Multivariable Calculus (Math 214) Assignment

due September 15, 1999

1. Let A and B be the points in Cartesian 3-space given by

   A = (2, -3, 5) , B = (0, 1, -1) .

   1. Find the scalar (dot) product A . B.

   2. Find the angle at the origin in the triangle with vertices A, 0, and B.

   3. Verify that the lengths of the three sides of this triangle satisfy the classical ``Law of Cosines''.

   4. Find the point where the three medians of this triangle meet.

2. Let f be the function that is defined by the formula

   f(x, y) = y^{2} - y + xy + x^{3} - 2 x + 1 .

   1. Evaluate the two partial derivatives of f at (0, 0).

   2. Use the first-year technique of ``implicit differentiation'' to find the slope of the curve f(x, y) = 1 at the point (0, 0).

   3. What connection exists between the first and second parts of this exercise?

3. Let

   f(x, y, z) = x^{2} + y^{2} + z^{2} .

   1. For each fixed r > 0 describe the set of points in space for which f(x, y, z) = r^{2}.

   2. Compute the three partial derivatives of f.

   3. Compute all nine second order partial derivatives of f.

   4. What points in space satisfy both f(x, y, z) = 1 and also x + y + z = 1 ?

4. Let

   f(x, y, z, t) = x^{2} - 2 x y cos (t) + y^{2} - z^{2} .

   1. What is the geometric significance of the set of 4-tuples (x, y, z, t) that satisfy the equation f(x, y, z, t) = 0 ?

   2. Find the four partial derivatives of f.