Tutorial 5

# Tutorial 5 - 3.4 no.4 F = z cos e y 2 i x √ z 2 1 j e 2 y...

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MATH1211/10-11(1)/Tu5 THE UNIVERSITY OF HONG KONG DEPARTMENT OF MATHEMATICS MATH1211 Multivariable Calculus 2010-11 First Semester: Tutorial 5 1. 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. 2. Verify that the path x ( t ) = (sin t, cos t, 2 t ) is a ﬂow line of the vector ﬁeld F = ( y, - x, 2). Justify geometrically with an appropriate sketch. 3. Calculate the divergence of the vector ﬁelds: (a) F = x 2 i + y 2 j (b) ( §
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Unformatted text preview: 3.4 no.4) F = z cos ( e y 2 ) i + x √ z 2 + 1 j + e 2 y sin3 x k . 4. Find the curl of the vector ﬁeld F = ( x + y ) i + x 2 z j-3 y k . Verify that div(curl F ) = 0. 5. Establish the identity ∇ · ( f F ) = f ∇ · F + F · ∇ f . ( Hint : Write F = ( F 1 ,F 2 ,...,F n ) if F is a vector ﬁeld in R n .) 6. Find the Taylor polynomial p 3 of order 3 of the function f ( x ) = √ x at the point a = 1. 1...
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