Explanation Given information: The equation of ellipsoid is x 2 a 2 + y 2 b 2 + z 2 c 2 = 1 Consider the value of x is au , y is bv , and z is cw . Calculation: Provide the equation of ellipsoid as follows: x 2 a 2 + y 2 b 2 + z 2 c 2 = 1 (2) The transformation takes the ellipsoid region x 2 a 2 + y 2 b 2 + z 2 c 2 ≤ 1 in xyz space into the space region. Consider the value of x is au , y is bv , and z is cw . Calculate the value of Jacobian as shown below: The Jacobian determinant or Jacobian of the coordinate transformation x = g ( u , v , w ) , y = h ( u , v , w ) , = z = k ( u , v , w ) . J ( u , v , w ) = | ∂ x ∂ u ∂ x ∂ v ∂ x ∂ w ∂ y ∂ u ∂ y ∂ v ∂ y ∂ w ∂ z ∂ u ∂ z ∂ v ∂ z ∂ w | (3) Find the partial derivatives as follows: Find ∂ x ∂ u : Differentiate x with respect to u . ∂ x ∂ u = ∂ ∂ u ( a u ) = a Find ∂ x ∂ v : Differentiate x with respect to v . ∂ x ∂ v = ∂ ∂ v ( a u ) = 0 Find ∂ x ∂ w : Differentiate x with respect to w . ∂ x ∂ w = ∂ ∂ w ( a u ) = 0 Find ∂ y ∂ u : Differentiate y with respect to u . ∂ y ∂ u = ∂ ∂ u ( b v ) = 0 Find ∂ y ∂ v : Differentiate y with respect to v . ∂ y ∂ v = ∂ ∂ v ( b v ) = b Find ∂ y ∂ w : Differentiate y with respect to w . ∂ y ∂ w = ∂ ∂ w ( b v ) = 0 Find ∂ z ∂ u : Differentiate z with respect to u

University Calculus: Early Transcendentals, Books a la Carte Edition (3rd Edition)

3rd Edition

ISBN: 9780321999610

Author: Joel R. Hass, Maurice D. Weir, George B. Thomas Jr.

Publisher: PEARSON

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Chapter 14.8, Problem 25E

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Show that the center of mass of the semi ellipsoid lies on the z-axis three eighths of the way from the base towards the top.

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Chapter 14.8, Problem 24E

Chapter 14.8, Problem 26E

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Chapter 14 Solutions

University Calculus: Early Transcendentals, Books a la Carte Edition (3rd 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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Show that the center of mass of the semi ellipsoid lies on the z -axis three eighths of the way from the base towards the top.

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Show that the center of mass of the semi ellipsoid lies on the z-axis three eighths of the way from the base towards the top.

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Chapter 14 Solutions