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Satyabama niversity questions in vector

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Satyabama niversity questions in vector

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Satyabama niversity questions in vector

  1. 1. Satyabama niversity questions in Vector. 1. Find ∇ φ if φ = x3 y2 z4 . 2. Find the directional derivative of φ = x2 yz + 4xz2 + xyz at (1,2,3) in the direction of 2 i + j − k 3.State Gauss Divergence theorem 4.State Gauss-Divergence theorem. 5.Show that kyxzjzxizxyf ˆ)3(ˆ)3(ˆ)6( 223 −+−++= is irrotational. 6.Prove that grade 2 g fgradgfgradf g f − =      7. If u = x2 – y2 , prove that ∇2 u = 0. 8.Find grad ϕ at the point (1, -2, -1) when ϕ = 3x2 y – y3 z2 . 9. Show that F = (y2 – z2 + 3yz – 2x) i + (3xy + 2xy) J + (3xy – 2xz + 2z) K is solenoidal. 10.Show that F = (x + 2y) i + (y + 3z) j + (x – 2z) k is solenoidal . 11. State stoke’s theorem. 12.Find the directional derivative of x2 + 2xy at (1, –1, 3) in the direction of x axis. 13. ∫ rdF.  is independent of the path when? Prove that div grad f = ∇2 f. 14.. State Stoke’s theorem. 15.Find the directional derivative of φ = x2 y2 z2 at the point (1, 1, 1) in the direction .kji ∧∧∧ ++ 16. Find the value of ‘a’ so that the vector kji zxzayyxF ∧∧∧ =+−++= )3()3()3( is solenoidal 17.Define solenoidal vector function and irrotational motion. 18. If ( ) ∫−+−= 4 2 32 )(,7635)( dttFfindthenktjtitttF Find grad ∅ if ∅ = xyz at (1,1,1). 19. Show that kzjyixF  222 ++= is a conservative vector field 20.Find a unit normal vector ‘n’ of the cone of revolution z2 = 4(x2 + y2 ) at the point (1, 0, 2). 21.. Is the flow of a fluid whose velocity vector v = [secx, cosecx, 0] is irrotational? 22.Determine the constant a so that the vector kzxjayxizxF  )5()3()( −++++= is solenoidal. 23. State Green’s theorem. 24.If r is the position vector of the point (x, y, z), a is a constant vector and φ = x2 + y2 + z2 , then find (a) grad ( r ο a ) and (b) r °gradφ.
  2. 2. 25.. If F is a solenoid vector, find the value of curl (curl(curl(curl F ))) Show that F = (y2 - z2 + 3yz - 2x)i + (3xz + 2xy) j + (3xy – 2xz + 2z) k is irrotational. 26.. State Green’s theorem in a plane. 27. Define irrotational vector 28.State Guass divergence theorem 29.If FcurldivfindkZjYiXF  333 ++= 30.Prove that 0. =∫ c rdr  31.Prove that (3x + 2y + 4z) → i + (2x + 5y + 4z) → j + (4x + 4y – 8z) → k is irrotational. 32. State Gauss-Divergence theorem. 33.Find the directional derivative of φ = x2 y2 z2 at the point (1, 1, 1) in the direction .kji ∧∧∧ ++ 34. Find the value of ‘a’ so that the vector kji zxzayyxF ∧∧∧ =+−++= )3()3()3( is solenoidal. 35.Determine the constant a so that the vector kzxjayxizxF  )5()3()( −++++= is Solenoidal. 36.State Gauss Divergence theorem. 36.If uand v   are vector point functions, then prove that ( ) ( ) ( )u v v u u v∇ × = ∇× − ∇×       g g g . 38.State Green’s theorem in a plane. 39.Find a and b such that → F = 3x2 → i + (ax3 -2yz2 ) → j +(3z2 -by2 z) → k is irrotational. 40.Prove that div → r = 3 , if → r = x → i + y → j + z → k . Using Green’s theorem prove that the area enclosed by the simple closed curve ‘C’ is ( )∫ − . 2 1 ydxxdy 41.. Prove r nn nrr → − =∇ 2 I. Verify Green’s theorem in the plane for ∫ (3x2 – 8y2 ) dx + (4y – 6xy) dy where C is the boundary of the C region defined by x = 0, y = 0, x + y =1. II. (a) Show that the vector F = (3x2 + 2y2 + 1) i + (4xy – 3y2 z – 3) j + (2-y3 ) k is irrotational and find its scalar potential. (b) If F = xy i + (x2 + y2 ) j, find ∫ F. dr where C is the arc of the parabola y = x2 – 4 from (2, 0) to (4, 12).
