More volunteers
These students have volunteered (well, they are in 291, darn it!) to answer questions next time about the 251 assigned homework problems in these sections:

Textbook section	Names
12.4	Thomas Ehrlich Evan Fitzgerald
12.5	Jacelyn Gerges Andrew Harrison
13.1, 13.2	Pawan Harvu David Hsiung
13.3	Gibson Kim Kevin Kobilinski
13.4	Cosmo Kwok

Motion in a straight line
The bug doesn't feel the sides.

Motion in a circle
The moving bug always feels a transverse force.

Differentiation and three kinds of multiplication for vector-valued functions
Scalar multiplication
Dot product
Cross product

An ongoing example: the right circular helix

Space curves: the beginning
Tangent vector, unit tangent vector, speed

Trick #1: defining the principal normal

The definition of curvature

The binormal

The Frenet frame

How the binormal changes

The definition of torsion
Bending out of a plane

The derivative of the normal

The Frenet-Serret formulas

Check everything with the helix!

The volume of a parellelopiped

The vector triple product equals a determinant equals a volume

Curves in space and in the plane

A differentiable curve has differentiable coordinates

The derivative is velocity

Length is speed

Direction is tangent to the curve

Tangent vector and tangent line

Distance=rate·time as an approximation

Definition of distance traveled

And the length of a curve

And the intractibility of computation

How a plane curve bends

What's more; what's less

The bewildering effect of speed on "perceived" bending

Test cases
circles, line, parabola

The rate of change of angle

The rate of change of angle with respect to arc length

The formula defining curvature

Too big! There must be other ways ...

More vector algebra

Is a point on a plane determined by three given points?

Description of a plane with a given point and a given normal vector

Description of a line through a given point in a given direction

Distance from a point to a plane

Other distances

Geometry, chirality, and prejudice

Flatland

Cross product

A multiplication table

A formula for the cross product

Deception? Lie? Well, only inferentially ....

Me, dog, lamp post

More generally

Now with angles (a robot arm?)

Orthogonality

Resolving into parallel and perpendicular components

Volunteers
These students have volunteered (well, they are in 291, darn it!) to answer questions next time about the 251 assigned homework problems in these sections:

Textbook section	Names
12.1	Valentyn Boginskey Joshua Bryan
12.2	Albert Chen Wei Chen
12.3	Alexander Crowell Christophe Doe

I thank them in advance.

Vector space

{Inner|scalr|dot} product

{Length|norm|magnitude}

A function

A quadratic function

The Cauchy-Schwarz inequality

The triangle inequality

Examples (at last!)
From quantum mechanics.

R³
Vectors are directed line segments. Vectors are forces. Why?

Components of vectors

Dot product

Law of cosines
(messed up in class!)

Word of the day
bravura Showy; ostentatious.

Calc 1

• The definition of derivative
  • As a limit
  • As another function
  • Computing the darn thing (algebraic algorithms, the chain rule)
• The Mean Value Theorem
  • Connection between {in|de}creasing and sign of the derivative
  • Detecting extreme values
  • Concavity
• Definition of the definite integral
  • As a limit of Riemann sums (a very strange kind of limit!)
  • Elementary properties: linearity in functions, positivity, additivity over adjoining intervals
• The Fundamental Theorem of Calculus
  • Computing definite integrals via antiderivatives
  • Definite integral as accumulated change
  • Simple numerical approximation schemes for the definite integral
  • Initial value problems for simple ODE's.

Calc 2
Lots and lots of detailed schemes for antidifferentiation. Wow!
More intricacies about sequences and series. More wow!!
This stuff is all worthwhile, but everyone would agree that the highlights of the course are:

• Integration by parts has wonderful, perhaps amazing, consequences. Certainly it allows computation of antiderivatives in unexpected ways. This is then applied to show that an arbitrary "signal" has less and less high frequency vibrations in it (Fourier series or integrals). Also integration by parts is important in the study of what are called boundary value problems, a different way of studying naturally occurring problems in mathematical physics which use differential equations and specify information at both an initial point and an endpoint.
• Taylor's Theorem which allows a sufficiently differentiable function to be approximated by a polynomial, with an error that can be described in several interesting ways, is also terrific. Polynomials can be evaluated very quickly ("Horner's method") and so any result which allows a polynomial to be used instead of a more "nasty" function is very useful.

Several variable calculus
Several here means "more than one".

• Continuity and differentiability The actual definitions have to be done more carefully than one might suspect. A century or so of experience has enabled mathematicians to write the definitions so that they look the same in Rⁿ as in r¹. This tendency towards compact writing is sometimes misleadingly terse. In this case, the definitions are made in certain ways which don't really show that phenomena occur which don't happen in one variable. Sigh. So part of the job is to take apart the definitions and display the rather weird examples.
• Integration in one variable is defined (in a calc 1 course, anyway!) by chopping up an interval into small subintervals. But in even two variables, if a function is defined over a region in the plane, the "chopping" can be done in many different ways, using pieces that aren't necessarily congruent (equal subintervals) or even similar (always two intervals are similar, but two little chunks of the plane are not). How can such a definition be made? And looking at different chopping strategies can be important in specific examples, to take advantage of the symmetries in physics or geometry.
• Relating the integral over a region to an integral on the boundary
  In the one-variable case, there seems to be only one FTC. Things are quite a bit more complicated in several variables. In Math 251, for example, maybe three (really, three) different analogues of FTC are given towards the end of the semester. Things get complicated. Certainly the theory is complicated But as well, the phenomena which are being described are complicated. For example, I cited the problem of trying to understand what happens to the interior temperature of a thin metal plate which is subject to some boundary heat distribution. In order to understand the isothermals in the plate, and the heat flow in the plate, we need some method of transferring information from the boundary to the inside of the plate. And that is what the higher dimensional versions of FTC will do. (Yeah, yeah, the plate also expands, but, y'know, for steel the coefficient of thermal expansion is about 1.24·10^-5cm per degree C, maybe not too large and not the first thing to worry about.)

