Math 127(3) - Warren D. Smith - Homework #6 - due Tues 11 Mar 2003
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My last lecture (content on differential forms such as curl, Jacobian
matrix, div) was unfortunately not too synchronized with the book.
The "triangle" operator (the book says to call it "del" but other
books call it "nabla") is discussed in section 17.8, which I
of temporarily skipped ahead to. This includes curl, div, and (what we
already knew) grad.  (The Jacobian matrix is in section 16.11.)
We'll get back to that material later and do some exercises on it now.
We are now about to start INTEGRAL vector calculus, in ch 16.

HOMEWORK EXERCISES:
(for 1,2,3 and 4) Find all the flat spots (book calls these
"stationary points"; other books call them "critical points") of
the following 4 functions and for each say if it is a local min, local
max, or saddlepoint.  (See 15.7 and 15.6.)

(1)              2   2
     5 x y exp(-x - y )

(2)   2
     x  + sin(y)

(3)   (x+y)(xy+5)

       2   2       2    2
(4)  (x + y ) exp(x  - y ).

(5) Compute the curl and the div of
   (x, y, z)

(6) Compute the curl and the div of
   (3X+Y+Z, 5Y+Z, X+Y).

(7) do book (section 17.8) ex 16 (gravity)

(8) do book (section 17.8) ex 30 (identities - hint, consider
 analogous plain vector identities before considering the
 pseudo-vector differential operator TRIANGLE)