GRIET(AUTONOMOU) POPULAR QUETION IN ADVANED ALULU UNIT-. If u = f(e z, e z, e u u u ) then prove that. z. If z u, Prove that u u u. zz. If r r e cos, e sin then show that r u u e [ urr u ]. 4. Find J, J ' v v for e sec u, e tan u and hence show that ' JJ. 5. how that the dependent variables in the following transformation are functionall dependent and also establish the relation 6. If u e sin z, u z, v z z, w z find 7. Evaluate 8. ompute (, ) J given u e cos, v e sin ( u, v) ( u, v, w) (,, ) given u, v v e cos z, w e ( u, v, w) (,,z), w 9. (a) Locate the stationar points and eamine their nature of the following function 4 4 4 (b) Find the stationar points and the etreme values of the function 9 + 6 (c) Find the critical points and stationar values of + 5 5 + 7. (a) The sum of three positive integers is.find the maimum of the product of the first, square of the second and the cube of the third (b) A rectangular bo open at the top has constant surface area 8 sq.ft.find its dimensions such that its volume is maimum (c) A rectangular bo open at the top is to be designed to have a fied capacit 4 cft. Determine its dimensions such that its surface area is a minimum using Lagrange s multiplier method (d) The temperature T at an point (,,z) in space is on the surface of the unit sphere + + z = T 4z.Find the highest temperature
(e) Locate the points on the sphere z which are nearest and farthest from the point (, 4 ) (e) Find the volume of the largest parallelepiped that can be inscribed in the ellipsoid a + b + z c = UNIT-II cos. (a) Find the perimeter of the hper ccloid, sin (b) Find the length of the curve a = ( a) (c) Find the length of the arc of the parabola = 4a cut off b the line = 8 (d) Find the length of the curve ( a ) 8a. (e) Find the perimeter of the curve r = a( + cosθ) and show that the arc of the upper half is bisected b θ = π/ (f) Find the length of the arc of the curve = log ( e ) from = to =. e + (g) how that the total length of the arc ( a )/ + ( b )/ = is 4(a +ab+b ). a+b (h) Find the length of the ccloid given b = a(θ + sinθ), = a( + cosθ). (a) Find the volume of the solid formed b the revolution of one arch of the ccloid = a(θ sinθ), = a( cosθ) about the base (b) Find the volume of solid formed b revolution of the area enclosed b the loop of the curve (a ) = (a + ) about -ais (c) Find the volume of the solid generated b revolution of one loop of the lemniscate r = a cosθ about (a) the initial line (b)about θ = π/ (d) Find the volume of solid generated b the revolution of the curve (a + ) = a about its asmptote (e) Find the volume of the solid generated b revolution of, about its asmptote a (f) Find the volume of revolution of the curve r = a( + cosθ) about the initial line (g) Find the volume of revolution of the hper ccloid / + / = a / about (a) X ais (b) Y ais (h) Find the volume of the solid generated b the revolution of the curve = asmptote a +a about its. (a) Find the surface area of the solid generated b the revolution of the asteroid
/ + / = a / about the -ais (b) Find the surface area generated b the arc of the ccloid = a(θ sinθ), = a( cosθ) revolving about (i) X ais (ii) the line = a (c) Find the surface area generated b revolution of loop of the curve 9a (a ) (d) Obtain the surface area of the solid of revolution of the curve r a( cos) about the initial line (e) Find the surface area generated b the revolution of an arc of the catenar = c cosh ( )about the X ais c UNIT-III. Evaluate the following double integrals (a) ( + )dd over the region R bounded b = and = log 8 log (b) e + d d (c) Evaluate d d where R is the region bounded b the parabola = 4 and = 4 (d) Evaluate e + d d over the triangle bounded b =, = and + = (e) Evaluate d d where R is the region in the first quadrant bounded b the hperbola = 6and the lines =, = and = 8 (f) Evaluate ( + ) d d throughout the area enclosed b the curves = 4, + =, = and = (g) Evaluate d d over the triangle with vertices at (,), (,)and (,). Evaluate the following double integrals b changing the order of integration (a) e d d b) d d c) dd a a a d) e d d e) 4 e 4 d d π π f) sin d d
