Here is the syllabus for the remainder of the semester. (Now that
we've had the first hourly and I have some idea what the class can
accomplish, I propose we can cover these topics in Kaplan:)
1. Definitions of grad, div, curl as in 3.1-3.5. (2hrs)
2. 3D integral calculus as in 5.8-5.13 (with review of Chapter 4
as needed.) (10hrs).
3. Polar, cylindrical, spherical and special coordinate systems,
as in 3.7-3.8 and also in 2.17. (4hrs)
4. Applications to the sciences, 5.14-5.18. (8hrs)
Note, we use Cartan's calculus of differentials all through these
sections. Kaplan has added a section (5.19). This may be considered
as supplementary reading.
5. As part of the grand review for the early part of the course,
we will also cover LaGrange multipliers (2.19-2.20) in the final
weeks of the semester.
Summary of Chapter 3:
Here are abstract definitions of all three vector differential operators.
Since we'll be spending the remainder of the semester understanding their
physical and geometrical meaning, we give here a bare bones definition only.
The differential of a scalar field in Cartesian coordinates may be considered
the "dual" of the gradient vector field, since both have the same components,
namely the partials of f with respect to x,y,z.
In this connection we define the ``directional derivative'' of f to be
the dot product of its gradient vector with the direction specified.
Remember that a ``direction'' is uniquely specified by a unit vector.
While the many different notations used for this concept in the various
engineering and scientific disciplines, this definition is adequate until
we explore the geometrical meaning of the gradient.
Assigned for Friday 19th Feb: Problem 1 on p186. I gave the example of
the shear field F(x,y,z) = [0,x,0] = x \hat j , and three different ways
of drawing a picture of it. I also solved Kaplan p186 #4 in the following,
preferred, way:
Note that the quotient \fat r/r is just \hat r. Moreover, using the
definition of r = \sqrt(x^2+y^2+z^2) to computer dr, we see that the
gradient of r is just the radial direction \hat r. (Which, on a moment's
reflection, is obvious.) So that allows us to find a potential function
for the gravitational field F = -kMn/r^2 \hat r by simply integrating
df = kMn ( - dr/r^2) = kMn d(1/r), so that F becomes the gradient of
the scalar field f = kMn/r.
To obtain a simple definition the divergence and curl of a vector field F
expressed in Cartesian coordinates, copute it's Jacobian matrix of 9
partial derivatives. The trace (sum down the diagonal) is called the
divergence of F (and it is a scalar). The curl of F is another vector field
that measures by how much the Jacobian fails to be a symmetric matrx. It
components are the differences of the 3 pairs of off diagonal elements. Since
there are 24 different ways of forming this vector, it is wise to learn one
way of remembering which is the correct one of these.
Using the Gibbs' method of expressing the \del as a vector differential operator,
then apply it to F in the manner of a cross product.
At the end of the hour I computed the curl of the shear field and found that it
is constant .... everywhere. Pete Krawczyk correctly answered the question as to
why this should be so even at a place where the flow of the shear field is not
stationary. Namely, if we imagine ourselves in a boat floating down the river,
and observe the difference of the shear field on either side of our flow line from
our own, it would look precisely as the shear flow looks to an outside observe,
at the origin.
It is essential that you keep a menagerie of particular examples in your notes
and in your head as we race ahead into the vector differential and integral
calculus. The shear field is our first denizen.
Notes 19th February.
Applying the curl to a gradient field requires NO calculation. Since the
components of the gradient are the 3 partials of the potential, and the
components of the Jacobian of the gradient are the 9 double partials, this
Jacobian is symmetric. Hence the curl is zero. Moreover, the divergence,
being the trace, is the sum of the second partials known as the LaPlacian
of the potential.
On the other hand, applying the divergence to a curl also yields zero.
This fact can be memorized, it can be deduced from Gibb's notation, because
the divergence is ``\del dot field'', and so
div curl F = \del . \del x F = \del x \del . F = det[\del,\del,F] = 0
Finally, just as the components of the gradient can be read off from computing
the differential mindlessly, the components of the curl and div can similarly
be mindlessly computed by taking differentials in the style of Cartan.
We will develop the Cartan algebra next week.