MA6251-Mathematics II.pdf

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VALLIAMMAI ENGINEERING COLLEGE SRM NAGAR, Kattangulathur. DEPARTMENT OF MATHEMATICS MATHEMATICS-II(MA6251) Unit I Vector Calculus Part-A 1. Find the unit normal vector to the surface z y x 2 2 at (1,-2,5). 2. Prove that 0 ) ( grad curl . 3.Define Solenoidal vector function.If k z x j z y i y x V ) 2 ( ) 2 ( ) 3 ( is solenoidal, find the value of . 4. State Green’s theorem. 5. Prove that k xy j zx i yz F is irrotational. 6. State Gauss Divergence theorem. 7. Find such that k z y x j z y x i z y x F ) 2 ( ) 4 ( ) 2 3 ( is solenoidal. 8. Prove that the area of the region R bounded by C is  R C ydx xdy dxdy ) ( 2 1 . 9. Find the value of a so that the vector k az x j z y i y x F ) ( ) 2 ( ) 3 ( is solenoidal. 10. State the physical interpretation of the line integral B A r d F . .

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11. Prove that 0 , 3 r curl r div , where r is the position vector of a point ( x,y,z) in space. 12.State Stoke’s theorem. 13. Is the position vector k z j y i x r irrotational? Justify. 14. Find ‘a’ so that the vector j y xy i x y ax A ˆ 2 ˆ 2 2 is irrotational. 15.If the directional derivative of the function 2 2 2 z y x at the point (1, 2, 3) in the direction of k j i ˆ ˆ ˆ is 2, find α. 16. Find ‘a’ such that k z y x j z ay x i z y x ˆ ) 2 ( ˆ ) 4 ( ˆ ) 2 3 ( is solenoidal. 17. Determine f(r) so that the vector r r f ) ( is solenoidal. 18. If j xy i x F ˆ ˆ 2 2 , evaluate the line integral r d F from (0, 0) to (1, 1) along the path y = x. 19. Evaluate C ydx xdy , where C is the circle . 4 2 2 y x 20. Find the work done by the force k z x j x i z xy F 2 2 3 3 2 when it moves a particle from (1 ,-2 , 1) to (3 ,1 , 4) along any path. PART-B 1. (a)Verify Guass divergence theorem for k z j y i x F 2 2 2 taken over the cube bounded by the planes x=0,y=0,z=0,x=1,y=1 and z=1. (b)Find the value of n such that the vector r r n is both solenoidal and irrotational. 2. (a) Verify Stokes theorem for j xy i y x F 2 ) ( 2 2 in the rectangular region of z=0 plane bounded by the lines x=0,y=0,x=a and y=b. (b)Show that the vector field j y x y i xy x F ) ( ) ( 2 2 2 2 is irrotational. Find its scalar potential.
3. (a)Verify Stokes theorem for j xy i y x F 2 ) ( 2 2 taken around the rectangle formed by the lines x=-a,x=a,y=0,y=b. (b)Find the values of a and b so that the surfaces 2 2 3 ) 3 ( x a z by ax and 11 4 3 2 z y x may cut orthogonally at(2,-1,-3).

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