Math 13, Winter 2011
Schedule
| Week | Day | Date | Section in text |
| Week 1 | Day 1 | January 5 | Terminology and concepts |
| Day 2 | January 7 | Ch. 13: review (minus 13.6) |
| Week 2 | Day 3 | January 10 | Ch. 14: review (minus the last parts of 14.3 and 14.4) |
| Day 4 | January 12 | Ch. 15: review |
| Day 5 | January 14 | 16.1, 16.2 |
| Week 3 | Day 6 | January 17
(switch to x-hour) | 16.2, 16.3 |
| Day 7 | January 19 | 16.4 |
| Day 8 | January 21 | 16.4 |
| Week 4 | Day 9 | January 24 | 16.5 |
| Day 10 | January 26 | 16.6 |
| Day 11 | January 28 | 16.7 |
| Week 5 | Day 12 | January 31
(previous week's x-hour) | 16.8 |
| Day 13 | February 2
(now Jan. 31) | 16.9 |
| Day 14 | February 4
(now Feb. 2) | 17.2 |
| Week 6 | Day 15 | February 7 | 17.1, 17.2 |
| Day 16 | February 9 | 17.3 |
| Day 17 | February 11
(switch to x-hour) | 17.4 |
| Week 7 | Day 18 | February 14 | 17.4 |
| Day 19 | February 16 | 17.5 |
| Day 20 | February 18 | 17.6, 17.7 |
| Week 8 | Day 21 | February 21 | 17.6, 17.7 |
| Day 22 | February 23 | 17.7 |
| Day 23 | February 25 | 17.7, 17.9 |
| Week 9 | Day 24 | February 28 | 17.9 |
| Day 25 | March 2 | 17.8 |
| Day 26 | March 4 | 17.8 |
| Week 10 | Day 27 | March 7 | Interesting topic TBD |
| Day 28 | March 9 | Review and questions |
Today we mainly discussed the topics that we'll cover in the course. At the end, I mentioned some ways in which mathematicians' language may differ from standard English. A summary can be found here.
I didn't get to this example in lecture, so you might want to work it out on your own: Find the equation of the plane parallel to the plane 5x + 2y - z = 12 that passes through the point (3,-4,7). (Tip: how would you find the normal vector used to define the plane whose equation is given?)
As you noticed in class, I couldn't get the software to work properly. I'm still not sure what went wrong, but I've tested everything on the library computers and it works there. If you want to try it yourself, here's what you do:
- Open the right program. It's called Maple. The computers in the lecture halls have Maple 14; the computers in the library have Maple 11. The commands below should work on both versions.
- First, type
with(plots):
and press Enter. This makes sure the graphing package is loaded.
- Now you can plot a vector function. Suppose you're trying to plot an upward-spiraling helix x=cos(t), y=sin(t), z=t, and you want to plot it for all t between -1 and 1. You will type
spacecurve([cos(t),sin(t),t],t=-1..1,axes=normal,labels=[x,y,z]);
The only necessary arguments are the first two. You have to give a vector function in the form [f(t),g(t),h(t)], and you have to give the domain for t (if it's the interval [a,b], type in t=a..b). The last two parts are optional. They just tell the computer to show the axes and label the axes with x, y, and z. If you don't use them, you get a nice vector curve, but since the computer won't draw the axes, you can't get an idea of its placement within R^3. It often looks very pretty like that, though!
If you're having trouble with this, I'd be happy to show you some tricks with Maple in office hours.
I didn't get to give an example of finding a tangent plane to a surface, so here is one: Find the tangent plane to the surface z=x sqrt(y) at the point (1,4,2).
f_x(x,y) = sqrt(y) and f_y(x,y) = x/(2sqrt(y)), so f_x(1,4)=2 and f_y(1,4)=1/4. Then the equation for the tangent plane is
z-2 = 2(x-1) + 1/4(y-4).
If you want to draw the surface z=f(x,y) in Maple, the general form for the command is
plot3d(f(x,y),x=a..b,y=c..d,axes=normal,labels=[x,y,z]);
So if you want to plot the surface z=xy for x between -1 and 2 and y between -2 and 5, you would type
plot3d(x*y,x=-1..2,y=-2..5,axes=normal,labels=[x,y,z]);
Here are instructions for drawing level surfaces in Maple.
Before you start, type
with(plots):
Suppose you want to plot only one level surface (for instance, the sphere x^2 + y^2 + z^2 = 4. Type
implicitplot3d(x^2 + y^2 + z^2 = 4,x=-2..2,y=-2..2,z=-2..2,axes=boxed,scaling=constrained);
The numbers after the x=, y=, and z= give the domains for these variables.
However, often you'll want to draw multiple level surfaces so you find out more about the function generating them. In this case, you may want to name the level surfaces and then plot them all together. Suppose we want to name the level surfaces x^2 + y^2 + z = 1 and x^2 + y^2 + z = 4. We would type
s1 := implicitplot3d(x^2 + y^2 + z = 1, x=-4..4,y=-4..4,z=-6..6,color=blue):
s2 := implicitplot3d(x^2 + y^2 + z = 4,x=-4..4,y=-4..4,z=-6..6,color=red):
Note that we don't include any information about the axes, but we do say which color we want the resulting surfaces to be. Then, when we type
display([s1,s2],axes=boxed,scaling=constrained);
we get a plot of both of these surfaces, the first in blue and the second in red.
