6 M

2(c) Verify Green's Theorem for ∮_c (y-sinx)dx+cosxdy where C is the plane triangle bounded by the lines y=0, \( x=\dfrac{\pi}{2} \), \( y=\dfrac{2x}{\pi} \)

8 M

3(a) If \( \overrightarrow {F}=2xyzt+(x^2z+2y))\hat{j}+(x^2y)\hat{k} \) then
I) Prove that \( \overrightarrow {F} \) is irrotational
II) Find its scalar potential φ
III) Find the work done in moving a particle under this force field from (0, 1, 1) to (1, 2, 0).

6 M

3(b) The vibrations of an elastic string is governed by the partial differential equation \( \dfrac{\partial^2 u}{\partial t^2}=C^2\dfrac{\partial^2 u}{\partial x^2}.A \) string is stretched and fastened to two points l apart. Motion is started by displacing the string in the form \( y=asin\left ( \dfrac{\pi x}{l} \right ) \) from which it is released at time t=0. Show that the displacement of any point at a distance x from one end at time is given by \( y(x,t)=a\sin\frac{\pi x}{l}\cos \frac{\pi ct}{l} \).

6 M

3(c) Find the Fourier series of
f(x)=πx, 0≤x<1;
  =0, x=1
  =π(x-2), 1 Hence deduce that \( \dfrac{\pi}{4}=1-\dfrac{1}{3}+\dfrac{1}{5}-\dfrac{1}{7}+\cdots \)

8 M

4(a) Find the complex form of Fourier series of f(x)=sinax in the interval (-π, x) where a is not an integer.

6 M

4(b) Evaluate \( \int \int _S x^3dydz+x^2ydzdx+x^2zdxdy \) where S is the closed surface consisting of the circular cylinder x² +y² =a², z=0 and z=b.

6 M

4(c) The equation of one dimensional heat flow is given by \( \dfrac{\partial u}{\partial t}=C^2 \dfrac{\partial^2 u}{\partial x^2} \).
A bar of 10cm long with Insulated sides has its ends A and B maintained at temperature 50°C and 100°C respectively, until steady-state conditions prevall. The temperature A is suddenly to raised 90°C and at the same time at B is lowered to 60°C. Find the temperature distribution in the bar at time t.

8 M

5(a) Evaluate by Green's theorem ∮_c (y³ -xy)dx+(xy+3xy²)dy where C is the bounded by the square with vertices (0,0), \( \left ( \dfrac{\pi}{2},0 \right ),\left ( \dfrac{\pi}{2},\dfrac{\pi}{2} \right ),\left ( 0,\dfrac{\pi}{2} \right ) \).

6 M

5(b) Find the Fourier sine integral of the function
f(x)=x, 0   =2-x, 1   =0, x>2

6 M

5(c) Solve \( \dfrac{\partial^2 u}{\partial x^2}+\dfrac{\partial^2 u}{\partial y^2}=0 \) for 02x.

8 M

6(a) Using Stokes' theorem find the work done in moving a particle once around the perimeter of the triangle with vertices at (2, 0, 0), (0, 3, 0) and (0, 0, 0) under the force field
\[\overrightarrow {F}=(x+y)\hat{i}+(2s-z)\hat{j}+(y+z)\hat{k}\].

6 M

6(b) Find the half range sine series of
f(x)=x, 0≤ x ≤ 2;
  =4-x, 2≤ x≤ 4

6 M

6(c) Find the Fourier cosine transform of \( f(x)=\dfrac{1}{1+x^2} \). Hence derive the Fourier sine transform of \( f(x)=\dfrac{x}{1+x^2} \)

8 M