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Sma2110:Real Analysis Ii Question Paper

Sma2110:Real Analysis Ii

Course:Bachelor Of Computer Science

Institution: Meru University Of Science And Technology question papers

Exam Year:2014

QUESTION ONE (30 MARKS)
a) Show that × = sin where is the angle between (2 Marks) b) Let , , be fixed vectors in space. The equation = 2 + 2 where 2 2. Define a vector function of t with domain 2 2 and = 1 + 22, = 2 3, = 1 3. (3 Marks) c) Find the length of the arc segment of the helix = cos)1 + (sin2 + 3 for 0 2. (5 Marks) d) Obtain a natural representation of the helix given by = cos)1 + (sin2 + 3. (3 Marks) e) Find the tangent line given by = 1 + 22 + 33 at t=3. (4 Marks) f) Show that the curvature ? ? 1 ? of the spherical indicatrix = of the tangent is given by 1 2 ? = (2+2) 2 (5 Marks) g) Given that = cos)1 + (sin2 + 3 find (3 Marks) h) Prove that along the curve = we have that = 2 + + (5 Marks)
2
QUESTION TWO (20 MARKS)
a) Show that the first fundamental form on the surface of revolution = ()cos1 + ()sin2 + ()3 is given by = 22 + ('2 + '2)2 (5 Marks) b) Show that the torsion of the Binormal indicatrix = of a sufficiently regular curve is given by 3 = (2+2) (15 Marks)
QUESTION THREE (20 MARKS)
a) Show that along the curve = ; = 2 (8 Marks) b) Prove that at a point on a curve , where 0 the torsion is given by
=
' '' '''
' × ''2
(12 Marks)
QUESTION FOUR (20 MARKS)
a) Find the normal curvature and the normal curvature of the curve = 2 , = on the surface given by = 1 + 2 + (2 + 2)3 at the point where = 1 (10 Marks) b) Prove that if and are the curves in space such that = () and = () for all s, then and are the same except for their positions in space (10 Marks)

Sma2110:Real Analysis Ii Question Paper

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