Q1.
(a) Q1. Define a differentiable function f at a point a in a domain D ⊂ Rn. Prove that every differentiable function is continuous at the point.
(b)
Define normal subgroup N of a group G and also define quotient group G/N. Prove that a mapping φ: G → G/N given by φ
(x) = Nx, ∀ x ∈ G, is a homomorphism.
(c) If f(z) = u(x, y) + iv(x, y) is differentiable at any point z = x + iy in ℂ-domain D, then prove that its real part u(x, y) and imaginary part v(x, y) are also differentiable at (x, y) and ux = vy and uy = -vx.
(d) Let x = a cos nt + b sin nt be the position of a particle moving in a straight line. Prove that it executes Simple Harmonic Motion (SHM) of period 2π/n and amplitude sqrt(a2 + b2).
(e)
Prove that a set A is closed if and only if its complement B is open in real number system R.
(f) Show that the power series ∑ an zn is either convergent ∀ z, or converges for z = 0 only, or converges for z in some region of the complex plane.
(g) For three vectors a, b, c, prove that a × (b × c) = (a·c) b − (a·b) c.
