For those of you who missed Monday and/or Wednesday, here
are the minutes.
Next Friday, 5mar, there will NOT be a quiz, in honor of EOH.
Those of you who will miss class by reason of EOH, please
email me the location of the exhibit you are participating in.
(I will try to come by sometime and appreciate it.)
On Friday this will happen:
1. Hand back the quiz from last week.
2. Hand out a study guide for the Midterm next Wednesday.
3. Answer question on the problems I suggested, which check
that your review of Math 242 is adequate for the midterm.
Here is the location in Kaplan of what you learned in 242.
On page 229 review 1,2,4 (arclenght), 5 (averages), 6
skip 3,7 -- end.
Skip Section 4.2
Page 244 review 1 (area), 2 (volume), 5 (order of integration)
skip 4, 7--10.
Eventually you'll need to review volumes, masses, moments of
inertia, but not for next week.
Notice that Section 4.6 on "change of variable" is the
Cartan calculus in disguise. So stick to the class notes here.
Page 259 1,2,3,4 (application)
But skip 5,6 which we cover in class.
Skip 4.8-9.
Section 5 is material to be covered in 280 because it is very
unlikely that you had an adequate treatment of Green's, Stokes
and Gauss' theorems.
So, on Monday I gave three partial introductions to the rest of
the semester. Science (until the advent of the computer) was
expressed in terms of its continuous models, governed by differential
equations, either ordinary, partial, or both. Thus the solution of
differential equations, or rather, an intelligent application of
what we could say about the solutions without solving them! was the
principal activity of scientists and engineers, and the principal
reason the advanced calculus and its cousins was invented.
The remainder of the semester will concern itself with the Fundamental
Theorem of the Calculus (FTC) and its vectorial interpretations, known as
Green's two theorem in the plane, and the theorems of Stokes and Gauss
in space.
Half of the FTC was discovered by Barrow (1670) who was Newton's (1687)
teacher. It said that the derivative d/dx of the (definite) integral
\int_a^x f(u)du returned the value of the integrand. In other words,
the definite integral was the way to compute the antiderivative if you
couldn't guess one. Newton, however, realized the significance of this
theorem, especially if you turned it around. The definite integral
\int_a^b f'(x)dx = f(b) - f(a) of the derivative of a function need
not be computed at all, just evaluate the function at both end points
in an oriented way (subtract f(lower) add f(upper) limit).
Of course, it was Leibniz (1684) who invented a way of building the
FTC into the notation, and we still use it. For f'(x)dx = df, and
so the FTC becomes \int_[a,b] df = f(b)-f(a). Thus f'(x)dx is called
an "exact" differential form, and f is called its anti-differential.
The generalization of the FTC becomes, in words:
The integral over a patch of an exact differential form equals
the integral of the anti-differential over the border of the patch.
When the patch is 1-dimensional (a curve, closed or not) the LHS
is integral of the gradient field along a curve. The theorem says
that the work done is the change in potential energy from the initial
to the final position.
Underlying this how to interpret the cumulative "action" of a
force field along a path as the integral of a Pfaffian 1-form,
\omega = a(x,y,z)dx + b(x,y,z)dy + c(x,y,z)dz
along a parametrized path x(t),y(t),z(t),a