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) Explain each of the following; (i) Linearly dependent vectors (1 mark) (ii) Curvative vector of the curve (1 mark) (iii) Osculating plane (1 mark)
b) Given the vector
~ a=
~ i
~ j +
~ k ,
~ b =2
~ i +
~ j +2
~ k and ,
~~~~ 2 k jic ? ??? verify that
? ? ) ()( ~~~~~~~~~ cbacabcba ?????? (4 marks)
c) The position vector of a particle in space at time t is given by
~
~
~~ 3)3()( k tjtiittr ???? . Find the angle between the velocity and acceleration vectors at time t=0 (4 marks)
d) Show that the locus of the curve of curvative is an evolute only when the curve is a plane curve (4 marks)
e) Define the term ‘vectifiable arc’ and hence compute the length of the arc
~ x = ??????????????1 +
??????????????2 + ??????3 0 = ?? = ?? (5 marks)
2
f) Find the parametric equation of a line through the vector
~ a=??1+2??2-4??3parallel to both
vectors
~ u =2??2-??3 and
~ v=??1 + ??2+??23 (3 marks)
g) Show that the curve generated by
~ r = -2 + ????????
~ i + ??2 + 2
~ j + -2 + ??2 + 2????????
~ k
lies on the plane
~ a=
~ j + 2
~ k and normal to ?? = 2
~ i + ?? - ?? (4 marks)
h) If
~ x=
~ x(t) is a regular parametric representation on the interval I, proof that for all ???????? there exists a neighbourhood or to in which ~ xx is one to one. (3 marks)
QUESTION TWO (20 MARKS)
a) given the curve
~ x= ?????? 3????1 + sin?? 3????2 + 3????3, find the equation of the tangent line and the normal plane at ?? (6 marks)
b) Define a regular representation of a curve and show that the representation ?? = 2???????? - 1 = ?? = 2?? is regular. (6 marks)
c) Find the equation of the principal normal line at the point ?? = ?? 2
on the helix
~ x = ?????? ????1 + ?????? ????2 + ????3 (8 marks)
QUESTION THREE (20 MARKS)
a) Show that the arc length of a curve on the surface
?? = F(??1??) = ??1(??1??)
~ i + ??2(??1??)
~ j + ??3(??1??)
~ k is given by
dt
dt dv
G
dt dv
dt dv
F
dt dv
E
a
b
2
1
22 2? ? ? ? ? ? ? ? ? ? ? ? ? ? ????? ? ? ? ? ?
(5 marks)
b) Find the second fundamental form on the surface
~ v= ??
~ i + ??
~ j + (??2 - ??2)
~ k (8 marks)
3
c) Show that if the space curve x=x(s) has a constant tortion then the curve ??1 = -?? ??
+ ?????? has a constant curvative ±?? (7 marks)
QUESTION FOUR (20 MARKS)
a) show that the equation
~ x=(u+v)e1+(u-v)e2+(u2+v2)e3 defines a mapping of the uv-plane on the elliptic paraboloid x3=1 2 (??1 2+??2 2) and hence u-parameter and v-parameter curves
b) Find the equation of the binomial line and the rectifying plane along the curve
~ x= (3t-t3)e1+3t2e2+(3t+t3)e3 at the point t=1 c) Find the curvative vector k and the curvative k on the curve
~ x=te1+1 2
t2e2+1 3 t3e3 at the point
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