  3. 3. III.(a) Show that F = (6xy + z3 ) i + (3x2 – z) j + (3xz2 –y) k is solenoidal and hence find the scalar potential. (b) Find the directional derivative of f(x,y,z) = x2 yz + 4xz2 at (1,–2, –1) in the direction of 2 i – j – k IV. Verify Gauss divergence theorem F = (x2 – yz) i + (y2 – zx) j + (z2 – yx) k and s is the surface of the rectangular parallelepiped bounded by x=0, x=a, y=0, y=b, z=0, z=c. V. (a) Prove that 22 )1( − +=∇ nn rnnr where kzjyixr ˆˆˆ ++= b) Prove kxzjxizxyF ˆ3ˆˆ)2( 223 +++= is a conservative force. Find φ so that F=∇φ . VI. Given the vector kxyyjxyiyxF ˆ)(ˆ2ˆ)( 222 −++−= verify Gauss-Divergence theorem over the cube with centre at the origin and of side length a. VII. Verify the Gauss divergince theorem for kyzjyixzF +−= 2 4 over the cube bounded by x = 0, x = 1, y = 0, y = 1, z = 0 and z = 1. VIII. (a) Find the directional derivative of f = xyz at (1, 1, 1) in the directions of i + j + k, i, -i. (b) Evaluate kzyjxizFwheredsnF s 22 ,. −+=∫∫ and S is the surface of the cylinder x2 + y2 = 1, included in the first octant between the planes z = 0 and z = 2. IX. Verify Stoke’s theorem for KyxJxzizyF 222 ++= where S is the open surface of the cube formed by the planes x = ± a, y = ± a, and z = ± a in which the plane z = -a is cut. X. Verify Gauss divergence theorem for KzJyIxF 222 ++= where S is the surface of the cubold formed by the planes x = 0, x = a, y = 0, y = b, z = 0 and z = c. XI. Show that F = (y2 + 2xz2 ) i + (2xy – z) j + ( 2x2 z – y + 2z) k is irrotational and hence find its scalar potential. XII. Verify Gauss divergence theorem F = x2 i + y2 j + z2 k, where S is the surface of the Cuboid formed by the planes x = 0, x = a, y = 0, y = b, z = 0, z = c. XIII. (a) Prove that 22 )1( − +=∇ nn rnnr . (b) Find ∫ c rdF.  , kzjyixF  22 24 +−= where S is the upperhalf of the surface of the sphere 1222 =++ zyx , C is its boundary. XIV. a) Find the workdone in moving a particle in the force field F = 3x2 i + (2xz-y)j + zk along (i) the straight line from (0,0,0) to (2,1,3). (ii) the curve defined by x2 = 4y, 3x3 = 8z from x=0 to x = 2. (b) Prove that ∇r n = nr n-2 r  , where .zkyjxir ++=  XV. (a) Prove that ∇ x (∇ x V) = ∇ (∇.V) - ∇2 V. (b) Verify Gauss divergence theorem, for f = 4xzi – y2 j + yzk taken over the cube bounded by x = y = z = 0 of x =y=z= 1. XVI. Show that       =∇+=∇ − .0 1 )1( 222 r duceandhencedernnr nn
  4. 4. XVII.a.) Verify Gauss divergence theorem for kji yzyxzF ∧∧∧ +−= 2 4 over the cube bounded by x = 0, x = 1, y = 0, y = 1. z = 0 and z = 1. b.) Using Stoke’s theorem evaluate ∫c [(x + y) dx + (2x – z) dy + (y + z) dz] where c is the boundary of the triangle with vertices (2, 0, 0), (0, 3, 0) and (0, 0, 6). XVIII.a,) Verify Divergence theorem for ( ) ( ) ( ) kxyzjxyiyzxF −+−+−= 222 3 taken over the rectangular parallelopiped 0 ≤ x ≤ a, 0 ≤ y ≤ b, 0 ≤ z ≤ c. b.)Verify Stokes theorem for kzxjyzixyF  −−= 2 where S is the open surface of the rectangular parallelepiped formed by the planes x = 0, x=1, y=0, y=2 and z = 3 above the XOY plane. XIX.. Find the values of the constants a,b,c so that kyxzjczxibzaxyF  )3()3()( 223 −+−++= may be irrotational. For these values of a,b,c find also the scalar potential of F  . XX.Verify Green’s theorem for ∫C [(xy + y2 ) dx + x2 dy], where