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tutorial4

# tutorial4 - x t =(sin t,e-t cos t is a ﬂow line of the...

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MATH1211/10-11(2)/Tu4/TNK THE UNIVERSITY OF HONG KONG DEPARTMENT OF MATHEMATICS MATH1211 Multivariable Calculus 2010-11 2nd Semester: Tutorial 4 Date of tutorial classes: February 24–25. (The section/problem numbers in the following refer to those in the textbook.) 1. (Variation of § 3.2 no.2) Calculate the length of the path x ( t ) = 3 t 2 i + 2(2 t + 1) 3 / 2 j , 0 t 2. 2. ( § 3.2 no.14) Determine the moving frame [ T , N , B ], and compute the curvature and torsion for the path x ( t ) = (sin t - t cos t ) i + (cos t + t sin t ) j + 2 k , t 0. 3. Verify that the path
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Unformatted text preview: x ( t ) = (sin t,e-t , cos t ) is a ﬂow line of the vector ﬁeld F = ( z,-y,-x ). 4. Find the curl of the vector ﬁeld F = z i + ( x + y 2 ) j + x 2 yz k . Verify that div (curl F ) = 0. 5. ( § 3.4 no.23) Establish the identity ∇ · ( f F ) = f ∇ · F + F · ∇ f , where f : R n → R is a function, and F is a vector ﬁeld in R n . ( Hint : Write F = ( F 1 ,F 2 ,...,F n ) if F is a vector ﬁeld in R n .) 1...
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