Using the recurrence relation, show that

Using appropriate figures, show that the group S₃ is not commutative.

Consider a rod of length L, whose ends are kept at a constant temperature and its lateral surface is insulated. The heat flow is described by the 1-D heat diffusion equation subject to the conditions:

Obtain a unique solution. 

Using Cauchy’s residue theorem, evaluate the integral

Obtain the Fourier sine transform of the function

 a>0,0x<∞

Obtain the eigenvalues and the corresponding eigenvectors of the matrix:

b) Diagonalized the matrix:

The eigenvalues of A are: λ₁ = 2, λ₂  =1 and λ₃ = -1.

c) Show that g^ij is a contravariant tensor of rank 2.

d) Show that the vectors  are linearly independent.

i) Show that the series  converges for |z|< 1 and find its sum.

ii) Obtain the analytic function whose real part u(x, y ) e^x cos y.