Explanation Given information: using the change of variables formula,and derive the formula for integration in spherical coordinates. Formula Used: Change of variables formula; ∬ D f ( x , y ) d x d y = ∬ D ∘ f ( x ( u , v ) , y ( u , v ) ) | ∂ ( x , y ) ∂ ( u , v ) | d u d v And, J a c ( G ) = | ∂ ( x , y ) ∂ ( u , v ) | = | ∂ x ∂ u ∂ x ∂ v ∂ y ∂ u ∂ y ∂ v | Proof: The spherical coordinates ( ρ , θ , ϕ ) . To transform the cartesian coordinates to spherical coordinates. x = ρ s i n ϕ c o s θ y = ρ s i n ϕ s i n θ z = ρ c o s ϕ ρ = x 2 + y 2 + z 2 t a n θ = y x t a n ϕ = x 2 + y 2 z Now by considering the change of value formula, ∭ W f ( x , y , z ) d x d y d z = ∭ W f ( x ( ρ , θ , ϕ ) , y ( ρ , θ , ϕ ) , z ( ρ , θ , ϕ ) ) | ∂ ( x , y , z ) ∂ ( ρ , θ , ϕ ) | d ρ d θ d ϕ = ∭ W f ( ρ s i n ϕ c o s θ , ρ s i n ϕ s i n θ , ρ c o s ϕ ) | ∂ ρ s i n ϕ c o s θ ∂ ρ ∂ ρ s i n ϕ c o s θ ∂ θ ∂ ρ s i n ϕ c o s θ ∂ ϕ ∂ ρ s i n ϕ s i n θ ∂ ρ ∂ ρ s i n ϕ s i n θ ∂ θ ∂ ρ s i n ϕ s i n θ ∂ ϕ ∂ ρ c o s ϕ ∂ ρ ∂ ρ c o s ϕ ∂ θ ∂ ρ c o s ϕ ∂ ϕ | d ρ d θ d ϕ Now solving the determinant; | ∂ ρ s i n ϕ c o s θ ∂ ρ ∂ ρ s i n ϕ c o s θ ∂ θ ∂ ρ s i n ϕ c o s θ ∂ ϕ ∂ ρ s i n ϕ s i n θ ∂ ρ ∂ ρ s i n ϕ s i n θ ∂ θ ∂ ρ s i n ϕ s i n θ ∂ ϕ ∂ ρ c o s ϕ ∂ ρ

3rd Edition

ISBN: 9781464125263

Author: Jon Rogawski, Colin Adams

Publisher: W.H. Freeman & Co

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Chapter 16.6, Problem 43E

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deriving the formula for integration in spherical coordinates using the change of variables formula

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Chapter 16.6, Problem 42E

Chapter 16.6, Problem 44E

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To proof: deriving the formula for integration in spherical coordinates using the change of variables formula

Solution Summary: The author explains how to deduce the formula for integration in spherical coordinates using the change of variables formula.

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To proof: deriving the formula for integration in spherical coordinates using the change of variables formula

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Chapter 16 Solutions

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deriving the formula for integration in spherical coordinates using the change of variables formula