- Using Stoke theorem evaluate integral sinz dxcosxdy sinydz where C is the boundary of rectangle Using Stoke theorem evaluate integral sinz dx-cosxdy sinydz where C is the boundary of rectangle 0
- Using Stoke theorem evaluate integral over C Fdr where C is a circle with centre Using Stoke theorem evaluate integral over C F.dr where C is a circle with centre (0,0,3) and radius 5 in the plane z =3. F = (2xy-2z)I (2x-4yy^2)j(x-2y-z^2)k >
- Using Stoke theorem find out integral over C Fdr where Using Stoke theorem find out integral over C F.dr where F = y^2i x^2j -(xz)k and C is the traignel with vertices (0,0,0) (1,0,0) and (1,1,0) >
- Solve 3x22 y 33x2y36y 3x24x1 Solve (3x2)^2 y" 3(3x_2)y-36y = 3x^24x1 >
- Solve D21 y x sinx by method of variation of parameters Solve (D^21) y = x sinx by method of variation of parameters. >
- Solve y 2y2y ex tanx by variation of parameters method Solve y"-2y2y = e^x tanx by variation of parameters method. >
- y y sec2 Solve by variation of parameters y"y = sec^2 x Solve by variation of parameters. >
- Find the image of the circle z 2 under the transformation w 3z Find the image of the circle |z|=2 under the transformation w=3z >
- Transform integral fxy dxdy by the substitution xy u and y uv Transform integral f(x,y) dxdy by the substitution xy = u and y =uv >
- Find the volume of the portion of the sphere x2y2z2 1 lying inside the cylinder Find the volume of the portion of the sphere x^2y^2z^2 =1 lying inside the cylinder x^2y^2 =x >
- Evaluate triple integral dxdydzx2y2z2 throughout the volume of the sphere Evaluate triple integral dxdydz/(x^2y^2z^2) throughout the volume of the sphere x^2y^2z^2 = a^2 >
- Find triple integral over the volume cut off the sphere x2y2z2 a2 by the cone x2y2 z2 Find triple integral over the volume cut off the sphere x^2y^2z^2 = a^2 by the cone x^2y^2 =z^2 >
- Find the equation of the tangent plane and normal to the surface Find the equation of the tangent plane and normal to the surface xz^2x^2 z = z-1 at (1,-3,2) >
- Find double integral over a square with vertices 10 21 12 and 01 for the function Find double integral over a square with vertices (1,0) (2,1) (1,2) and (0,1) for the function (x-y)^4 e^(xy) dxdy >
- Show that the level surfaces of scalars fxyz y2z2 x and gxyz ln y2z2 4x form orthogonal families Show that the level surfaces of scalars f(x,y,z) = y^2z^2 -x and g(x,y,z) = ln (y^2z^2)4x form orthogonal families. >
- Find a and b if the two surfaces cut orthogonally Find a and b if the two surfaces cut orthogonally.
