Explanation Given information: The spherical coordinates p , ϕ , and θ take the place of u , v , and w . The transformation from Cartesian p ϕ θ -space to Cartesian xyz space is given by x = ρ sin ϕ cos θ y = ρ sin ϕ sin θ z = ρ cos ϕ Calculation: Calculate the value of Jacobian as shown below: J ( u , v ) = | ∂ x ∂ ρ ∂ x ∂ ϕ ∂ x ∂ θ ∂ y ∂ ρ ∂ y ∂ ϕ ∂ y ∂ θ ∂ z ∂ ρ ∂ z ∂ ϕ ∂ z ∂ θ | (1) Find the partial derivatives as follows: Find ∂ x ∂ ρ : Differentiate x with respect to ρ . ∂ x ∂ ρ = ∂ ∂ ρ ( ρ sin ϕ cos θ ) = sin ϕ cos θ Find ∂ x ∂ ϕ : Differentiate x with respect to ϕ . ∂ x ∂ ϕ = ∂ ∂ ϕ ( ρ sin ϕ cos θ ) = ρ cos ϕ cos θ Find ∂ x ∂ θ : Differentiate x with respect to θ . ∂ x ∂ θ = ∂ ∂ θ ( ρ sin ϕ cos θ ) = − ρ sin ϕ sin θ Find ∂ y ∂ ρ : Differentiate y with respect to ρ . ∂ y ∂ ρ = ∂ ∂ ρ ( ρ sin ϕ sin θ ) = sin ϕ sin θ Find ∂ y ∂ ϕ : Differentiate y with respect to ϕ . ∂ y ∂ ϕ = ∂ y ∂ ϕ ( ρ sin ϕ sin θ ) = ρ cos ϕ sin θ Find ∂ y ∂ θ : Differentiate y with respect to θ . ∂ y ∂ θ = ∂ ∂ θ ( ρ sin ϕ sin θ ) = ρ sin ϕ cos θ Find ∂ z ∂ ρ : Differentiate z with respect to ρ . ∂ z ∂ ρ = ∂ ∂ ρ ( ρ cos ϕ ) = cos ϕ Find ∂ z ∂ ϕ : Differentiate z with respect to ϕ

University Calculus: Early Transcendentals (4th Edition)

4th Edition

ISBN: 9780134995540

Author: Joel R. Hass, Christopher E. Heil, Przemyslaw Bogacki, Maurice D. Weir, George B. Thomas Jr.

Publisher: PEARSON

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Chapter 14.8, Problem 23E

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Prove that the Jacobian transformation from Cartesian pϕθ-space to Cartesian xyz-space is p2sinϕ.

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Chapter 14.8, Problem 22E

Chapter 14.8, Problem 24E

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Example(1): (Adiabatic humidification and cooling of air). Air has to be humidified and cooled adiabatically in a honzontal spray chamber with recirculated water. The active part of the chamber is Im #2m #15 m long. Under the operating conditions, the coefficient of heat transfer is expected to be 1300 kcal/(hr)(m2)(°C). 200 m3/min of air at 60 °C and 1 atm pressure with a humidity of 0.018 kg water/kg dry air is to be blown through the spray chamber. Calculate the following (a) the temperature and hunudity of the exit air (b) make-up water to be supplied, windage and blow down are neglected (c) the expected gas-phase mass transfer coefficient, kya (d) the temperature and humidity of the exit air if an identical spray chamber is added in series with the existing one O

Chapter 14 Solutions

University Calculus: Early Transcendentals (4th Edition)

Ch. 14.1 - In Exercises 1-14. evaluate the iterated...Ch. 14.1 - Prob. 2E Ch. 14.1 - Prob. 3E Ch. 14.1 - In Exercises 1-14, evaluate the iterated...Ch. 14.1 - In Exercises 1-14, evaluate the iterated...Ch. 14.1 - In Exercises 1-14, evaluate the iterated...Ch. 14.1 - In Exercises 1-14, evaluate the iterated integral....Ch. 14.1 - In Exercises 1-14, evaluate the iterated...Ch. 14.1 - In Exercises 1-14, evaluate the iterated integral....Ch. 14.1 - Prob. 10E

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Prove that the Jacobian transformation from Cartesian p ϕ θ -space to Cartesian xyz- space is p 2 sin ϕ .

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Prove that the Jacobian transformation from Cartesian pϕθ-space to Cartesian xyz-space is p2sinϕ.

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Chapter 14 Solutions