This document provides 15 important questions on vector calculus concepts including directional derivatives, unit normals, solenoidal and irrotational vectors, and verification problems for Gauss's divergence theorem, Green's theorem, and Stokes' theorem. Example problems include finding directional derivatives, unit normals, determining if a vector is solenoidal or irrotational, evaluating line integrals, and verifying the vector calculus theorems for different bounding shapes and regions.

1. UNIT-II VECTOR CALCULUS
IMPORTANT QUESTIONS:
PART-A
1. S.T
2. If is solenoidal find .
3. Find the directional derivatives of at (1,-2,-1) in the direction
4. Find the unit normal to the surface at (-1,1,1).
5. Find the angle between the surface (i) at (2,-1,2).
(ii) at (1,1,1).
6. Find a’ and b’ such that the surface a cut orthogonally at
(1,-1,2).
7. If then find the value of .
8. If ,then find
9. Find
10. Prove that Curl(curl ) = grad(div ) - .
11. P.T the vector = is solenoidal.
12. If = is solenoidal find ‘a’.
13. Determine f(r) so that the vector f(r) is solenoidal.
14. P.T = is irrotational.
15. Find the ‘a,’b,and’c so that is irrotational.
16. Prove is irrotational.and also find scalar potential.
17. If = .Evaluate form (0,0,0) to (1,1,1) along the curve
x=t, y=
18. Define GAUSS DIVERGENCE THEOREM?
19. Define STOKE’S THEOREM?
20. Define GREEN’S THEOREM?
21. Using Greens theorem to find area of the ellipse and area of the circle?

2. UNIT-II VECTOR CALCULUS
PRAT-B
1. Evaluate where =z
include in the first octant z = 0 and z = 5.
2. Evaluate where =
which is the first octant.
3. Evaluate where and S is the surface of the plane 2x+y+2z=6
in the first octant.
4. Verify Green’s theorem in the XY plane for where C is the
Boundary of the region given by x = 0,y = 0,x+y = 1.
5. Verify Green’s theorem in the XY plane for where C is the boundary of the
region given by y = x and y = .
6. Verify Green’s theorem in the XY plane for where C is the
boundary of the region given by x = ,y= .
7. Verify Green’s theorem in the XY plane for taken round the circle
8. Evaluate where C is the square formed by the line x = ±1, y = ±1.
9. Verify the G.D.T for over the cube bounded by x = 0,x = 1,y = 0, y = 1, z =0
,z = 1.
10. Verify the G.D.T for taken over the rectangular
parallelepiped 0 ≤ x ≤ a, 0 ≤ y ≤ b,0 ≤ z≤ c.(or) x = 0, x =a, y = 0,y = b, z =0 , z = c.
11. Verify G.D.T for the function over the cylinder region bounded by ,
z = 0 and z = 2.
12. Using the G.D.T of where and S is the sphere .
13. Verify Stokes theorem for the vector in the rectangular region bounded by
XY plane by the lines x = 0,x = a, y = 0, y = b.
14. Verify stokes theorem for where S is the surface bounded by the plane
x=0,x=1,y=0,y=1,z=0,z=1 above XY plane.
15. Verify stokes theorem for taken around the rectangle bounded by the lines
x = ±a, y = 0, y = b.
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