91Ó°ÊÓ

Evaluate the line integral \(\oint(x+2 y) d x-2 x d y\) along each of the following closed paths, taken counterclockwise: (a) the circle \(x^{2}+y^{2}=1 ;\) (b) the square with corners at \((1,1),(-1,1),(-1,-1),(1,-1) ;\) (c) the square with corners \((0,1),(-1,0),(0,-1),(1,0) .\)

The force scting on a moving charged particle in a magnetic field \(\mathbf{B}\) is \(\mathbf{F}=q(\mathrm{v} \times \mathbf{B})\) where 4 is the electric charge of the particle, and \(v\) is its velocity. Suppose that a particle moves in the \((x, y)\) plane with a uniform \(\mathrm{B}\) in the \(z\) direction. Assuming Newton's second law, \(m d v / d t=\mathbf{F}\), show that the force and velocity are perpendicular and that both have constant magnitude. \(\operatorname{Hint}:\) Find \((d / d t)(\mathrm{v} \cdot \mathrm{v})\)

If the temperature is \(T=x^{2}-x y+z^{2}\), find (a) the direction of heat flow at \((2,1,-1)\); (b) the rate of change of temperature in the direction \(j-k\) at \((2,1,-1)\).

For the force field \(F=-y i+x j+z k\), calculate the work done in moving a particle from \((1,0,0)\) to \((-1,0, \pi)\) (a) along the helix \(x=\cos t, y=\sin t, z=t ;\) (b) along the straight line joining the points. Do you expect your answers to be the same? Why or why not?