Explanation Calculation: The paraboloid equation is z = 1 − x 2 − y 2 We substitute x = r cos θ and y = r sin θ in the right-hand side of the equation, and verify that the result is 1 − r 2 1 − x 2 − y 2 = 1 − ( r cos θ ) 2 − ( r sin θ ) 2 = 1 − r 2 ( cos 2 θ + sin 2 θ ) = 1 − r 2 = z Hence G ( r , θ ) = ( r cos θ , r sin θ , 1 − r 2 ) parametrizes the paraboloid z = 1 − x 2 − y 2 . Moreover, for every x, y, z satisfying z = 1 − x 2 − y 2 there are suitable values of r and θ so that x = r cos θ , y = r sin θ and z = 1 − r 2 ...

4th Edition

ISBN: 9781319050733

Author: Rogawski

Publisher: MAC HIGHER

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Chapter 17.4, Problem 2E

To determine

To Show: That $G (r, θ) = (r \cos θ, r \sin θ, 1 - r^{2})$ parametrizes the paraboloid $z = 1 - x^{2} - y^{2}$ . And describe the grid curves of the parametrization.

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Chapter 17.4, Problem 1E

Chapter 17.4, Problem 3E

Chapter 17 Solutions

CALCULUS (CLOTH)

Ch. 17.1 - Prob. 1PQ Ch. 17.1 - Prob. 2PQ Ch. 17.1 - Prob. 3PQ Ch. 17.1 - Prob. 4PQ Ch. 17.1 - Prob. 1E Ch. 17.1 - Prob. 2E Ch. 17.1 - Prob. 3E Ch. 17.1 - Prob. 4E Ch. 17.1 - Prob. 5E Ch. 17.1 - Prob. 6E

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That G ( r , θ ) = ( r cos θ , r sin θ , 1 − r 2 ) parametrizes the paraboloid z = 1 − x 2 − y 2 . And describe the grid curves of the parametrization.

Solution Summary: The author concludes that G ( r, ) parametrizes the whole paraboloid z = 1

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To Show: That $G (r, θ) = (r \cos θ, r \sin θ, 1 - r^{2})$ parametrizes the paraboloid $z = 1 - x^{2} - y^{2}$ . And describe the grid curves of the parametrization.

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Chapter 17 Solutions

That G ( r , θ ) = ( r cos θ , r sin θ , 1 − r 2 ) parametrizes the paraboloid z = 1 − x 2 − y 2 . And describe the grid curves of the parametrization.

Solution Summary: The author concludes that G ( r, ) parametrizes the whole paraboloid z = 1

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To Show: That G(r,θ)=(rcosθ,rsinθ,1−r2) parametrizes the paraboloid z=1−x2−y2. And describe the grid curves of the parametrization.

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Chapter 17 Solutions

To Show: That $G (r, θ) = (r \cos θ, r \sin θ, 1 - r^{2})$ parametrizes the paraboloid $z = 1 - x^{2} - y^{2}$ . And describe the grid curves of the parametrization.