Download OU (Osmania University) B.Sc Computer Science 6th Sem Vector Calculus Important Question Bank For 2021 Exam
I S
Subject Title: Vector calculus
Prepared by: B.Lalitha
Semester: VI
Updated on: 20-02-2020
Unit - I: LINE INTEGRALS AND SURFACE INTEGRALS
1.
Define line integral.
2.
Define Surface integral.
3.
If F=xyi-zj+x2k and C is the curve x=t2 , y=2t, z=t3 from t=0 to t=1.Evaluate
cF.dr.
4.
If F= (3x2+6y)-14zj+20xzk then evaluate the line integral c F.dr from (0,0,0) to
(1,1,1) along x=t, y=t ,z= t3 .
5.
If F= x2 y 2 i+yj then evaluate c F.dr where C is the curve y2=4x in the XY plane
from (0,0) to (4,4).
6.
Prove that the work done by a force F depends on the end points and not on the
path in a conservative field.
7.
Find the line integral c rxdr where the curve C is the ellipse x2/a2 +y2/b2 =1
taken in anti clock wise direction .what do you notice about the magnitude if the
answer?
8.
If F= (5xy-6x2)i + (2y-4z)j Evaluate c F.dr along the curve c in the xy-plane
given by y=x3 from the point (1,1) to (2,8).
9.
Compute the line integral (y2 dx-x2dy)around the triangle whose vertices are
(1,0),(0,1) and (-1,0).
10.
Find the line integral of F= (y,-x,0) along the curve consisting of the two st.line
segments
a) X=1,1y2 b) y=1,0x1
11.
Evaluate s A.n ds where A=18zi-12j+3yk and S is that of the plane
2x+3y+6z=12 which is located in the first octant.
12.
Defined work done by force
Unit - II: VOLUME INTEGRALS,GRADIENT,DIVERGENCE AND CURL.
13.
Define Volume integral.
14.
If z=f(x+ay)+(x-ay), prove that 2z/y2=a22z/x2
15.
Define Gradient.
16.
Define Divergence.
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17.
Define Curl.
18.
Compute the gradient of the scalar function f(x,y.z) = exy (x+y+z) at (2,1,1).
19.
Find a unit normal vector to the surface x2 +y2 +2z2 =26 at the point(2,2,3).
20.
Find the unit normal to xy=z2 at(1,1,-1).
21.
Find the angle between the two surfaces x2+y2 +z2 =9, x2 +y2-z =3 at (2,-1,2).
22.
Find the directional derivative of 2xy+z2 at (1,-1,3) in the direction of i+2j+3k.
23.
Find the volume of the tetrahedron with vertices at (0,0,0),(a,0,0),(0,b,0) and
(0,0,c).
24.
If F=(2x2 -3z)i-3xyj-4xk, evaluate .F dv and xF dv where v is the closed region
bounded bt x=0,y=0,z=0,2x+2y+z=4.
25.
Show that the vector field F=(x2+xy2 )i+ (y2 +x2y)j is the conservative and find
the scalar potential function.
Unit - III: DIVERGENCE AND CURL OF A VECTOR FIELD
26.
If A is a vector function find div(curlA).
27.
If f=x3 i+y3 j+z3k then find div curl F.
28.
Show that the vector ex+y-2z (i+j+k) is solenoidal.
29.
Prove that F=yz+zx+yxk is irrotational
30.
Find the value of a,b.c such that the following vector is irrotational F=
(x+2y+az)i+(bx-3y-z)j+(4x+cy+2z)k.
31.
If F is a conservative vector field show that curl F=0
32.
Find divF, where F=rn r. find n if it is solenoidal.
33.
Evaluate 2 log r where r=(x2+y2+z2).
34.
Show that . (. F) = X(Xf) + 2 f
35.
Show that the vector field F=(x2 -yz)I +(y2 -zx)j +(z2-xy)k is conservative and find
the scalar potential function corresponding to it.
36.
Find the curl f=grad (x3 +y3 +z3 -3xyz).
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This post was last modified on 23 January 2021