Download OU B.Sc Computer Science 6th Sem Vector Calculus Important Questions

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Subject Title: Vector calculus Prepared by: B.Lalitha

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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=t² ,z=t³.

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5. If F= x² y² i+yj then evaluate ?_CF.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 ? r x dr 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)

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10. Find the line integral of F=(y,-x,0) along the curve consisting of the two st.line segments a) X=1, 1=y=2 b) y=1, 0=x=1
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)+f(x-ay), prove that ?²z/?y²=a²?²z/?x²

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15. Define Gradient.
16. Define Divergence.
17. Define Curl.
18. Compute the gradient of the scalar function f(x,y.z) = e^(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).

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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.

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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=x² i+y² j+z³k then find div curl F.
28. Show that the vector e^x?² (i+j+k) is solenoidal.
29. Prove that F=yz+zx+yxk is irrotational

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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=v(x²+y²+z²).
34. Show that ?. (?. F)= ?X(?Xf) + ?² f

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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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