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