Explanation Given: 1) Vector F = 〈 y 2 , x 2 , z 2 〉 . 2) Need to prove ∫ C 1 F . d r = ∫ C 2 F . d r Where C 1 and C 2 are two closed curves lying on a cylinder whose central axis is z axis. 3) Given figure is as follows Formula used: 1) Stoke’s Theorem for a surface bounded by two curve is as follows ∫ C 1 F . d r − ∫ C 2 F . d r = ∬ S ( ∇ × F ) . n d s 2) General parametric equation of cylinder is as follows x = r cos ( θ ) y = r sin ( θ ) z = z Calculation: For any two closed curve lying on the cylinder having same central axis , the following formula holds true ∫ C 1 F . d r − ∫ C 2 F . d r = ∬ S ( ∇ × F ) . n d s ... ( i ) [Stokes theorem] Lets calculate right hand side of above equation that is as follows ∬ S ( ∇ × F ) . n d s ... ( i i ) ∇ × F = ∇ × 〈 y 2 , x 2 , z 2 〉 [given that F = 〈 y 2 , x 2 , z 2 〉 ] = | i ^ j ^ k ^ ∂ ∂ x ∂ ∂ y ∂ ∂ z y 2 x 2 z 2 | = i [ ∂ ∂ y ( z 2 ) − ∂ ∂ z ( x 2 ) ] − j [ ∂ ∂ x ( z 2 ) − ∂ ∂ z ( y 2 ) ] + k [ ∂ ∂ x ( x 2 ) − ∂ ∂ y ( y 2 ) ] = i [ 0 − 0 ] − j [ 0 − 0 ] + k [ 2 x − 2 y ] ∇ × F = k ( 2 x − 2 y ) ... ( i i i ) General parametric equation of cylinder is as follows x = r cos ( θ ) y = r sin ( θ ) z = z Placing parametric values of x and y in equation ( i i i ) ∇ × F = k ( 2 r cos ( θ ) − 2 r sin ( θ ) ) Let’s calculate unit vector n n = ( r cos ( θ ) , r sin ( θ ) , 0 ) n = ( r cos ( θ ) , r sin ( θ ) , 0 ) [since the surface in our case is lying on cylinder having center on z axis] Calculate ( ∇ × F ) ...

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ISBN: 9781319050733

Author: Rogawski

Publisher: MAC HIGHER

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Chapter 18 Solutions

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Prove the given condition for a vector field with respect to particular surface.

Solution Summary: The author explains the Stoke's Theorem for a surface bounded by two curves.

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Chapter 18 Solutions