1994

1) Show that \(r^{n} \vec{r}\) is an irrotational vector for any value of \(n\), but is solenoidal only if \(n =-3\).

[10M]

2) If \(\vec{F}=y \vec{i}+(x-2 x z) \vec{j}-x y \vec{k},\) evaluate \(\iint_{S}(\nabla \times \vec{F}) \cdot \vec{n} d S\),
where \(S\) is the surface of the sphere \(x ^{2}+ y ^{2}+ z ^{2}= a ^{2}\) above the xy plane.

[10M]

1993

1) Prove that the angular velocity of rotation at any point is equal to one half of the curl of the velocity vector \(V\).

[10M]

–

2) Evaluate \(\iint_{s} \nabla \times F h d S\) where \(S\) is the upper half surface of the unit sphere \(x^{2}+y^{2}+z^{2}=1\) and \(\vec{F}=z \hat{i} +x \hat{j} +y \hat{k}\).

[10M]

1992

1) If \(\vec{f}(x, y, z)=\left(y^{2}+z^{2}\right) \vec{i}+\left(z^{2}+x^{2}\right) \vec{j}+\left(x^{2}+y^{2}\right) \vec{k}\), then calculate \(\int_{c} \vec{f} \overrightarrow{d x},\) where \(\mathrm{C}\) consists of (i) the line segment from \((0,0,0)\) to \((1,1,1)\) (ii) the three line segments \(\mathrm{AB}, \mathrm{BC}\) and \(\mathrm{CD},\) where \(\mathrm{A}, \mathrm{B}, \mathrm{C}\) and \(\mathrm{D}\) are respectively the points \((0,0,0)\), \((1,0,0)\), \((1,1,0)\) and \((1,1,1)\) (iii) the curve \(\bar{x}+\vec{u}+\overline{u^{2}} j+\overline{u^{2} k} u\), from 0 to 1

[10M]

2) If \(\vec{a}\) and \(\vec{b}\) are constant vectors, show that

(i) \(\operatorname{div}\{x \times(\vec{a} \times \vec{x})\}=-2 \vec{x} \vec{a}\)
(ii) \(\operatorname{div}\{(\vec{a} \times \vec{x}) \times(\vec{b} \times \vec{x})\}=2 \vec{a} \cdot(\vec{b} \times \vec{x})-2 b \cdot(\vec{a} \times x)\)

[10M]

3) Obtain the formula

[10M]

1991

1) If \(\phi\) be a scalar point function and \(\mathrm{F}\) be a vector point function, show that the components of \(\mathrm{F}\) normal and tangential to surface \(\phi=0\) at any point there of are \(\dfrac{(F . \nabla \phi) \nabla \phi}{(\nabla \phi)^{2}}\) and \(\dfrac{\nabla \phi \times(F \times \nabla \phi)}{(\nabla \phi)^{2}}\).

2) Find the value of \(\int\) curl F. dS taken over the portion of the surface \(x^{2}+y^{2}-2 a x+a z=0,\) for which \(z \geq 0\) when \(F=\left(y^{2}+z^{2}-x^{2}\right) \hat{i}+\left(z^{2}+x^{2}-y^{2}\right) \hat{j}+\left(x^{2}+y^{2}-z^{2}\right) \hat{k}\)

1989

1) Define the curl of a vector point function.

2) Prove that \(\nabla \times\left(\dfrac{\vec{r}}{r^{2}}\right)=0\) where \(\vec{r}=(x, y, z)\) and \(r=\vert \vec{r} \vert\).