. Evaluate the following double integrals b transforming into polar coordinates a) b) R d d dd where R is the region enclosed b,,. b with b > a over the annular region between the circles a and a a c) e ( + ) a a dd d) ( + ) d d e) e ( + ) dd f) a d d a a 4. Evaluate the following double integrals b change of variables ( ) a) e dd R ( ) where R is triangular Region bounded b =, = and =. Use = u-uv, =uv b) B using transformation + = u, = uv evaluate e /(+) dd c) ( + ) dd where R is the parallelogram in the -plane with vertices (,), (,), R (,), and (,) b using transformation u = +, v = 5. Evaluate the following triple integrals a) ( + + z) d d dz over the tetrahedron =, =, z = and the plane + + z = b) d d dz over the positive octant of the sphere z + + z = c) dzdd. d) (+++z) d d dz over the tetrahedron bounded b =, =, z = and the plane + + z = 6. Find the following volumes b triple integrals a) the clinders + = a and z = a b) the ellipsoid a + b + z c = c) the cone z and the Paraboloid z d) The cone z, and the sphere z a
UNIT-IV. Find the equations of the tangent plane and normal line to the surface z + = z at the point (,-,). Find the directional derivative of = 4z z at the point (,-,) in the direction of A = i j + 6k. Find the values of a, b and c so that the directional derivative of = a + bz + cz at (,,-)has a maimum of magnitude of 64 in a direction parallel to z-ais 4. Find the constants a and b so that the surface a bz = (a + ) will be orthogonal to the surface 4 + z = 4 at the point (,-,) 5. Find directional derivative of div( 5 i + 5 j + z 5 k) at the point P(,, ) in the direction of the outer normal to sphere + +z = 9 6. Prove that the field F = z i + ( z + z cos z)j + ( z + cos z)k is irrotational and hence determine the scalar potential such that = F 7. how that the vector field F = ( cos + z )i + (sin 4)j + (z + )k is irrotational and hence determine the scalar potential such that = F 8. Prove that ( A ) = (. A ) A 9. Evaluate the line integral F.dr where F = ( + )i + zj + (z )k and is (a) The space curve = t, = t, z = t in the range t (b) The line segments joining (,, ) to (,, ) then to (,, ) and then to (,, ) (c) The straight line joining (,, ) to (,, ). Evaluate the circulation ( ) d ( ) d directl where consists of the parabola = 8 and the line = in the anti-clock wise direction. how that the field F = ( cos + z )i + (sin 4)j + (z + )k is conservative. Find the potential and hence the work done in moving a particle in the field from (,, -) to ( π,, ). Evaluate the surface integral F. nˆ d where F = 6zi + ( + )j k and is the surface of the clinder + z = 9, =, =, z = and = 8. Evaluate F. nd ˆ for the field F = 6zi + 6j + k over the surface of the plane + + 4z = in the first octant 4. Find the flu of the field F = i + 4zj + k across the part of the clindrical surface + z = 5in the first octant bounded b = = z =, = 5
5. Evaluate + = 6, F. nd ˆ for the field F = z i + j k over the surface of the clinder z 4 in the first octant UNIT-V. Verif Greens Theorem for ( sin ) d cos d Where is the triangle in the XY plane with vertices at (,),(,) and(,). Verif Greens Theorem ( ) d ( ) d around the boundar of the region b 8 and. Verif Gauss (divergence) theorem for F = ( z)i + ( z)j + (z )k and is the urface of the parallelepiped bounded b a, b, z c 4. Verif Gauss theorem for F = i + j + z k over the surface of the clinder + = 9,z = and z = 5. Evaluate F. nˆ d b divergence theorem where F = z i + ( ) j + k and is the closed surface of the clinder bounded b + z = 6 and the planes =, = 5 6. Evaluate F. n ds b Gauss theorem for the field F = i + j + z k and is the surface of the sphere + + z = a 7. Verif tokes theorem for F = i + j where is the boundar of the rectangle whose sides are = =, =, = 4 in the plane z = 8. Verif tokes theorem for F = zi + ( + z)j + k where is the boundar of the triangle with vertices at (,, ), (,, ) and (,, ) 9. Verif tokes theorem for F = ( )i z j z k where is the surface of the hemisphere + + z = 6, z >. Evaluate d zd dz where is the curve of intersection of the sphere + + z = a and the plane + z = a *******