Verify Cauchy-Riemann equations for ( f(z) = e^z ) and find ( f'(z) ). (7 marks)
Solve using Laplace transform: [ y'' + 4y = 8t, \quad y(0) = 0, \quad y'(0) = 2 ] (7 marks) higher engineering mathematics b s grewal
Solve the Laplace equation ( \frac\partial^2 u\partial x^2 + \frac\partial^2 u\partial y^2 = 0 ) for a rectangular plate with boundary conditions: ( u(0,y)=0, u(a,y)=0, u(x,0)=0, u(x,b) = \sin\left(\frac\pi xa\right) ). (7 marks) Unit – D: Laplace Transforms Q7 (a) Find the Laplace transform of: (i) ( t^2 e^-3t \sin 2t ) (ii) ( \frac1 - \cos att ) (7 marks) Verify Cauchy-Riemann equations for ( f(z) = e^z
Prove that ( \nabla \times ( \nabla \times \vecF ) = \nabla(\nabla \cdot \vecF) - \nabla^2 \vecF ). Hence find ( \nabla \times (\nabla \times \vecr) ) where ( \vecr = x\hati + y\hatj + z\hatk ). (7 marks) Unit – C: Fourier Series & Partial Differential Equations Q5 (a) Find the Fourier series expansion of ( f(x) = x^2 ) in ( (-\pi, \pi) ). Hence deduce that: [ \frac11^2 + \frac12^2 + \frac13^2 + \cdots = \frac\pi^26 ] (7 marks) Hence find ( \nabla \times (\nabla \times \vecr)
Trace the curve ( r = a(1 + \cos\theta) ) (Cardioid) and find the area enclosed. (7 marks) Unit – B: Multiple Integrals & Vector Calculus Q3 (a) Evaluate: [ \int_0^1 \int_0^\sqrt1-x^2 \int_0^\sqrt1-x^2-y^2 \fracdz , dy , dx\sqrt1-x^2-y^2-z^2 ] (7 marks)