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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Question

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

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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Prob. 43E Ch. 17.2 - Prob. 44E Ch. 17.2 - Prob. 45E Ch. 17.2 - Prob. 46E Ch. 17.2 - Prob. 47E Ch. 17.2 - Prob. 48E Ch. 17.2 - Prob. 49E Ch. 17.2 - Prob. 50E Ch. 17.2 - Prob. 51E Ch. 17.2 - Prob. 52E Ch. 17.2 - Prob. 53E Ch. 17.2 - Prob. 54E Ch. 17.2 - Prob. 55E Ch. 17.2 - Prob. 56E Ch. 17.2 - Prob. 57E Ch. 17.2 - Prob. 58E Ch. 17.2 - Prob. 59E Ch. 17.2 - Prob. 60E Ch. 17.2 - Prob. 61E Ch. 17.2 - Prob. 62E Ch. 17.2 - Prob. 63E Ch. 17.2 - Prob. 64E Ch. 17.2 - Prob. 65E Ch. 17.2 - Prob. 66E Ch. 17.2 - Prob. 67E Ch. 17.2 - Prob. 68E Ch. 17.2 - Prob. 69E Ch. 17.2 - Prob. 70E Ch. 17.2 - Prob. 71E Ch. 17.2 - Prob. 72E Ch. 17.2 - Prob. 73E Ch. 17.2 - Prob. 74E Ch. 17.2 - Prob. 75E Ch. 17.3 - Prob. 1PQ Ch. 17.3 - Prob. 2PQ Ch. 17.3 - Prob. 3PQ Ch. 17.3 - Prob. 4PQ Ch. 17.3 - Prob. 1E Ch. 17.3 - Prob. 2E Ch. 17.3 - Prob. 3E Ch. 17.3 - Prob. 4E Ch. 17.3 - Prob. 5E Ch. 17.3 - Prob. 6E Ch. 17.3 - Prob. 7E Ch. 17.3 - Prob. 8E Ch. 17.3 - Prob. 9E Ch. 17.3 - Prob. 10E Ch. 17.3 - Prob. 11E Ch. 17.3 - Prob. 12E Ch. 17.3 - 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Prob. 12E Ch. 17.5 - Prob. 13E Ch. 17.5 - Prob. 14E Ch. 17.5 - Prob. 15E Ch. 17.5 - Prob. 16E Ch. 17.5 - Prob. 17E Ch. 17.5 - Prob. 18E Ch. 17.5 - Prob. 19E Ch. 17.5 - Prob. 20E Ch. 17.5 - Prob. 21E Ch. 17.5 - Prob. 22E Ch. 17.5 - Prob. 23E Ch. 17.5 - Prob. 24E Ch. 17.5 - Prob. 25E Ch. 17.5 - Prob. 26E Ch. 17.5 - Prob. 27E Ch. 17.5 - Prob. 28E Ch. 17.5 - Prob. 29E Ch. 17.5 - Prob. 30E Ch. 17.5 - Prob. 31E Ch. 17.5 - Prob. 32E Ch. 17.5 - Prob. 33E Ch. 17.5 - Prob. 34E Ch. 17.5 - Prob. 35E Ch. 17.5 - Prob. 36E Ch. 17.5 - Prob. 37E Ch. 17.5 - Prob. 38E Ch. 17 - Prob. 1CRE Ch. 17 - Prob. 2CRE Ch. 17 - Prob. 3CRE Ch. 17 - Prob. 4CRE Ch. 17 - Prob. 5CRE Ch. 17 - Prob. 6CRE Ch. 17 - Prob. 7CRE Ch. 17 - Prob. 8CRE Ch. 17 - Prob. 9CRE Ch. 17 - Prob. 10CRE Ch. 17 - Prob. 11CRE Ch. 17 - Prob. 12CRE Ch. 17 - Prob. 13CRE Ch. 17 - Prob. 14CRE Ch. 17 - Prob. 15CRE Ch. 17 - Prob. 16CRE Ch. 17 - Prob. 17CRE Ch. 17 - Prob. 18CRE Ch. 17 - Prob. 19CRE Ch. 17 - Prob. 20CRE Ch. 17 - Prob. 21CRE Ch. 17 - Prob. 22CRE Ch. 17 - Prob. 23CRE Ch. 17 - Prob. 24CRE Ch. 17 - Prob. 25CRE Ch. 17 - Prob. 26CRE Ch. 17 - Prob. 27CRE Ch. 17 - Prob. 28CRE Ch. 17 - Prob. 29CRE Ch. 17 - Prob. 30CRE Ch. 17 - Prob. 31CRE Ch. 17 - Prob. 32CRE Ch. 17 - Prob. 33CRE Ch. 17 - Prob. 34CRE Ch. 17 - Prob. 35CRE Ch. 17 - Prob. 36CRE Ch. 17 - Prob. 37CRE Ch. 17 - Prob. 38CRE Ch. 17 - Prob. 39CRE Ch. 17 - Prob. 40CRE Ch. 17 - Prob. 41CRE Ch. 17 - Prob. 42CRE Ch. 17 - Prob. 43CRE Ch. 17 - Prob. 44CRE Ch. 17 - Prob. 45CRE Ch. 17 - Prob. 46CRE Ch. 17 - Prob. 47CRE Ch. 17 - Prob. 48CRE Ch. 17 - Prob. 49CRE Ch. 17 - Prob. 50CRE Ch. 17 - Prob. 51CRE Ch. 17 - Prob. 52CRE Ch. 17 - Prob. 53CRE Ch. 17 - Prob. 54CRE Ch. 17 - Prob. 55CRE Ch. 17 - Prob. 56CRE Ch. 17 - Prob. 57CRE Ch. 17 - Prob. 58CRE Ch. 17 - Prob. 59CRE Ch. 17 - Prob. 60CRE Ch. 17 - Prob. 61CRE

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01 - What Is an Integral in Calculus? Learn Calculus Integration and how to Solve Integrals.; Author: Math and Science;https://www.youtube.com/watch?v=BHRWArTFgTs;License: Standard YouTube License, CC-BY

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) $F . n = r^{3}$ 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.