# Math 2130Q-001, Fall 2011

## Schedule

Revised Thursday, September 1
Week Sections in text (estimated) Quiz/Test Administrivia
Week 1: Aug. 29 - Sept. 2 1.1-1.4 None!
Week 2: Sept. 5-9 1.4-1.6 Quiz #1 No class on Monday (Labor Day)
Week 3: Sept. 12-16 1.6, 4.1-4.3 Quiz #2 Monday: last day to drop without a W
or change grading option to pass/fail
Week 4: Sept. 19-23 4.3-4.4, 5.1-5.2 Quiz #3
Week 5: Sept. 26-30 5.3-5.4 Midterm #1
Week 6: Oct. 3-7 6.1-6.4 Quiz #4
Week 7: Oct. 10-14 6.4-6.5, 7.1 Quiz #5
Week 8: Oct. 17-21 7.2-7.4 Quiz #6
Week 9: Oct. 24-28 7.4-7.5 Quiz #7
Week 10: Oct. 31 - Nov. 4 7.6, 8.1-8.2 Midterm #2 Monday: last day to drop
or change a pass/fail option to letter grade
Week 11: Nov. 7-11 8.3-8.4, 9.1 Quiz #8
Week 12: Nov. 14-18 9.1-3 Quiz #9
Week 13: Nov. 21-25 Thanksgiving break
Week 14: Nov. 28 - Dec. 2 9.3, 9.6, 9.4 Quiz #10
Week 15: Dec. 5-9 9.5 and review Quiz #11

#### August 31

Welcome to Math 2130Q! The problem I left you with at the end of class was this: try to find a set of vectors in R^2 that is not a basis because at least one vector can be written in more than one way as a linear combination of the vectors you chose.

#### September 2

Think very carefully about how different equations can define the same line!

#### September 7

I suggest that you read the proof of the Triangle Inequality in the book (p. 28). I didn't have time to do it in class, but it's a neat little proof of a result you've been using since you were old enough to understand what a shortcut is.

#### September 9

One of your classmates gave me a nice little tip after class for drawing points in 3-space: put a little 3D box around each point. It's a useful reminder for your brain that you're looking at something that's supposed to be 3D.

You should also be aware of the difference between the way I used the dot product to calculate distance in class and the way the book does it (p. 35) and understand the relationship between the two.

#### September 12

I suggest that you take a look at 1.6, #15: it's a guide to showing that |a cross b| = |a||b|sin(theta).

I also promised you some identities involving the dot product and the cross product. Here are two:

(a cross b) dot (c cross d) = (a dot c)(b dot d) - (b dot c)(a dot d)
(a cross b) cross (c cross d) = (a dot (b cross d))c - (a dot (b cross c))d

#### September 14

The moral of today's story is that if you want to do anything basic to a vector function like take its limit, find out if it's continuous, or take its derivative, you just have to pay attention to the coordinate functions and do whatever you want to do to them.

#### September 16

Suppose a duck is flying, and its acceleration vector is < t, 0, -g >. Further suppose its initial velocity was < 0, 0, 0 > and its initial position was < 4, 1, 0 >. Find the duck's position vector.

#### September 19

One thing to be aware of about the quadric surfaces we talked about (elliptic cones, etc.) is that their equations won't always have precisely the form given in class and the book. For instance, x^2 - y^2 +z^2 = 5 will still be a hyperboloid of one sheet, even though the y-term is negative instead of the z-term. It's just symmetric around the y-axis and not the z-axis. When looking at an equation, you need to do one thing and then ask yourself two questions. The "one thing" is to write the equation with all the variables on one side, the constant on the other, and to multiply by -1 if necessary to ensure that at least 2 of the terms are positive.
• Are all the terms squares? If one term is linear, it's a paraboloid. (If one of the positive terms is the linear term, it's a hyperbolic paraboloid. Multiply by -1 to get an equation more like the one in the book.) Otherwise, it's one of the other four.
• Do all the variables' signs match? If so, it's an ellipsoid. If not, you have an elliptic cone or a hyperboloid, and you need to check the sign of the constant to figure out what you've got.

#### September 21

Here's another parametrized surface for you to think about:
f(u,v) = < cos(u)sin(v), sin(u)sin(v), cos(v) >
Try to imagine what this would look like!

#### September 23

Here are some sets. Think about what their limit points, interior points, and boundary points are!
• The union of (-infinity,2] and [pi,infinity) in R
• { (x,y) | 0 < x < 2 and -1 < y <= 0 } in R^2
• The xy-plane in R^3
• { (x,y,z) | x^2 + y^2 -z^2 > 5 } in R^3

#### September 26

I recommend looking at Examples 5 and 6 on p. 229 to better understand why we want to think about functions only being differentiable on open sets instead of sets in general.

#### September 28

Make sure you understand the difference in finding the directional derivative in the direction of a vector u and finding the derivative with respect to u!

#### October 3

To practice using the gradient to get a tangent approximation to the level set of a function at a certain point, try finding the tangent approximation to the function f(x,y,z) = ze^(2x)cos(y) at f(x,y,z)=e^4 at the particular point (2,0,1).

