2 HW 2

1. Find the components of the acceleration in polar coordinates by differentiating the expression for the velocity in these coordinates. Identify the components $a_r$ and $a_\theta$.

2. 1st Ed: p78, q46, 2nd Ed: p91, q46. $r=\sqrt{x^2+y^2+z^2}$

3. 1st Ed: p78, q54, 2nd Ed: p92, q54. You may want to refresh your memory on total derivatives.

4. The height of the ground above sea level is $\sin(x)\sin(2y)$.
   1. Draw the contour lines.
   2. Consider the point $x=0.5$ and $y=1.5$. Find the gradient of height at that point and draw it in the graph.
   3. If I want to climb at the fastest rate, for a given speed, in which direction should I move at that point? In particular, what is ${\rm d}y/{\rm d}x$?
   4. If I am traveling along the line $y=3 x$ with speed 60, (ignoring the vertical component of velocity), how rapidly am I changing height?

5. (6 points). 1st Ed: p78, q60, 2nd Ed: p92, q60. Also find two scalar equations that describe the line through P that crosses the surface normally at P.

   Find the unit normal $\vec n$ to the surface at P. Now assume that the surface is reflective, satisfying Snell's law. An incoming light beam parallel to the $x$-axis hits the surface at P. Find a vector equation that describes the path of the reflected beam.

   Hint: let $\vec v$ be a vector along the light ray. The component of $\vec v$ in the direction of $\vec n$ is $\vec n\cdot\vec v$. The component vector in the direction of $\vec n$ is defined as $\vec v_1=\vec n(\vec n\cdot\vec v)$. Sketch this vector along with vector $\vec n$. In which direction is the remainder $\vec v_2 =\vec v - \vec v_1$? Now figure out what happens to $\vec v_1$ and $\vec v_2$ during the reflection. Take it from there.