Using Green’s Theorem evaluate the integral where C is a circle of radius 3 units centered at the origin.

Calculate the divergence of the vector function

Using the divergence theorem evaluate where  and S is the surface of the cube defined by 

Calculate the work done by a force  in moving a particle counter clockwise along the circle x²+y²=4 from the point (2, 0) to the point (0, -2).

Obtain a solution of the wave equation

Obtain the Fourier series expansion for the following periodic function which has a period of π:

Obtain the divergence and curl of the following vector field:

Evaluate .

b) Show that for any scalar field :

.

Reduce the following PDE to a set of three ODEs by the method of separation of variables.

Obtain a unit tangent vector to any point on the curve defined by the parameteric equations:

x = sin 3t; y = 2 cos 3t;z = 4t.

b) Obtain the directional derivative for a scalar field at the point (1,-2,-1) in the direction .

Obtain all the first and second order partial derivatives of the function:

f (x, y) = sin xy

Obtain a unit vector perpendicular to the plane of the vectors

 and 

b) For any four vectors   and   determine:

Show that the function

i) u(x,y)=  is a solution of the two-dimensional Laplace’s equation.

ii) u(x,t)=e^{-6x cos2x} is a solution of the one-dimensional heat equation.

Solve the following ODE using the power series method:

(x+2)y"+xy'-y=0

A block of mass 1 kg is attached to a spring of force constant  .It is pulled 0.3 m from its equilibrium position and released from rest. This spring‐block apparatus is submerged in a viscous fluid medium which exerts a damping force of – 4 v (where v is the instantaneous velocity of the block). Determine of the position x(t) of the block at time t. 

Newton’s Law of Cooling states that the rate at which the object's temperature decreases is directly proportional to the difference between the temperature of the object and the ambient temperature. A cup of tea (temperature = 90°C) is placed in a room whose temperature is 30°C. After five minutes, the temperature of the tea has dropped to 70°C. After how much time would the temperature of the tea be 50°C?

Solve the following ordinary differential equations:

i) 

ii) 

iii) 

Consider two cylindrical pipes of equal length. One of these acts as a closed organ pipe and the other as open organ pipe. The frequency of the third harmonic in the closed pipe is 200 Hz higher than the first harmonic of the open pipe. Calculate the fundamental frequency of the closed pipe.

A stretched string of mass 20 g vibrates with a frequency of 30 Hz in its fundamental mode and the supports are 40 cm apart. The amplitude of vibrations at the antinode is 4 cm. Calculate the velocity of propagation of the wave in the string as well as the tension in it.

Design an astable multivibrator using IC 555 timer to generate a rectangular wave of 60% duty cycle at 10 kHz frequency