- Find phix if ph122 4 and del phi 2xyz3 i x2 z3 j 3x2 yz2 k Find phi(x) if ph(-1,2,2) = 4 and del phi = 2xyz^3 i x^2 z^3 j 3x^2 yz^2 k >
- Show that vector F3y4z2i4x3z2j3x2y2k is solenoidal Show that vector F=3y^4z^2i4x^3z^2j-3x^2y^2k is solenoidal >
- Show that F 6xyz3I 3x2zj 3xz2y k is irrotational Show that F = (6xyz^3)I(3x^2-z)j (3xz^2-y)k is irrotational. >
- Prove that div r 3 and curl r 0 where r is the position vector of a point xyz in space Prove that div r =3 and curl r =0 where r is the position vector of a point (x,y,z) in space. >
- Find the directional derivative of phi xy2 z3 at the point 111 along the normal to the surface Find the directional derivative of phi = xy^2 z^3 at the point (1,1,1) along the normal to the surface x^2xyz^2 =3 at (1,1,1) >
- If delphi yzi xz jxy k find phi If delphi = yzi xz jxy k find phi. >
- Find div gradphi and curl grad phi at 111 for phi Find div (gradphi) and curl (grad phi) at (1,1,1) for phi =x^2y^3 z^4 >
- If vector v cross product of vectors w and r then prove that vector w curl of vector If vector v = cross product of vectors w and r then prove that vector w = curl of vector v /2 >
- Prove that the area enclosed by a closed curve C is 12 of integral xdy ydx Hence find area of the ellipse Prove that the area enclosed by a closed curve C is 1/2 of integral xdy-ydx. Hence find area of the ellipse. >
- Evaluate using Green theorem integral over a circle x2y2 4 for the function y2xy1dx x2xy1dy Evaluate using Green theorem integral over a circle x^2y^2 =4 for the function y(2xy-1)dx x(2xy1)dy >
- Using Green theorem find the area of the hypocycloid x23y23 a23 a Using Green theorem, find the area of the hypocycloid x^(2/3)y^(2/3) = a^(2/3), a>0 >
- Find the area between the curves y2 4x and x2 4y using Green theorem Find the area between the curves y^2 =4x and x^2 =4y using Green theorem. >
- Show that the function fz x iy is no where differentiable Show that the function f(z) = x-iy is no where differentiable. >
- Check whether the function w sinz is analytic Check whether the function w=sinz is analytic >
- Check whether the function 2xyix2y2 is analytic Check whether the function 2xyi(x^2-y^2) is analytic >
- Prove that fz ez is analytic Prove that f(z) = e^z is analytic >
- If uiv is analytic prove that v iu is also analytic If uiv is analytic prove that v-iu is also analytic. >
- Solve D44D38D28D4y Solve (D^44D^38D^28D4)y =0 >
- Solve D25D6y 11e5x Solve (D^25D6)y = 11e^5x >
- Prove that u x33xy23x23y2 1 satisfies Laplace equation and determine the corresponding analytic function Prove that u = x^3-3xy^23x^2-3y^21 satisfies Laplace equation and determine the corresponding analytic function f(z) = uiv >
- Construct an analytic function fz for which real part is ex cosy Construct an analytic function f(z) for which real part is e^x cosy >
- Find Lft if ft 3 0t5 0 elsewhere Find L{f(t)} if f(t) = 3, 0
- Integral 0 t0 infty e2t t sin 3t dt using Laplace Integral 0 t0 infty e^(-2t) t sin 3t dt using Laplace >
- Using integration find out Laplace cos2t cos3 Using integration find out Laplace {(cos2t-cos3t)/t} >
- If u log x2y2 find v such that fz uiv is analytic If u = log (x^2y^2) find v such that f(z) = uiv is analytic. >
- Find the critical points for the transformation w2 Find the critical points for the transformation w^2 = (z-a)(z-b) >
- If x uv and y uv show that JJ1 If x =uv, and y =u/v show that JJ=1 >
- If x uv cos v y u sin v then find J and verify it is reciprocal of J If x =uv cos v , y = u sin v then find J and verify it is reciprocal of J. >
- If u xyz and uv yz and uvw z find Jacobian If u = xyz and uv = yz, and uvw =z find Jacobian
- f x v2w2 y w2u2 and z u2v2 find J and use that to find J inverse Jacobian transformation f x = v^2w^2, y = w^2u^2 and z = u^2v^2 find J and use that to find J (inverse Jacobian transformation)
- If the given matrix A has two eigen values equal to 1 Find the eigen values of A inverse If the given matrix A has two eigen values equal to 1. Find the eigen values of A inverse. 2 2 1 1 3 1 1 2 2 >
- Consider the matrix A 6 2 2 2 3 1 2 1 3 If product of two eigen values are 16 find the third eigen value Consider the matrix A 6 -2 2 -2 3 -1 2 -1 3 If product of two eigen values are 16 find the third eigen value. >
- Evaluate using Contour integration integral of 454sin t where t varies from 0 to Evaluate using Contour integration integral of 4/54sin t where t varies from 0 to 2pi >
- Find integral 0 to 2pi dtheta12xsintheta x2 0z1 using contour integration Find integral 0 to 2pi dtheta/(1-2xsintheta x^2), 0