- 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
- Prove that integral 0 to 2pi the function dt2cos t is 2pirt3 using contour integration Prove that integral 0 to 2pi the function dt/2cos t is 2pi/rt3 using contour integration. >
- The circle on the sphere x2y2z24x2y8z6 0 has centre 212 Find the equation of the circle The circle on the sphere x^2y^2z^24x-2y8z6 =0 has centre (2,1,2). Find the equation of the circle. >
- Show that the line passing through 673 and having direction ratios 345 touches the sphere Show that the line passing through (6,7,3) and having direction ratios (3,4,5) touches the sphere x^2y^2z^2-2x-4y-4=0. Find the point of contact. >
- A sphere is inscribed in the tetrahedran whose faces are 3 axes and 2x6y3z 14 Find its centre radius and its equation A sphere is inscribed in the tetrahedran whose faces are 3 axes and 2x6y3z =14. Find its centre, radius and its equation. >
- Find the equation of the cone with vertex at 111 and which passes through the curve given by Find the equation of the cone with vertex at (1,1,1) and which passes through the curve given by x^2y^2 =4 and z =2 >
- Evaluate double integral xyx2y232 over the positive quadrant of the circle Evaluate double integral xy(x^2y^2)^(3/2) over the positive quadrant of the circle x^2y^2 =k^2 >
- If F x3iy3jz3 k find div curl F If F =x^3iy^3jz^3 k find div curl F >
- If v x3yi y2zjxazk is solenoidal find the value of a If v = (x3y)i (y-2z)j(xaz)k is solenoidal, find the value of a. >
- Prove that del rn nrn2 r Prove that del r^n = nr^(n-2) r >
- If F 3xy3x2z2i 2x2 j 2x3z k check whether the integral over C Fdr is independent of the path If F = (3xy-3x^2z^2)i 2x^2 j - 2x^3z k , check whether the integral over C, F.dr is independent of the path C >
- If u xyyzzx v x2y2z2 and w xyz determine the function relationship between If u =xyyzzx, v = x^2y^2z^2 and w = xyz determine the function relationship between u,v,w. >
- If u xyxy and v xyxy2 prove that u and v are functionally dependent If u =(xy)/)x=y) and v = xy/(x-y)^2 prove that u and v are functionally dependent. >
- Find the constants a and b such that the matrix a 4 1 b has 3 and 2 as its eigen values Find the constants a and b such that the matrix a 4 1 b has 3 and -2 as its eigen values. >
- If the system of equations x2yz 0 5xyz 0 and x5ylemdaz 0 has non trivial solution find the value of If the system of equations x2yz =0, 5xyz =0 and x5y(lemda)z =0 has non trivial solution find the value of c. >
- If one of the eigen values of A is 9 find the other two values If one of the eigen values of A is -9, find the other two values. A = 7 4 4 4 -8 -1 4 -1 -8 >
- Prove that the eigen values of 3 A inverse are the same as those of Prove that the eigen values of 3 A inverse are the same as those of A = 1 2 2 1 >
- If the sum of two eigen values and trace of 3x3 matrix A are equal find the value of determinant If the sum of two eigen values and trace of 3x3 matrix A are equal find the value of determinant A >
- Find curl F where F grad x3y3z33xyz Find curl F where F = grad (x^3y^3z^3-3xyz) >
- Find the radius of curvature at xpi2 on the curve y 4sinx sin 2x Find the radius of curvature at x=pi/2 on the curve y = 4sinx -sin 2x >
- Show that matrix A 2 1 1 1 2 1 1 1 2 satisfies the Cayley Hamilton theorem and hence find A inverse Show that matrix A 2 -1 1 -1 2 -1 1 -1 2 satisfies the Cayley Hamilton theorem and hence find A inverse. >
- Find the angle between the normals to the surface xyz2 at the points Find the angle between the normals to the surface xy=z^2 at the points (1,1,1) and (4,1,2) >
- Find a such that 3x 2yz i 4xayzjxy2zk is solenoidal Find a such that (3x-2yz)i (4xay-z)j(x-y-2z)k is solenoidal >
- Given that F x2yi y2z jz2 x k find del F at the point Given that F = x^2yi y^2z jz^2 x k find del.F at the point (0,1,1) >
- Prove that F is irrotational if Fm 3x2y4zi 2x564zj4x4y8kk Prove that F is irrotational if Fm= (3x_2y4z)i (2x564z)j(4x4y-8k)k >
- If two vectors A and B are irrotational prove that AxB is solenoidal If two vectors A and B are irrotational prove that AxB is solenoidal >
- Find the directional derivative of xyz at 211 in the direction of jk Find the directional derivative of xyz at (2,1,1) in the direction of jk >
- Find curl axr where a is a constant vector Find curl (axr) where a is a constant vector. >
- Find the unit normal to the surface x2xyz2 4 at Find the unit normal to the surface x^2xyz^2 =4 at (1,-1,2) >
- What is the greatest rate of increase of phi xyz^2 at What is the greatest rate of increase of phi = xyz^2 at (1,0,3) >
- Find the particular integral of D12 y sin h 2x Find the particular integral of (D-1)^2 y = sin h 2x >