Explanation Given: F = 〈 y , − x , x 2 + y 2 〉 and let S be the portion of the paraboloid z = x 2 + y 2 , where x 2 + y 2 ≤ 3 Proof: (a) The given surface is represented in the polar coordinates as G ( r , θ ) = ( r cos θ , r sin θ , r 2 ) 0 ≤ r ≤ 3 , 0 ≤ θ ≤ 2 π Thus, the tangent vectors are T r = 〈 cos θ , sin θ , 2 r 〉 T θ = 〈 − r sin θ , r cos θ , 0 〉 Normal vector is n = T r × T θ = | i j k cos θ sin θ 2 r − r sin θ r cos θ 0 | = i ( sin θ .0 − 2 r ( r cos θ ) ) − j ( cos θ .0 − 2 r ( − r sin θ ) ) + k ( r cos θ . cos θ − sin θ ( − r sin θ ) ) = i ( − 2 r 2 cos θ ) − j ( 2 r 2 sin θ ) + k ( r cos 2 θ + r sin 2 θ ) = − 2 r 2 cos θ i − 2 r 2 sin θ j + r k Therefore, F . n = F ( G ( r , θ ) ) . n ( r , θ ) = 〈 r sin θ , − r cos θ , r 2 〉 〈 − 2 r 2 cos θ , − 2 r 2 sin θ , r 〉 = − 2 r 3 sin θ cos θ + 2 r 3 sin θ cos θ + r 3 = r 3 Thus, it is proved that F ...

4th Edition

ISBN: 9781319050733

Author: Rogawski

Publisher: MAC HIGHER

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Chapter 17.5, Problem 2E

To determine

To prove:

(a) $F . n = r^{3}$ if $S$ is parameterized in polar variables $x = r \cos θ, y = r \sin θ$ .

(b) $\int \int_{S} F . d S = \int_{0}^{2 π} \int_{0}^{\sqrt{3}} r^{3} d r d θ$ and evaluate.

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Chapter 17.5, Problem 1E

Chapter 17.5, Problem 3E

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Chapter 17 Solutions

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To prove: (a) F . n = r 3 if S is parameterized in polar variables x = r cos θ , y = r sin θ . (b) ∫ ∫ S F . d S = ∫ 0 2 π ∫ 0 3 r 3 d r d θ and evaluate.

Solution Summary: The author explains how the given surface is represented in the polar coordinates as G = cos, sin, 2 r

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To prove:

(a) $F . n = r^{3}$ if $S$ is parameterized in polar variables $x = r \cos θ, y = r \sin θ$ .

(b) $\int \int_{S} F . d S = \int_{0}^{2 π} \int_{0}^{\sqrt{3}} r^{3} d r d θ$ and evaluate.

Want to see the full answer?

Chapter 17 Solutions

To prove: (a) F . n = r 3 if S is parameterized in polar variables x = r cos θ , y = r sin θ . (b) ∫ ∫ S F . d S = ∫ 0 2 π ∫ 0 3 r 3 d r d θ and evaluate.

Solution Summary: The author explains how the given surface is represented in the polar coordinates as G = cos, sin, 2 r

Concept explainers

Videos

To prove: (a) F.n=r3 if S is parameterized in polar variables x=rcosθ,y=rsinθ. (b) ∫∫SF.dS=∫02π∫03r3drdθ and evaluate.

Want to see the full answer?

Chapter 17 Solutions

To prove:

(a) $F . n = r^{3}$ if $S$ is parameterized in polar variables $x = r \cos θ, y = r \sin θ$ .

(b) $\int \int_{S} F . d S = \int_{0}^{2 π} \int_{0}^{\sqrt{3}} r^{3} d r d θ$ and evaluate.