Explanation Given Info: Given G ( θ , ϕ ) = ( cos θ sin ϕ , sin θ sin ϕ , cos ϕ ) . f ( x , y , z ) = 1 . Formula Used: Let G ( u , v ) be a parametrization of a surface S with parameter domain D . Assume that G is continuously differentiable, one-to-one, and regular (except possibly at the boundary of D ). Then, ∬ S f ( x , y , z ) d S = ∬ D f ( G ( θ , ϕ ) ) ‖ N ( θ , ϕ ) ‖ d θ d ϕ . Where N ( θ , ϕ ) = T ϕ × T θ , T θ = ∂ G ∂ θ , T ϕ = ∂ G ∂ ϕ Calculation: T θ = ∂ ( cos θ sin ϕ , sin θ sin ϕ , cos ϕ ) ∂ θ T θ = ( − sin θ sin ϕ , cos θ sin ϕ , 0 ) T ϕ = ∂ ( cos θ sin ϕ , sin θ sin ϕ , cos ϕ ) ∂ ϕ T ϕ = ( cos θ cos ϕ , sin θ cos ϕ , − sin ϕ ) N ( θ , ϕ ) = i j k − sin θ sin ϕ , cos θ sin ϕ 0 cos θ cos ϕ sin θ cos ϕ − sin ϕ N ( θ , ϕ ) = − cos θ sin 2 ϕ i − sin θ sin 2 ϕ j + sin ϕ cos ϕ k

4th Edition

ISBN: 9781319050733

Author: Rogawski

Publisher: MAC HIGHER

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Question

Chapter 17.4, Problem 12E

To determine

To estimate: The area $\iint_{S} d S$ of small patch of surfacewhose spherical coordinates satisfy by $\frac{π}{2} - 0.15 \leq θ \leq \frac{π}{2} + 0.15, \frac{π}{4} - 0.1 \leq ϕ \leq \frac{π}{4} + 0.1$ .

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Chapter 17.4, Problem 11E

Chapter 17.4, Problem 13E

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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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To estimate: The area ∬ S d S of small patch of surfacewhose spherical coordinates satisfy by π 2 − 0.15 ≤ θ ≤ π 2 + 0.15 , π 4 − 0.1 ≤ ϕ ≤ π 4 + 0.1 .

Solution Summary: The author explains how to estimate the area S d S of a small patch of surface with spherical coordinates.

Concept explainers

Videos

To estimate: The area $\iint_{S} d S$ of small patch of surfacewhose spherical coordinates satisfy by $\frac{π}{2} - 0.15 \leq θ \leq \frac{π}{2} + 0.15, \frac{π}{4} - 0.1 \leq ϕ \leq \frac{π}{4} + 0.1$ .

Want to see the full answer?

Chapter 17 Solutions

To estimate: The area ∬ S d S of small patch of surfacewhose spherical coordinates satisfy by π 2 − 0.15 ≤ θ ≤ π 2 + 0.15 , π 4 − 0.1 ≤ ϕ ≤ π 4 + 0.1 .

Solution Summary: The author explains how to estimate the area S d S of a small patch of surface with spherical coordinates.

Concept explainers

Videos

To estimate: The area ∬SdS of small patch of surfacewhose spherical coordinates satisfy by π2−0.15≤θ≤π2+0.15,π4−0.1≤ϕ≤π4+0.1.

Want to see the full answer?

Chapter 17 Solutions

To estimate: The area $\iint_{S} d S$ of small patch of surfacewhose spherical coordinates satisfy by $\frac{π}{2} - 0.15 \leq θ \leq \frac{π}{2} + 0.15, \frac{π}{4} - 0.1 \leq ϕ \leq \frac{π}{4} + 0.1$ .