To determine Verify the conclusion of Green’s Theorem for the field F = 2 x i − 3 y j . Explanation Given information: The vector field is F = 2 x i − 3 y j . The domain of integration is the disk R : x 2 + y 2 ≤ a 2 and its bounding circle C : r = ( a cos t ) i + ( a sin t ) j , 0 ≤ t ≤ 2 π . Green’s Theorem (Circulation-Curl or Tangential Form): Let C be a piecewise smooth, simple closed curve enclosing a region R in the plane. Let F = M i + N j be a vector field with M and N having continuous first partial derivatives in an open region containing R . Then the counter clockwise circulation of F around C equals the double integral of ( curl F ) ⋅ k over R . ∮ C F ⋅ T d s = ∮ C M d x + N d y = ∬ R ( ∂ N ∂ x − ∂ M ∂ y ) d x d y Green’s Theorem (Flux-Divergence or Normal Form): Let C be a piecewise smooth, simple closed curve enclosing a region R in the plane. Let F = M i + N j be a vector field with M and N having continuous first partial derivatives in an open region containing R . Then the outward flux of F across C equals the double integral of div F over the region R enclosed by C . ∮ C F ⋅ n d s = ∮ C M d y − N d x = ∬ R ( ∂ M ∂ x + ∂ N ∂ y ) d x d y Calculation: Write the vector field. F = 2 x i − 3 y j (1) C : r = ( a cos t ) i + ( a sin t ) j , 0 ≤ t ≤ 2 π x = a cos t , y = a sin t (2) From Equation (1) and Equation (2), M = 2 x = 2 a cos t (3) Find the derivative of d x . d x = − a sin t d t Find the derivative of d y . d y = a cos t d t Find the derivative of ∂ M ∂ x . ∂ M ∂ x = 2 Find the derivative of ∂ M ∂ y . ∂ M ∂ y = 0 From Equation (1) and Equation (2), N = − 3 y = − 3 a sin t (4) Find the derivative of ∂ N ∂ x . ∂ N ∂ x = 0 Find the derivative of ∂ N ∂ y . ∂ N ∂ y = − 3 Verify the Green’s Theorem (Flux-Divergence or Normal Form). ∮ C M d y − N d x = ∫ 0 2 π [ ( 2 a cos t ) ( a cos t ) − ( − 3 a sin t ) ( − a sin t ) ] d t = ∫ 0 2 π ( 2 a 2 cos 2 t − 3 a 2 sin 2 t ) d t = a 2 ∫ 0 2 π ( 2 cos 2 t − 3 sin 2 t ) d t = a 2 ∫ 0 2 π [ 2 ( cos 2 t + 1 2 ) − 3 ( 1 − cos 2 t 2 ) ] d t ∮ C M d y − N d x = a 2 [ ( sin 2 t 2 + t ) − 3 2 ( t − sin 2 t 2 ) ] 0 2 π = a 2 [ ( ( sin 2 ( 2 π ) 2 + ( 2 π ) ) − 3 2 ( ( 2 π ) − sin 2 <

Thomas' Calculus, Books a la Carte Edition, plus MyLab Math with Pearson eText -- Access Card Package (14th Edition)

14th Edition

ISBN: 9780134768755

Author: Hass

Publisher: PEARSON

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Chapter 16.4, Problem 9E

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Verify the conclusion of Green’s Theorem for the field F=2xi−3yj.

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Chapter 16.4, Problem 8E

Chapter 16.4, Problem 10E

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Chapter 16 Solutions

Thomas' Calculus, Books a la Carte Edition, plus MyLab Math with Pearson eText -- Access Card Package (14th Edition)

Ch. 16.1 - Match the vector equations in Exercises 1–8 with...Ch. 16.1 - Match the vector equations in Exercises 1–8 with...Ch. 16.1 - Match the vector equations in Exercises 1–8 with...Ch. 16.1 - Match the vector equations in Exercises 1–8 with...Ch. 16.1 - Match the vector equations in Exercises 1–8 with...Ch. 16.1 - Prob. 6E Ch. 16.1 - Prob. 7E Ch. 16.1 - Match the vector equations in Exercises 1–8 with...Ch. 16.1 - Evaluate ∫C (x + y) ds, where C is the...Ch. 16.1 - Evaluate ∫C (x − y + z − 2) ds, where C is the...

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