Spring 2006

# Mathematics Math21a Spring 2006

## Multivariable Calculus

Office: SciCtr 434
Email: knill@math.harvard.edu
Harvard Mathematics

 Weekly checklists [PDF]: "Somewhere, over the rainbow ..." Week1 Week2 Week3 Week4 Week5 Week6 Week7 Week8 Week9 Week10 Week11 Week12

 Send questions of potential general interest to math21a@fas.harvard.edu.
 Question: Q: What is the curve? Answer: You see the curve to the left. There are about three constraints: there should be around 40 percent A and A-, the average should be in the B+ range and the standard deviation window should include only A and B grades. Then, according to the distribution, the cutoffs are set. This also depends on gaps in the distribution, i.e. avoiding too flat parts of the curve for a cut. The cuts also depend at the analysis of grades of a few "gauge" students near the threshold.
Question: Q: Do you have statistics about resurrection? Answer: Sure. Resurrection usually takes place for about 10 percent of students. Because the HW score were in general high, beating the average grade with the final grade is not so easy. But resurrection happend, for example, with hard work. The tougher material of the third part could make a difference. The graphics to the left shows the correlation between average score and final score. The graphics to the right shows the difference of the final score minus the average score.
 Question: Q: What does ```rnyyl? Nyy qbar? Erzrzore gung ceboyref ner rnfl vs lbh xabj gur nafjref ``` on the exams page mean? Answer: You have to find out yourself. Hint: it is is a Cesar cypher The one used is quite special and has its own name.

 Question: Q: In many of the problems I encountered, it mentioned to assume the region is a simple solid. I understand that this is because the theorems only work under these conditions. However, the book only gives two examples of simple solids. It says a simple solid is something like a rectangular box or an ellipsoid, but it never says how to determine if something is a simple solid-it never mentions what something that isn't a simple solid is. What determines if a solid is "simple"? What happens when solids aren't simple? How would one go about calculating things like flux? Answer: Stewart calls a region simple, if it is both Type I, Type II (p 934). In three dimensions, it should be os that you can set up a triple integral with any given variable order and the integrals of the third bound are given by functions (p.966). The definition of "simple solid curve" not so important. Dont mix it up with "simply connected". That refers to the property that you can pull together any closed loop in the region to a point. For the integral theorems, the notion of simple does not matter. It is just simpler to setup both the triple integral as well as the surface integrals if you have a simple region. But it can also be no problem for other regions like the doughnut, which is not simple but still, it is still (relatively) simple to parametrize the boundary. If you should have a solid, for which you have several surfaces as a boundary, you will have to parametrize all the boundaries. An example is a swiss cheese. You have to parametrize the outside, as well as the holes. A swiss cheese is not a simple region. The notion of simply connected only enters at one place in our course. If a region is simply connected and a vector field has curl(F)=0 in that region, then there is a function f such that F = grad(f).

 Question: Q: Will we be expected to derive any results e.g. how to get from rectangular to polar coordinates in a double integral. Or will we be expected to just apply these results? Answer: In general, you don't know to know the proofs. But note that in order to remember the r factor for example, when going to polar coordinates, it is good to have in mind the little wedge of size dr and r dtheta . The area of this gives you the distortion factor r dr dtheta While you have not to know the derivations of such things in an exam, it will help you to remember this in the long term. Similar with other topics of this midterm: to remember the Lagrange equations, it is good to have in mind that these equations just express that two gradients have to be perpendicular and if the two gradients were not perpendicular, then the rate of change of the function along the constraings is nonzero. The second derivative theorem is a result which is probably best remembered for now without proof.. One needs linear algebra to give an intuitive proof. The proof you find in the book is a hack, not intuitive.

 Question: Q: In chapter 9 about curves, there are a lot of formulas. Which do we have to know? Answer: Velocity, speed, acceleration, the unit tangent vector, how to get r back from r', arc length have to be known Curvature formula, the formulas for the Frenet frame have not to be known by heart, but you have to know what curvature means. You will not have to memorize the curvature formula. But when given, you can compute the curvature like here in the case of the helix: ``` r[t_]:={Cos[t],Sin[t],t}; v=Cross[ r'[t],r''[t]]/Sqrt[r'[t].r'[t]]^3; Simplify[Sqrt[v.v]] ``` where the curvature is constatn. Here is a proof of the curvature formula with mathematica ``` r[t_]:={x[t],y[t],z[t]}; s[t_]:=Sqrt[r'[t].r'[t]]; T[t_]:=r'[t]/s[t]; kappa1[t_]:=Sqrt[T'[t].T'[t]]/s[t] v[t_]:=Cross[r'[t],r''[t]]; kappa2[t_]:=Sqrt[v[t].v[t]]/s[t]^3; kappa1[t]^2==kappa2[t]^2 ```
 Question: Q: Can I use the old edition of Stewards book? Answer: We need the third edition. The problems are different from edition to edition.