C is bounded by y = x; and y = x2 . XXI..a.) Using Stoke’s theorem evaluate ∫C [(x + y) dx + (2x – z) dy + (y + z) dz] where C is the boundary of the triangle with vertices (2, 0, 0), (0, 3, 0) and (0, 0, 6). b.)Prove that kxzjxyizxyF  )23()4sin2()cos( 232 ++−++= is irrotational and find its scalar potential. XXII. Verify Stroke’s theorem for a vector field defined by jxyiyxF  2)( 22 +−= in the rectangular region in the XOY plane bounded by the lines x=0, x=a, y=0 and y=b. XXIII. (a) Prove the relation: curl(curl F ) = ∇(∇° F ) - ∇2 F . (b) Verify Gauss divergence theorem for F = x2 i + y2 j + z2 k , where S is the surface of the cuboid formed by the planes x=0, x=a, y=0, y=b, z=0 and z=c. XXIV.. (a) Verify Stoke’s theorem for F = - y i + 2yz j + y2 k, where S is the upper half of the sphere x2 + y2 + z2 = a2 and C is the circular boundary on the xoy plane. XXV. Verify Stokes theorem when F = (2xy – x2 )i – (x2 – y2 )j and C is the boundary of the region enclosed by the parabolas y2 = x and x2 = y. XXVI. Verify Gauss divergence theorem for F = x2 i + y2 j + z2 k where S is the surface of the cuboid formed by the planes x = 0, x = a, y = 0, y = b, z = 0 and z = c. XXVII.(a) Find the angle between the surfaces x2 +y2 =2 and x+2y- z=2 at (1,1,1) (b) Find the value of m, if ( ) ( ) ( )kyxz3jzmxizxy6F 223 −+−++= is irrotational. Find also φ such that F = grade φ XXVIII. Verify Divergence theorem for ( ) ( ) ( )kxyzjzxyiyzxF 222 −+−+−= taken over the rectangular parallelopiped bounded by x=o,x=a,y=o,y=b,z=o,z=c
  5. 5. XXIX. (a) Find directional derivative of Φ = 3x2 +2y-3z at (1,1,1) in the direction 2 kji  −+ 2 (b) Show that kyxzjzxizxyF  )3()3()6( 223 −+−++= is irrotational vector and find the scalar potential function Φ∇=F  XXX. verify Gauss divergence Theorem for the function kzjxiyF 2 ++=  over the cylindrical region bounded by x2 +y2 =9, z=0 and z=2. XXXI. (a) Find the directional derivative of φ = xy + yz + zx at (1, 2, 3) in the direction of 3 → i +4y → j + 5 → k . (b) Find the work done in moving a particle by the force → F = 3x2 → i + (2xz-y) → j + z → k along the line joining (0, 0, 0) to (2, 1, 3). XXXII. Verify Green’s theorem in the plane for dyxyydxxyx C )2()( 232 −+−∫ where C is a sqare with the vertices (0, 0), (2, 0), (2, 2), (0, 2). XXXIII, A light horizontal strut AB is freely pinned at A and B. It is under the action of equal and opposite compressive forces P at its ends and it carries a load ‘w’ at its center. Then for ., 2 cos sin 2 Pr. 2 000. 2 1 , 2 0 2 2 2 EI P n x nl n nx p w ythat ove l xat dx dy andxatyAlsox w Py dx yd EI L x =           − = ====−=+<< Show that       =∇+=∇ − .0 1 )1( 222 r duceandhencedernnr nn XXXIV.a.) Verify Gauss divergence theorem for kji yzyxzF ∧∧∧ +−= 2 4 over the cube bounded by x = 0, x = 1, y = 0, y = 1. z = 0 and z = 1. b.)Prove that kxyzjzxyiyzxA  )6()4()2( −−+++= is solenoidal as well as irrotational. Also find the scalar potential of A. XXXV.Verify Green theorem in the plane for ∫ +− C xydydxyx ],2)[( 22 where C is the closed curve of the region bounded by y = x2 and y2 = x. XXXVI.(a) Verify Green’s theorem in a plane for the integral ( 2 ) C x y dx x dy− +∫ taken around the circle C: 2 2 1x y+ = . (b) Determine ( )f r so that ( )f r r  is Solenoidal.
