IAS PYQs 1
2000
1) In what direction from the point (-1,1,1) is the directional derivative of \(f=x^{2} y z^{3}\) a maximum \(?\) Compute its magnitude.
[10M]
2) Show that
(i) \((A+B) \cdot(B+C) \times(C+A)=2 A \cdot B \times C\)
(ii)\(\nabla \times(A \times B)=(B . \nabla) A-B(\nabla . A)-(A . \nabla) B+A(\nabla . B)\)
[10M]
3) Evaluate \(\iint_{S} \vec{F} \cdot \hat{n} dS\), where \(\mathbf{F}=2 x y \mathbf{i}+y z^{2} \mathbf{j}+x z \mathbf{k}\) and \(\mathrm{S}\) is the surface of the parallelcpiped bounded by \(\mathrm{x}=0\), \(\mathrm{y}=0, \mathrm{z}=0\), \(x=2\), \(y=1\) and \(z=3\).
[10M]
1999
1) If \(\vec{a},\vec{b},\vec{c}\) are the position vectors of \(A,B,C\) prove that \(\vec{a}\times\vec{b}+\vec{b}\times\vec{c}+\vec{c}\times\vec{a}\) is a vector perpendicuar to the plane \(ABC\).
[10M]
2) If \(\vec{F}=\nabla(x^3+y^3+z^3-3xyz),\) find \(\nabla\times\vec{F}.\)
[10M]
3) Evaluate \(\int_C (e^{-x}sin y \,dx+e^{-x}cos y \,dy)\) (Green’s theorem), where C is the rectangle whose vertices are \((0,0),(\pi,0),(\pi,\dfrac{\pi}{2})\) and \((0,\dfrac{\pi}{2})\).
[10M]
4) If \(X,Y,Z\) are the components of contravariant vector in rectangular Cartesian coordinates \(x,y,z\) in a three dimensional space, show that the components of the vectors in cylinderal coordinates \(r,\theta,z\) are \(Xcos\theta+Ysin\theta,\dfrac{-X}{r}sin\theta+\dfrac{Y}{r}cos\theta,Z\)
[10M]
5) Show that \(\nabla^2\phi=g^{ij}(\dfrac{\partial^2\phi}{\partial x^i\partial x^j}-\dfrac{\partial\phi}{\partial x^l}\{\dfrac{1}{ij}\})\) where \(\phi\) is a scalar function of coordinates \(x^i\).
[10M]
1998
1) If \(r_1\) and \(r_2\) are the vectors joining the fixed points \(A(x_1,y_1,z_1)\) and \(B(x_2,y_2,z_2)\) respectively to a variable point \(P(x,y,z)\), then find the value of \(grad(r_1\cdot r_2)\) and \(curl(r_1\times r_2)\).
[10M]
2) Show that \((a\times b)\times c=a\times (b\times c)\) if either \(b=0\) (or any other vector is 0) or \(c\) is collinear with \(a\) or \(b\) is orthogonal to \(a\) and \(c\)(both).
[10M]
1997
1) Prove that if \(\vec{A}, \vec{B}\) and \(\dot{C}\) are three given non-coplanar vectors, then any vector \(\vec{F}\) can be put in the form \(F=\alpha \vec{B} \times \vec{C}+\beta \vec{C} \times \vec{A}+\gamma \vec{A} \times \vec{B}\). For given determine \(\alpha\), \(\beta\), \(\gamma\).
[10M]
2) Verify Gauss theorem for \(\vec{F}=4 x i-2 y^{2} \hat{j}+z^{2} \hat{k}\) taken over the region bounded by \(x^{2}+y^{2}=4\) \(z=0\) and \(z=3\).
[10M]
1996
1) If \(x \vec{i}+y \hat{j}+z \hat{k}\) and \(r=|\vec{r}|,\) show that:
(i) \(\vec{r} \times\) grad f \((r)=0\)
(ii) \(\operatorname{div}\left(r^{n} \vec{r}\right)=(n+3) r^{n}\)
[15M]
2) Verify Gauss’ divergence theorem for \(\vec{F}=x y \hat{i}+z^{2}\hat{j}+2 yz\hat{k}\) on the tetrahedron \(x=y=z=0, x+y+z=1\)
[15M]
1995
1) Let the region \(\mathrm{V}\) be bounded by the smooth surface \(\mathrm{S}\) and let \(\mathrm{n}\) denote outward drawn unit normal vector at a point on \(\mathrm{S}\). If \(\phi\) is harmonic in \(\mathrm{V}\), show that \(\int_{S} \dfrac{\partial \phi}{\partial n} d S=0\)
[20M]
2) In the vector field \(v(\mathrm{x})\), let there exist a surface on which \(v=0$.$ Show that, at an arbitrary point of this surface, curl\)v=0$$ is tangential to the surface or vanishes.
[20M]