Explanation Given: Let C be a curve in polar form r = f ( θ ) for θ 1 ≤ θ ≤ θ 2 , parameterized by r ( θ ) = ( f ( θ ) cos θ , f ( θ ) sin θ ) . Proof: a. The parameterized curve in the polar form: r ( θ ) = ( f ( θ ) cos θ , f ( θ ) sin θ ) .... ( 1 ) The polar coordinates are: x = r cos θ , y = r sin θ Thus, the vortex field in polar coordinates will be: F = 〈 − y x 2 + y 2 , x x 2 + y 2 〉 = 〈 − r sin θ r 2 , r cos θ r 2 〉 = r − 1 〈 − sin θ , cos θ 〉 Thus, it is proved that F ( r , θ ) = r − 1 〈 − sin θ , cos θ 〉 . b. The given curve is parameterized by r ( θ ) = ( f ( θ ) cos θ , f ( θ ) sin θ ) .... ( 1 ) for θ 1 ≤ θ ≤ θ 2 . We have to evaluate the derivative of r ( θ ) ...

4th Edition

ISBN: 9781319050733

Author: Rogawski

Publisher: MAC HIGHER

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Question

Chapter 17.2, Problem 63E

To determine

To Prove:

a. The vortex field in polar coordinates is written $F (r, θ) = r^{- 1} 〈 - \sin θ, \cos θ 〉$ .

b. $F \times r' (θ) d θ = d θ$ .

c. $\int_{C} F \times d r = θ_{2} - θ_{1}$ .

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Chapter 17.2, Problem 62E

Chapter 17.2, Problem 64E

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Chapter 17 Solutions

CALCULUS (CLOTH)

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To Prove: a. The vortex field in polar coordinates is written F ( r , θ ) = r − 1 〈 − sin θ , cos θ 〉 . b. F × r ' ( θ ) d θ = d θ . c. ∫ C F × d r = θ 2 − θ 1 .

Solution Summary: The author proves that the vortex field in polar coordinates is written F ( r, ) = 1.

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To Prove:

a. The vortex field in polar coordinates is written $F (r, θ) = r^{- 1} 〈 - \sin θ, \cos θ 〉$ .

b. $F \times r' (θ) d θ = d θ$ .

c. $\int_{C} F \times d r = θ_{2} - θ_{1}$ .

Want to see the full answer?

Chapter 17 Solutions

To Prove: a. The vortex field in polar coordinates is written F ( r , θ ) = r − 1 〈 − sin θ , cos θ 〉 . b. F × r ' ( θ ) d θ = d θ . c. ∫ C F × d r = θ 2 − θ 1 .

Solution Summary: The author proves that the vortex field in polar coordinates is written F ( r, ) = 1.

Concept explainers

Videos

To Prove: a. The vortex field in polar coordinates is written F(r,θ)=r−1〈−sinθ,cosθ〉. b. F×r'(θ)dθ=dθ. c. ∫CF×dr=θ2−θ1.

Want to see the full answer?

Chapter 17 Solutions

To Prove:

a. The vortex field in polar coordinates is written $F (r, θ) = r^{- 1} 〈 - \sin θ, \cos θ 〉$ .

b. $F \times r' (θ) d θ = d θ$ .

c. $\int_{C} F \times d r = θ_{2} - θ_{1}$ .