Homework should be turned in at the BEGINNING OF CLASS. All problem numbers refer to McCallum's book, the required text for the course. You should write up solutions neatly to all problems, making sure to SHOW ALL YOUR WORK. A nonempty subset will be graded. You are strongly encouraged to work on these problems with other classmates, although the solutions you turn in should be your own work.

**Section 13.4**

Problems: 5, 6, 7, 8, 11, 16, 19, 22, 27, 29, 33, 34, 38

*Hint:* For #33, compute the unit vectors of the two directions
given, from (4,5) to (5,6) and from (4,5) to (6,6). Then let grad f
at the point (4,5) be given by the expression a **i** + b **j**.
Set up two linear equations for a and b using the dot product formula for
the directional derivative.

**Section 13.5**

Problems: 2, 5, 9, 10, 11, 14, 20

*Hint:* Remember that the gradient of a 3-variable function is perpendicular to a level **surface**
of that function. Before you take a gradient, make sure you have an
equation of the form f(x,y,z) = constant from which you then compute grad f.
Write down the function f(x,y,z) so we can understand your thought process.

Problem #14 is an excellent conceptual problem. Part (a) involves looking at the
gradient of a two-variable function which is a vector in the plane. Then, part (b)
considers the graph of the function z = f(x,y) as a **surface** in space. How do you adjust
this equation so you think of it as a level surface to some function of three variables?

Problem #20 is a standard Celestial Mechanics problem. It shows that the gravitational
force field is really the gradient vector field of a particular function called the
*Newtonian potential*. Physicists think of this as a conservative system because
quantities such as energy are conserved.

**Section 13.6**

Problems: 1, 3, 4, 9, 10, 16, 17