I talked about integrating over rectangles, switching the order of integration, and introduced the idea of integrating over other shapes. Keep in mind that often it will be equally easy to integrate in either order but that sometimes one order will be much easier than the other. If you ever find yourself tempted to do an integration by parts, for example, try reversing the order of integration. It may not be necessary if you do that.
Someone asked me a good question after class: does Fubini's Theorem still hold for integrals over more general regions?
The answer is that although you can reverse the order of integration for more general regions for any function f(x,y) you're likely to see in this class, you have to be more careful with the bounds. If both sets of bounds are constant (a < x < b and c < y < d), we can use the same bounds when we reverse the order of integration. If one variable is bounded between two functions (say, g(x) < y < h(x)), we need to change both sets of bounds: now y will be between two constant bounds and x will be between two functions of y. Can you explain why that is?
This is a question I mentioned in both classes. It's problem #28 in 16.4:
A cylindrical drill with radius r_1 is used to bore a hole through the center of a sphere of radius r_2. Find the volume of the ring-shaped region that remains.
Express the volume in the first part entirely in terms of the height h of the ring.
Something to keep in mind when calculating masses and moments of inertia is that if (1) the lamina is symmetrical and (2) the density function is symmetrical enough, then you may be able to find a component of the center of mass by symmetry. For instance, if you have a lamina that is the bottom half of a circle and the density function is constant, you know that the x-coordinate of the center of mass will be 0. If the density function is not symmetric, though (like sqrt(x)+xy), you'll have to do all the calculations.
The best advice I can give you for learning how to find the bounds for triple integrals over general regions is simply to do lots of them. Look at the problems in 16.5, for instance. Or find the bounds of integration for the region inside the ellipsoid 9x^2 + 4y^2 + 36z^2 = 1. :)
If you want to plot in cylindrical coordinates in Maple, it's not any harder than plotting in rectangular coordinates: you just need to add the option coords=cylindrical. As an example, if you want to plot $z=theta$, you would enter the following.
with(plots):
implicitplot3d(z=theta,r=-3..3,theta=-3*Pi..3*Pi,z=-3*Pi..3*Pi,axes=boxed,coords=cylindrical);
It shouldn't be a surprise to find out that to plot in spherical coordinates in Maple, you just need to add the option coords=spherical. If you're still trying to sort out the roles of phi and theta, I recommend drawing a sphere and then adjusting the ranges for phi and theta:
with(plots):
implicitplot3d(rho=1,rho=-2..2,theta=0..2*Pi,phi=0..Pi/2,axes=boxed,coords=spherical,scaling=constrained);
That should give you the unit sphere. Now experiment!
If you want to plot a vector field in Maple, here's what you do:
with(Student[VectorCalculus]):
VectorField(< P(x,y),Q(x,y) >,output=plot,view=[a..b,c..d],scaling=constrained);
< P(x,y),Q(x,y) > will be the vector function defining the vector field, a and b will be the x-bounds, and c and d will be the y-bounds.
Think about all of the terms we've used to describe regions in the plane: open, connected, and simply connected. Try to think of regions that have each combination of these properties. Are there any combinations that aren't possible?
How would you restate Green's Theorem if the condition that C be positively oriented were changed to the condition that C be negatively oriented?
I strongly suggest that you do the half of the proof of Green's Theorem that I didn't do in class today. Also, if you are reading this before the second midterm, write the word "balloon" on the bottom of the second page of your exam.
If you're interested in the physical applications of div and curl, you might be interested in one of the books I put on reserve in Baker: Div, grad, curl, and all that. It's written from a physicist's point of view. The topics aren't covered in the same order we've been covering them in class, so parts of it might be tricky, but I'd be happy to go over them with you.
I really recommend taking a look at #12 in 17.5 to make sure you understand what kinds of functions div, grad, and curl can be applied to.
Here's the code for plotting parametric surfaces in Maple:
with(plots):
plot3d([x(u,v),y(u,v),z(u,v)],u=a..b,v=c..d,axes=normal,scaling=constrained);
where a and b are the u-bounds and c and d are the v-bounds.
I really recommend doing problems 13-18 in 17.6 so you can get a feel for associating plots of parametric surfaces with their parametric equations.
Take a look at the applications of surface integrals on pp. 1118-1119 of your book -- I didn't go over the formulas in class because they're so similar to the ones you already know, but you should review them.
Please think carefully about why we were able to avoid parametrizing the sphere in the last example today. We clearly had to be able to find a normal vector without parametrizing. What was special about the form of F.n that allowed us to do this?
I strongly encourage you to read the full proof of the Divergence Theorem in the book. I think the outline I did in class today should make it easy enough to follow.
Please check my claim from class that if E=(epsilon)(Q)r/|r|^3, div E = 0.
Remember that the positive orientation of the boundary curve of a parametrized surface is entirely dependent on the orientation of the surface itself. The only reason we can say that a curve bounding a region in the plane has positive orientation when it is traversed counterclockwise is that we always assume that the region in the plane is oriented upward.
It would probably be a good idea if you read through the proof of a special case of Stokes's Theorem in the book (pp. 1129-30).
Here's the example that I didn't get to today:
Verify Stokes's Theorem for F = < x, y, xyz > and the surface S that is the part of the plane 2x+y+z=2 in the first octant with positive orientation.
If you're interested in understanding the physical meaning of curl, I recommend reading p. 1132. They talk about a paddle wheel while I talked about a rubber duck in class, but the general idea is the same.