\sqrt{x^{2} + y^{2} + z^{2}}})$

Step by step solution

Step 1:Given information

Given $(x, y, z)$ are rectangular coodinates and $(p, θ, ϕ)$ are spherical coordinates

Simplification

$(x, y, z)$ are rectangular coordinates and $(p, θ, ϕ)$ are spherical coordinates

The relation is

From the figure

$\Rightarrow \cos ϕ = \frac{z}{r}, \sin ϕ = \frac{ρ}{r}$

Now

$x = ρ \cos ϕ$

$= r \sin θ \cos ϕ$ where $(ρ = r \sin θ)$

similarly

$y = ρ \sin ϕ$

$= r \sin θ \sin ϕ$

And $z = r \cos ϕ$

now

$x^{2} + y^{2} + z^{2} = r^{2} [\sin^{2} ϕ (\sin^{2} θ + \cos^{2} θ) + \cos^{2} ϕ] \Rightarrow x^{2} + y^{2} + z^{2} = r^{2} [\sin^{2} ϕ + \cos^{2} ϕ] \Rightarrow x^{2} + y^{2} + z^{2} = r^{2} \Rightarrow r = \sqrt{x^{2} + y^{2} + z^{2}}$

Similarly

$\Rightarrow θ = \tan^{- 1} \frac{y}{x}$

and we have $\cos ϕ = \frac{z}{r}$

$\Rightarrow ϕ = \cos^{- 1} \frac{z}{r} \Rightarrow ϕ = \cos^{- 1} (\frac{z}{\sqrt{x^{2} + y^{2} + z^{2}}})$

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. To convert from rectangular to spherical coordinates
$p =, θ =, ϕ =$

Short Answer

Step by step solution

Step 1:Given information

Simplification

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. To convert from rectangular to spherical coordinatesp=____,θ=____,ϕ=____

Short Answer

Step by step solution

Step 1:Given information

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One App. One Place for Learning.

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. To convert from rectangular to spherical coordinates
$p =, θ =, ϕ =$