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.