Distance in R¹
We looked at distance in the real line. The distance between a and b is defined by
     dist(a,b)=|a-b|
This is a great definition. Notice that:

1. dist(a,b) is always non-negative, and is equal to 0 exactly when a=b.
2. dist(a,b)=dist(b,a): this is called symmetry.

3. For any points a and b and c in R¹, dist(a,b)+dist(b,c)≥dist(a,c).

Distance in R²
Well, I went through the usual diagram to try to motivate the algebraic definition of distance in R². Look, please, at the diagram to the right. In the plane, points correspond to ordered pairs of numbers. So a point p might correspond to an ordered pair, (x₁,y₁), and q might correspond to (x₂,y₂). Then the point (x₁,y₂) is the vertex of a right triangle whose hypotenuse is a line segment connecting p and q. One leg of the right triangle is on a line where all the first coordinates are x₁, and the length of that leg is given by the one dimensional formula, |y₁-y₂|. The other leg is on the line where all the second coordinates are y₂, and the length of that leg is |x₁-x₂|. Then by Pythagoras, the hypotenuse has length sqrt(|x₁-x₂|²+|y₁-y₂|²). And usually the absolute values are discarded since we are squaring the quantities. Therefore we officially define:
dist(p,q)=sqrt((x₁-x₂)²+(y₁-y₂)²) if p has coordinates (x₁,y₁) and q has coordinates (x₂,y₂).

Are rules 1 and 2 and 3 above still valid? Well, 1 is true because, first, the square root is always non-negative. And if the square root defining dist is equal to zero, then since the terms inside it are also non-negative, the only way the square root can be zero is if both x₁-x₂=0 and y₁-y₂=0. The coordinates must be the same, so p and q are the same points.

Rule 2, symmetry, is also true and equally simple to verify, since squares of W and -W are the same. But rule 3 is a different sort of thing.

Rule 3 for distance in R²
Well, if p has coordinates (x₁,y₁) and q has coordinates (x₂,y₂) and r has coordinates (x₃,y₃), then I want to know if dist(p,q)+dist(q,r) is always ≥ dist(p,r).

The geometric point of view
This is what I personally prefer, and this is what a student in the class saw almost immediately. The length of one side of a triangle is less than or equal to the sum of the lengths of the other two sides. And that is exactly what Rule 3 states. So rule 3 is clearly true!
The algebraic point of view
Maybe other people don't necessarily accept the picture, and don't "see" the result. Let us try to verify algebraically that
sqrt((x₁-x₂)²+(y₁-y₂)²)+sqrt((x₂-x₃)²+(y₂-y₃)²) if p≥sqrt((x₁-x₃)²+(y₁-y₃)²)>
I don't think this is totally, immediately "clear". In fact, this inequality, if you want to worry, is a statement about six variables. Six! I asked if there was any way to transform it into another algebraic statement whose correctness we would accept immediately. We began working. We squared, we rearranged ... it was a mess. Then I suggested that maybe we should try to rewrite things a bit:
Change x₁-x₂ to A     Change y₁-y₂ to B     Change x₂-x₃ to C     Change x₂-x₃ to A    
Then x₁-x₃ will become A+C and y₁-y₃ will become B+D.

The inequality we would like to verify now is:
sqrt(A²+B²)+sqrt(C²+D²)≥sqrt((A+C)²+(B+D)²).
At least we have only four variables now, rather than six. Now square: A²+B²+2sqrt(A²+B²)sqrt(C²+D²)+C²+D²≥(A+C)²+(B+D)²
This step is reversible because we know that the inequality we began with has only non-negative terms, and square roots and squaring will preserve such inequalities. Now "expand":
(A+C)²+(B+D)² becomes A²+2AC+C²+B²+2BD+D².

We want to verify that:
A²+B²+2sqrt(A²+B²)sqrt(C²+D²)+C²+D²≥A²+2AC+C²+B²+2BD+D²
"Cancel" (subtract) everything that's equal on both sides. The result is:
2sqrt(A²+B²)sqrt(C²+D²)≥2AC+2BD

Divide by 2:
sqrt(A²+B²)sqrt(C²+D²)≥AC+BD

And let's square again (amazing!):
(A²+B²)(C²+D²)≥(AC+BD)²
But (AC+BD)²=(AC)²+2(AC)(BD)+(BD)²
and (A²+B²)(C²+D²)=A²C²+A²D²+B²C²+B²D²
Therefore the inequality we want is:
A²C²+A²D²+B²C²+B²D²≥(AC)²+2(AC)(BD)+(BD)²
Let's cancel stuff again. The result is:
A²D²+B²D²≥2(AC)(BD)

Well, uhhhh ... this is:
A²D²-2(AC)(BD)+B²D²≥0
and this is the same as:
(AC-BD)²≥0 and this is true because squares are non-negative. Sigh.

What is going on?
Any student who followed this is either crazy or brilliant or both. We have verified that the triangle inequality (rule 3 for distance) is true for the distance formula in R². These inequalities are all quite famous, actually. I just took a rather "low tech" way to verify them. The inequality
sqrt(A²+B²)sqrt(C²+D²)≥AC+BD
which we also verified in passing has its own name. It is called the Cauchy-Schwarz inequality. It is important.

The approach I just used is what could have been done maybe two or three hundred years ago. I will try to show you what the heck is going on in a more modern fashion Monday.

Read the text!
Please read the first few sections of chapter 12, as on the Math 251 syllabus.

Maintained by greenfie@math.rutgers.edu and last modified 9/7/2006.