Download OU (Osmania University) B.Sc Computer Science 6th Sem Vector Calculus Important Question Bank For 2021 Exam

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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