Mathematical Methods of Physics: For Undergraduate

Prove that the line joining the midpoints of two sides of a triangle is parallel and half to the third.

A particle is displaced from the point whose position vector is \(5\hat{i}-5\hat{j}-7\hat{k}\) to the point whose position vector is \(6\hat{i}+2\hat{j}-2\hat{k}\) under the action of a number of constant forces defined by \(10\hat{i}-\hat{j}+11\hat{k}\text{,}\) \(4\hat{i}+5\hat{j}+6\hat{k}\text{,}\) and \(-2\hat{i}+\hat{j}-8\hat{k}\text{.}\) Find the work done.

A force is represented in magnitude and direction by the line joining the point A(1,-2,4) to the point B(5,2,3). Find its moment about the point (-2,3,5).

A rigid body is rotating at the rate of 2.5 radians per second about an axis AB, where A and B are the points (1,-2,1) and (3,-4,2). Find the velocity of the point P at (5,-1,1) of the body.

A particle P is moving on a circle of radius \(r\) with constant angular velocity \(\omega = \frac{d\theta}{dt}\text{.}\) Show that its acceleration is \(-\omega^{2}\vec{r}\text{.}\)

If \(\phi=log\mid r \mid\text{,}\) then show that \(grad \phi=\frac{\vec{r}}{r^{2}}\text{.}\)

Prove that \(grad \frac{1}{r}=-\frac{r}{r^{3}}\text{.}\)

If \(\phi(x,y,z)=3x^{2}z-xy^{3}+5\text{,}\) find \(\phi\) at the point (a) (0,0,0) and (b) (1,-2,2).

Find the directional derivative of \(x^{2}+y^{2}+4xy\) at (1,-2,2) in the direction \((2\hat{i}-2\hat{j}+\hat{k})\text{.}\)

The temperature of the points in room is given by \(T(x,y,z) = x^{2}+y^{2}-z^{2}\text{.}\) A fly located at (1,1,1) desires to fly in such a direction that it will get warm as soon as possible. In what direction should it fly?

Evaluate \(\vec{\bigtriangledown}\cdot(r^{3}\vec{r})\)

If \(\vec{A}=\frac{\vec{r}}{r}\text{,}\) find \(\text{grad div} \vec{A}\)

Prove \(\nabla^{2}(\phi \psi) = \phi \nabla^{2}\psi +2\nabla \phi\cdot\nabla \psi +\psi \nabla^{2} \phi\text{.}\)

Prove that the vector \(\vec{A}=3y^{4}z^{2}\hat{i}+4x^{3}z^{2}\vec{j}-3x^{2}y^{2}\hat{k}\) is solenoidal.

If \(\vec{A}\) and \(\vec{B}\) are irrotational, prove that \(\vec{A}\times\vec{B}\) is solenoidal.

Prove that \(\nabla^{2}\left[\nabla\left(\frac{\vec{r}}{r^{2}}\right)\right]= 2r^{-4}\text{.}\)

Find the work done when a force \(\vec{F}=(x^{2}-y^{2}+x)\hat{i}-(2xy+y)\hat{j}\) moves a particle from origin to (1,1) along a parabola \(y^{2}=x\text{.}\)

Evaluate \(\iint\limits_{R}\sqrt{x^{2}+y^{2}}\,dx\,dy\) over the region R in the xy - plane bounded by \(x^{2}+y^{2}=36\text{.}\)

Evaluate \(\iiint\limits_{V}(2x+y) \,dV\text{,}\) where V is the closed region bounded by the cylinder \(z=4-x^{2}\) and the planes \(x=0,y=0,y=2,\) and \(z=0\text{.}\)

Use divergence theorem to evaluate \(\iint\limits_{S}\left(y^{2}z^{2}\hat{i}+z^{2}x^{2}\hat{j}+x^{2}y^{2}\hat{k}\right)\cdot\vec{ds}\text{,}\) where S is the upper part of the sphere \(x^{2}+y^{2}+z^{2}=9\) above the x-y plane.

Verify divergence theorem for \(\vec{F}=4xz\hat{i}+xyz^{2}\hat{j}3z\hat{k}\) over the region above XOY plane bounded by the cone \(z^{2}= x^{2}+y^{2}\) and the plane \(z=4\text{.}\)

Using Stoke’s theorem, evaluate \(\int\limits_{C}[(x+y)dx+(2x-z)dy+(y-z)dz]\text{,}\) where C is the boundary of the triangle with vertices (2,0,0), (0,3,0) and (0,0,6).

Represent the vector \(\vec{F}= 2y\hat{i}-z\hat{j}+3x\hat{k}\) in spherical coordinates and determine \(F_{r}, F_{\theta}, F_{\phi}\text{.}\)

Prove that a spherical coordinate system is orthogonal.

Express the velocity \(\vec{v}\) and acceleration \(\vec{a}\) of a particle in spherical coordinates.