POPULAR QUESTIONS IN ADVANCED CALCULUS

Transcription

1 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 (b) Find the stationar points and the etreme values of the function (c) Find the critical points and stationar values of (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

2 (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

3 / + / = 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

4 . 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

5 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

6 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 *******