  6. 6. xxxviii.(a) Evaluate µ S F n dS∫∫ u g where F yz i zx j xy k= + + u   u and S is the part of the surface of the sphere 2 2 2 1x y z+ + = which lies in the first octant. (b) Verify Stoke’s theorem 2 2 ( ) 2F x y i xy j= + − u   taken around the rectangle bounded by the lines , 0,x a y y b= ± = = . XXXIX. (a) What is the angle between the surfaces x2 + y2 + z2 = 9 and x2 + y2 + z2 – 3 = 0 at (2, -1, 2). (b) Show that → F = (2xy+z3 ) → i + x2 → j + 3xz2 → k is a conservative field. Find the scalar potential φ . XL.. Verify Stoke’s theorem for → F = xy → i + xy2 → j taken round the square bounded by the lines x = 1, x = -1, y = 1, y = -1. XLI.a)Prove that the given ( ) ( ) kzxjxyizxyF 232 34sin2cos +−++= is irrotational and find the scalar potential (b) Using Green’s theorem evaluate ( ) ( )∫ −+− c dyxyydxyx 6483 22 where C is the boundary of the region enclosed by 2 xyandxy == XLII. (a) If kzjyixr ++= then prove that div (grad (rn )) = n(n + 1) rn–1 . (b) Verify Gauss Divergence theorem for the given function kzjxiyF 2 ++= over the cylindrical region bounded by x2 + y2 = 9, z = 0 and z = 2.
  7. 7. xxxviii.(a) Evaluate µ S F n dS∫∫ ur g where F yz i zx j xy k= + + ur r r ur and S is the part of the surface of the sphere 2 2 2 1x y z+ + = which lies in the first octant. (b) Verify Stoke’s theorem 2 2 ( ) 2F x y i xy j= + − ur r r taken around the rectangle bounded by the lines , 0,x a y y b= ± = = . XXXIX. (a) What is the angle between the surfaces x2 + y2 + z2 = 9 and x2 + y2 + z2 – 3 = 0 at (2, -1, 2). (b) Show that → F = (2xy+z3 ) → i + x2 → j + 3xz2 → k is a conservative field. Find the scalar potential φ . XL.. Verify Stoke’s theorem for → F = xy → i + xy2 → j taken round the square bounded by the lines x = 1, x = -1, y = 1, y = -1. XLI.a)Prove that the given ( ) ( ) kzxjxyizxyF 232 34sin2cos +−++= is irrotational and find the scalar potential (b) Using Green’s theorem evaluate ( ) ( )∫ −+− c dyxyydxyx 6483 22 where C is the boundary of the region enclosed by 2 xyandxy == XLII. (a) If kzjyixr ++= then prove that div (grad (rn )) = n(n + 1) rn–1 . (b) Verify Gauss Divergence theorem for the given function kzjxiyF 2 ++= over the cylindrical region bounded by x2 + y2 = 9, z = 0 and z = 2.

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