UNIT-II VECTOR CALCULUS IMPORTANT QUESTIONS: PART-A1. S.T2. If is solenoidal find .3. Find the directional derivatives of at (1,-2,-1) in the direction4. 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 find9. Find10. 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?
UNIT-II VECTOR CALCULUS PRAT-B1. 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 circle8. 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. ############################