#### October 5

I recommend reading through examples 10 and 11 on pp. 271-273 to better understand when changing variables is an advantage.

#### October 7

Finding the extreme points of a function on a given domain is just like it was in your first calculus course. The only complication is that the boundary may not be as simple as two endpoints, so you may have to parametrize it to find your answer.

#### October 10

Try the problem I gave you at the end of class with two constraints: maximize f(x,y,z) = x + 2y + 3z on the curve where x^2 + y^2 = 4 and x - y + z = 1 intersect.

#### October 12

If the second derivative test doesn't give you any information, what techniques might you use to figure out whether you have a local max, a local min, or a saddle point?

#### October 14

Think about all the different kinds of equations you've seen in this course that define surfaces (elliptic paraboloids, ellipsoids, hyperboloids, etc.). Which of them could be usefully put into spherical coordinates or cylindrical coordinates?

#### October 17

Your book doesn't mention Fubini's Theorem, but you should know about it: it says that if you want to integrate a function over a region, iterated integrals will give you the right answer and that the order of integration doesn't matter.

Remember that the book does have integral tables on p. 737.

#### October 19

I want to emphasize here that the book's notation and my notation are different: the book will write dx and dy immediately after the corresponding integral signs, and I will write them all at the end in the correct order of integration. Both are legitimate, though mine is more common.

#### October 21

More notation: I also want to emphasize that the book will use r, phi, and theta for spherical coordinates and I will use rho, phi, and theta, which is more common. Now that that's out of the way, here's the challenge problem I mentioned towards the end of class: Suppose that you have a sphere of radius a and that a hole of radius b is drilled through the center of it. First, find the volume of the remaining solid. Then show that the volume you calculated can be written in terms of the height h of the solid without reference to a and b.

#### October 24

Remember what dV and dA turn into in different coordinate systems!

#### October 26

Honestly, the only way to learn how to do Jacobians is to do lots of them. To decide if using a Jacobian would help, check to see if the region you're integrating over is difficult to write down a single integral for or if the function you're integrating over would benefit from a change of variable.

#### October 28

The main applications I talked about today are
• finding the mass of a region given a density function,
• finding the center of mass of a region given a density function (or simply given the region if a centroid is requested), and
• finding probabilities given a probability density function and figuring out how to turn a function into a probability density function. (Recall that a p.d.f. is simply a function that is never negative and whose integral over the entire space is 1.)

#### October 31

From now on, we'll be integrating over lots of interesting curves and surfaces. Keep in mind that the underlying dimension of the object is what's important. A line is 1D, so a line integral is a standard single-variable integral, a surface is 2D, so a surface integral (when you meet one!) is a double integral, etc.

#### November 4

When we parametrize by arc length, we do so to create a "standard" parametrization. Instead of tracing the curve at nonconstant speed, we find a parametrization for the curve that lets us traverse it with constant speed (in fact, a speed of 1). When we've done this, we can use distance or time as our variable: since our speed is constant, the distance we've traveled is in some sense equivalent to the time we've been traveling.

#### November 7

Curvature is useful because it's an intrinsic feature of the curve: it's based on the arc length and the unit tangent vector at a point, which don't depend on the particular parametrization.

#### November 9

Although I've been unable to find a good YouTube video on div and curl, you could go have a look at the book Div, grad, curl, and all that by H. M. Schey in the library. It talks about multivariable calculus from a physics point of view, and you might especially like it if you're in science or engineering. If you make your own video explaining div or curl, show me!

#### November 11

How would you restate Green's Theorem if the condition that the boundary be traversed counterclockwise had to be changed to the condition that the boundary be traversed clockwise?

#### November 14

Make sure you understand how the integrals over "interior" boundary curves cancel each other out when applying Green's Theorem to more complicated regions!

#### November 16

Remember that if the derivative doesn't exist at some point in the region, we can't apply Green's Theorem. Most of the time it won't matter, but it's worth checking.

#### November 18

Here's a practice problem to enjoy over Thanksgiving break: find the surface area of the part of the hyperbolic paraboloid defined by the equation z = x^2 - y^2 that lies over the unit circle centered at the origin.

#### November 28

I'll go ahead and derive the vector form of Green's Theorem in class on Wednesday. In the meantime, try proving one of the equalities from Section 9.6 I gave you today!

#### November 30

Today we tried a few little tricks that you can do with the Divergence Theorem: adding surfaces to nonclosed surfaces to close them off, letting us apply the theorem, and using it when the surface is piecewise smooth but whose surfaces have obvious normal vectors. You should be able to recognize situations like these!

#### December 2

After a little bit of searching, I have to report that I haven't found an application of the Divergence Theorem or Stokes's Theorem that doesn't relate to physics. I'll ask my colleagues, though, and please let me know if any of you find one!

#### December 5

Now you know the Fundamental Theorem of Calculus, the Fundamental Theorem for Line Integrals, Green's Theorem, the Divergence Theorem, and Stokes's Theorem. It might be useful to make up a chart giving the hypotheses of each and identifying the regions and boundaries involved.

Remember your #2 pencil on Wednesday!