Vector Analysis Cat Ii Question Paper
Vector Analysis Cat Ii
Course:Bachelor Of Science In Analytical Chemistry
Institution: Jomo Kenyatta University Of Agriculture And Technology question papers
Exam Year:2013
VECTOR ANALYSIS
1. Determine the value of k so that the vectors u=<-3,k,4> and v=<3,-k,-4> are orthogonal. (3 Marks)
2. Find the parametric equation of a line passing through point (-1,2,7) and parallel to the vector r=(4,4, 2).
(3 Marks)
3. A particle move along a curve given by the position vector r=cos (?t)i+sin (?t)j. For ? a
constant. Find
(i) The velocity of the particle at any given time t. Show that the velocity is
perpendicular to the position vector r,
(ii) Acceleration of the particle and show that it’s directed towards the origin. (3,3 mks)
4. Show that the force field F= (y2-2xyz3)i+( 3+2xy-x2z3)j+(6z3-3x2yz2)k is a conservative forcefield.
Hence determine the potential F and the work done in moving a particle from (3,0,-2) to (-2,1,0) in
the force field . (For conservative forcefield curlF=0 ) (6mks)
5. i) Determine directional derivative of at P(-1, 0, 3) from P in the
direction of Q(2,1,-2). (4mks)
ii) A function f(x,y)=x2-y2+3xy.. Find the rate of change of f(x,y) from P(1,0) to
Q(-1,-2). What is the maximum rate of change of f(x,y)? (4mks)
iii) Find the equation of the tangent plane to z=x2+y2 +2xy at the point (1,-1,2). (4mks)
6. If find given that and at t=0.
(4mks)
7. Evaluate and C the curve given by C:
x=t ,y=t2+1, t=0 to t=1. (4mks)
8. i) Evaluate across a square bounded by the lines by use of
Green’s theorem. (6mks)
ii)Verify Greens theorem in a plane for the vector field for the region
bounded by and . (6mks)
iii) Verify stoke’s theorem for a portion of the parabloid z=4-x2-y2 that lies upon the
plane z=0 in the vector field . (6mks)
9. Find T, N and K i.e the unit tangent vector, the unit normal and the curvature of the helix given by the
parametric equations . (4mks)
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