IAS PYQs 3
1988
1) Define the divergence of a vector point function, prove that \(\operatorname{div}(\vec{u} \times \vec{v})=\vec{v} \cdot \operatorname{curl} \vec{u}-\vec{u} \operatorname{curl} \vec{v}\).
2) Using Gauss divergence theorem, evaluate \(\iint_{s}\left(x \hat{i}+y \hat{i}+z^{2} \hat{k}\right) \cdot \hat{n} d s \quad\) where \(\mathrm{S}\) is the closed surface bounded by the cone \(x^{2}+y^{2}=2\) and the plane \(\mathrm{Z}=1\) and \(\hat{n}\) is the outward unit normal to \(\mathrm{S}\).
1987
1) Show that for a vector field \(\vec{f},\) curl (curl \(\vec{f})\)=\(\operatorname{grad}(\operatorname{div} \vec{f})-\nabla^{2} \vec{f}\).
2) If \(\vec{r}\) is the position vector to a point whose distance from the origin is \(\mathrm{r},\) prove that \(\operatorname{div} \vec{f}=0\) if \(\vec{f}=\dfrac{\vec{r}}{r^{3}}\).
3) Prove that for a three vectors \(\vec{a}, \vec{b}, \vec{c}\) \(\vec{a} \times(\vec{b} \times \vec{c})=\vec{b}(\vec{a} \cdot \vec{c})-\vec{c}(\vec{a} \cdot \vec{b})\) and explain its geometric meaning.
1986
1) Let \(\vec{a}, \vec{b}\) be given vectors in the three dimensional Euclidean space \(E_{3}\) and let \(\phi(\vec{x})\) be a scalar field of the vectors \(\vec{x}\) also of \(E_{3}\). If \(\phi(\vec{x})=(\vec{x} \times \vec{a}) \cdot(\vec{x} \times \vec{b}),\) show that
grad\(\phi(i . e, \nabla \phi(\vec{x}))\)=\(\vec{b} \times(\vec{x} \times \vec{a})+\vec{a} \times(\vec{x} \times \vec{b})\)
2) If \(\vec{f}, \vec{g}\) are two vector fields in \(E_{3}\) and if ‘div’, ‘curl’ are defined on an open set \(S \subset E_{3}\) show that \(\operatorname{div}(\vec{f} \times \vec{g})=\vec{g} . \operatorname{curl} \vec{f}-\vec{f} \cdot \operatorname{curl} \vec{g}\).
1985
1) If \(\mathrm{P}\), \(\mathrm{Q}\), \(\mathrm{R}\) are points \((3,-2,-1)\), \((1,3,4)\), \((2,1,-2)\) respectively. Find the distance from \(\mathrm{P}\) to the plane \(\mathrm{OQR}\), where \('\mathrm{O}'\) is the origin.
2) Find the angle between the tangents to the curve \(\vec{r}=t^{2} \hat{i}-2 t \hat{j}+t^{3} \hat{k}\) at the points \(\mathrm{t}=1\) and \(\mathrm{t}=2\).
3) Find div \(\mathrm{F}\) and curl \(\mathrm{F}\), where \(F=\nabla\left(x^{3}+y^{3}+z^{3}-3 x y z\right)\).