Grewal [best] | Higher Engineering Mathematics B S

Verify Cauchy-Riemann equations for ( f(z) = e^z ) and find ( f'(z) ). (7 marks)

Verify Green’s theorem for ( \oint_C (xy , dx + x^2 , dy) ), where ( C ) is the triangle with vertices (0,0), (1,0), and (0,1). (7 marks) higher engineering mathematics b s grewal

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) Verify Cauchy-Riemann equations for ( f(z) = e^z

Solve the wave equation ( \frac\partial^2 y\partial t^2 = 4 \frac\partial^2 y\partial x^2 ) with boundary conditions ( y(0,t)=0, y(3,t)=0, y(x,0)=0, \frac\partial y\partial t(x,0) = 5 \sin 2\pi x ). (7 marks) Hence find ( \nabla \times (\nabla \times \vecr)

Find the inverse Laplace transform of: [ \fracs^2 + 2s + 3(s^2 + 2s + 2)(s^2 + 2s + 5) ] (7 marks)

Evaluate by Simpson’s 3/8 rule: [ \int_0^6 \fracdx1 + x^2 ] taking ( h = 1 ). (7 marks)