Theory and Applications

a. Give an example of a vector field F (x, y, z) that has value 0 at only one point and such that curl F is nonzero everywhere. Be sure to identify the point and compute the curl.

b. Give an example of a vector field F (x, y, z) that has value 0 on precisely one line and such that curl F is nonzero everywhere. Be sure to identify the line and compute the curl.

c. Give an example of a vector field F (x, y, z) that has value 0 on a surface and such that curl F is nonzero everywhere. Be sure to identify the surface and compute the curl.

Calculus notes for the week of 10/3/16 4.1 Maxima and Minima and 4.2 What Derivatives Tell Us 15 10 5 01 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 -5 -10 -15 f has a local maximum at c if f(c) > f(x) for all x sufficiently close to c. f has a local minimum at c if f(c) < f(x) for all x sufficiently close to c. We see that, if f is differentiable at a local...