At our first class meeting, we discussed partial differentiation
from the intuitive, computational point of view. We worked some example
problems from pages 4-5 of the textbook.
The homework is to read sections 1 and 2 of Chapter 1 and to write
solutions to exercises 2, 7, and 9 on pages 4-5 to hand in at the
beginning of the next class. [Can you check the answers using the
computer? (Maple, for example)]
We discussed a bit of plane topology (open sets), the class of
functions of two variables, and the mean-value theorem / linear
approximation formula for such functions.
For next Tuesday, read sections 3 and 4 in Chapter 1. Here are the
homework exercises that are due next Thursday:
Mainly we worked on an exercise on differentiability. We found that in all three examples, the partial derivatives exist at all points, but the partial derivatives fail to be continuous at the origin. Thus, all three functions fail to be class . We found that the function of Example A is continuous but not differentiable at the origin; the function of Example B is neither continuous nor differentiable at the origin; and the function of Example C is both continuous and differentiable at all points.
We discussed the chain rule and implicit differentiation. We reviewed Cramer's rule from linear algebra and saw its usefulness in computing partial derivatives.
Here are the homework exercises that are due next Thursday:
We worked on some more examples of computing partial derivatives using the chain rule and linear algebra.
We discussed Taylor's theorem in two variables.
Here are the homework exercises that are due next Thursday:
We worked on homework problems during class and observed that the multilinearity of the determinant could simplify some of the calculations.
Announcements: Dr. Straube will be substituting on Thursday, September 21. The first exam (which covers Chapter 1) will be on Thursday, September 28.
Dr. Straube covered the theorem about equality of mixed second partial derivatives and the statement of the implicit function theorem.
We continued the discussion of the implicit function theorem and reviewed for the exam to be held next class.
We began discussing integration of functions of more than one variable.
The homework due Thursday is exercises 4, 5, and 6 on page 187.
We discussed integration in polar coordinates and interchange of order in interated integrals.
Here are the homework exercises that are due next Thursday:
We discussed several topics related to the homework problems: the properties of continuous functions on compact subsets of the plane; an example of proving properties of double integrals; the proof that double integrals can be evaluated as iterated one-dimensional integrals; an example of integration in polar coordinates.
Here are the homework exercises that are due next Thursday:
We discussed surface area: in particular, the example of Schwarz showing that it is not correct to define surface area as the limit of sums of areas of inscribed triangles.
We discussed some of the homework problems and also the proof of the theorem stating that a continuous function on a compact planar region is Riemann integrable.
The homework is to begin reviewing for the second examination, which will be given on Tuesday, October 31. Here is a list of exercises to do for review (these are not to be handed in):
We discussed the notion of a path integral (or line integral), the notion of an exact differential, necessary and sufficient conditions for exactness, and the concept of simple connectivity.
The homework is to continue reviewing for the second examination, which will be given on Tuesday, October 31.
We discussed some of the review exercises in preparation for the second examination, which will be given on Tuesday, October 31.
We continued the discussion of path integrals, exact differentials, and Green's theorem.
Here are the homework exercises that are due next Thursday.
We continued the discussion of Green's theorem in the plane, and we worked on some of the homework problems.
We discussed harmonic conjugates, Green's theorem, and Stokes's theorem.
Here are the homework exercises that are due next Thursday.
We continued the discussion of Green's theorem, Stokes's theorem, and the divergence theorem.
We discussed change of variables in multiple integrals.
Some homework exercises to work, but not to hand in, are 3, 4, 8, and 9 on page 245.
Turning to applications of differentiation, we discussed first the classification of local extrema of functions of one variable in terms of the lowest-order derivative that is not zero at a critical point. Then we turned to the classification of critical points of functions of two variables in terms of the matrix of second partial derivatives.
Since Thursday this week is Thanksgiving Day, there is no homework assignment.
We discussed the classification of critical points of functions of two and three variables.
Some homework exercises to work, but not to hand in, are 1 and 10 on page 121; 1, 5, 6, 7, and 11 on pages 125-126; 7 and 13 on page 131.
We continued the discussion of critical points of multi-variable functions. We also discussed the problem of constrained extrema and the method of Lagrange.
Some homework exercises to work, but not to hand in, are 4, 6, and 9 on page 135 and 1, 2, and 8 on page 140.
This was the last class meeting for the semester. We completed the course evaluation forms and reviewed for the final examination to be given on Friday, December 8, from 12:30 to 2:30 in the afternoon.