2004

1) Evaluate \(\int_{e} \vec{F} \cdot \vec{d} r\) for the field \(\vec{F}=\) grad \(\left(x y^{2} z^{3}\right)\) where \(e\) is the ellipse in which the plane \(\mathrm{z}=2 \mathrm{x}+3 \mathrm{y}\) cuts the cylinder \(\mathrm{x}^{2}+\mathrm{y}^{2}=12\) counterclockwise as viewed from the positive end of the z-axis looking towards the origin.

[10M]

2) Show that \(\operatorname{div}(\vec{A} \times \vec{B})=\vec{B} \cdot \operatorname{curl} \vec{A}-\vec{A} \cdot \operatorname{curl} \vec{B}.\)

[10M]

3) Evaluate curl \(\left[\dfrac{(2 \vec{i}-\vec{j}+3 \vec{k}) \times \vec{r}}{r^3}\right]\), where \(\vec{r}=x\vec{i}+y\vec{j}+z \vec{k}\) and \(r^{3}=x^{3}+y^{3}+z^{3}\).

[10M]

4) Evaluate \(\iint_S (x \vec{i}+y \vec{j}+z \vec{k}) \cdot \vec{n} d s\), where \(S\) is the surface \(x+y+z=1\) lying in the first octant.

[10M]

5) Evaluate \(\nabla^{2} u\) in spherical polar coordinates.

[10M]

2003

1) Find expressions for curvature and torsion at a point on the curve \(x=a \cos \theta\), \(y=a \sin \theta\), \(z=a\theta \cot \beta\)

[10M]

2) If \(\bar{r}\) is the position vector of the point \((\mathrm{x}, \mathrm{y}, 2)\) with respect to the origin, prove that \(\nabla^{2} f(r)=f^{\prime \prime}(r)+\dfrac{2}{r} f^{\prime}(r)\)
Find \(f(r)\) such that \(\Delta^{2} f(r)=0\).

[10M]

3) If \(\vec{F}\) is solenoidal, prove that Curl Curl Curl Curl \(\vec{F}=\nabla^{4} \vec{F}\).

[10M]

4) Verify Stoke’s Theorem when
\(\overline{\mathbf{F}}=\left(2 x y-x^{2}\right) \bar{i}-\left(x^{2}-y^{2}\right) \bar{d}\)
and \(C\) is the boundary of the region enclosed by the parabolas \(y^{2}=x\) and \(x 2=y\)

[10M]

TBC

5) Express \(\nabla \times \vec{F}\) and \(\nabla^{2} \Phi\) in cylindrical co-ordinates,

[10M]

2002

1) Find the eurvature and torsion of the curve \(x=\dfrac{(2t+1)}{t-1}\), \(y=\dfrac{t^{2}}{t-1}\), \(z=t+2\). Interpret your answer.

[10M]

2) State Stoke’s theorem and then verify it for \(\bar{A}=\left(x^{2}+1\right) \bar{i}+x y \bar{j}\) integrated round the square in the plane

\(z=0\) whose sides are along the lines

[10M]

1. Prove that: (i) \(\vec{\nabla} \times(\vec{\mathbf{A}} \times \vec{\mathbf{B}})=\vec{\mathbf{A}}(\vec{\nabla} \cdot \vec{\mathbf{B}})-\vec{\mathbf{B}}(\vec{\nabla} \cdot \vec{\mathbf{A}})+(\vec{\mathbf{B}} \cdot \vec{\nabla}) \vec{\mathbf{A}}-(\vec{\mathbf{A}} \cdot \vec{\nabla}) \vec{\mathbf{B}}\)

[10M]

(ii) \(\operatorname{curl} \dfrac{\vec{a} \times \vec{r}}{r^{3}}\)= \(-\dfrac{\vec{a}}{r^{3}}+\dfrac{3 \vec{r}}{r^{3}}(\vec{a} \cdot \vec{r})\), \(\vec{a}\) = constant vector.

4) Show that if \(\vec{A} \neq \vec{0}\) and both of the conditions \(\vec{A} \cdot \vec{B}=\vec{A} \vec{C}\) and \(\vec{A} \times \vec{B}=\vec{A} \times \vec{C}\) hold simultancously then \(\vec{B}=\vec{C}\) but if only one of these conditions holds then \(\bar{B} \neq \bar{C}\) necessarily.

[10M]

5) Prove the following
(i) If \(u_{1}, u_{2}, u_{3}\) are general coordinates, then
\(\dfrac{\partial \vec{r}}{\partial u_{1}} \times \dfrac{\partial \vec{r}}{\partial u_{2}} \times \dfrac{\partial vec{r}}{\partial u_{3}}\) and \(\vec{\nabla} u_{1}, \vec{\nabla} u_{2}, \vec{\nabla} u_{3}\) are reciprocal system of vectors.

[5M]

(ii) \(\left(\dfrac{\partial \vec{r}}{u_{1}} \cdot \dfrac{\partial \vec{r}}{u_{2}} \times \dfrac{\partial \vec{r}}{u_{3}}\right)\left(\vec{\nabla} u_{1} \cdot \vec{\nabla} u_{2} \times \vec{\nabla} u_{3}\right)=1\).

[5M]

2001

1) Find an equation for the plane passing through the points \(P_{1}(3,1,-2)\), \(P_{2}(-1,2,4)\), \(P_{3}(2,-1,1)\) by using vector method.

[10M]

2) Prove that \(\nabla \times(\nabla \times \bar{A})=-\nabla^{2} \bar{A}+\nabla(\nabla \cdot \bar{A})\).

[10M]

3) If \(\nabla \cdot \bar{E}, \nabla \cdot \bar{H}, \nabla \times \bar{E}=-\dfrac{\partial \bar{H}}{\partial t}\), \(\nabla \times \bar{H}=\dfrac{\partial \bar{E}}{\partial t}\) Show that \(\bar{E}\) & \(\bar{H}\) satisfy \(\nabla^{2} u=-\dfrac{\partial^{2} \bar{u}}{\partial t^{2}}\)

[10M]

4) Given the space curve \(x=t\), \(y=t^{2}\), \(z=\dfrac{2}{3} t^{3}\). Find
(i) the curvature \(\rho\) (ii) the torsion \(\tau\).

[10M]

5) If \(\quad F=\left(y^{2}+z^{2}-x^{2}\right) i+\left(z^{2}+x^{2}-y^{2}\right) j+\left(x^{2}+y^{2}-z^{2}\right) k\) evaluate \(\iint_{S} \operatorname{curl} \bar{F} \cdot \hat{n} d s\), taken over the portion of the surface \(x^{2}+y^{2}+z^{2}-2 a x+a z=0\) above the plane \(z=0\) and verify Stokes’ theorem.

[10M]

2000

1) Solve for x the vector equation \(p x+x \times a=b, p \neq 0\)

[10M]

2) Prove the identities:
(i) Curl grad \(\phi=0, (ii)\) div curl \(f=0\).

If \(\overrightarrow{O A}=a i, \overrightarrow{O B}=a j, \overrightarrow{O C}=a k\) form three coterminous edges of a cube and \(\mathrm{s}\) denotes the surface of the cube, evaluate \(\left.\int_{1}^{\prime}\left(x^{3}-y z\right) i-2 x^{2} y j+2 k\right\}\) nds by expressing it as volume integral, Where \(n\) is the unit outward normal to \(ds\).

[20M]