Day 51 in MAT229

Last time: Exam 3

Next time: Section 13.1

Today:

Announcements:
- Hand in these homework problems from 12.4, pp. 635-, #1, 5, 8, 9, 18, 24, 26, for Monday, 12/8.
- Hey: there's another lunch coming up! On Friday. Be ready....
  Join us for the next Department of Mathematics Major/Minors Lunch.
  When: Friday December 5, 2008, 11:30am-1:30pm
  Where: ST 343 (the Calculus Lab)
  All mathematics majors and minors [and even prospective ones!;)] are welcome. This is a great opportunity to get together with other students and faculty on a very informal basis. Plus, the food is good.
So let's wrap up section 12.4: polar areas and arc lengths
- Let's convince ourselves of the derivation of the polar area formula:
  
  ${A=\frac{1}{2}\int_{\alpha}^{\beta}r^2d\theta}$
- And do some examples:
  - A relatively simple one: #4, p. 635.
  - A little more complicated: #16, p. 636.
  - Now you try: #13, p. 636.
- The arc length formula is also relatively easy: we get it by considering the polar representation as a parametric representation of the coordinates x and y:
  
  ${A=\int_{\alpha}^{\beta}\sqrt{\left(f(\theta)\right)^2+\left(f'(\theta)\right)^2}d\theta}$
- Examples:
  - Let's check the formula on the circumference of a circle.
  - What's the length of that limacon in #16, p. 636?
Vector geometry (Chapter 13)
- Objectives:
  1. Introduce the definitions and rules of vectors
  2. Use vectors to represent objects in two and three-d (e.g. lines, planes).
  3. Use vectors to represent motions in space, and the associated quantities such as accelerations, forces, etc.
  4. Use certain vector notions (e.g. dot and cross products) to represent things more easily.
- Sections 13.1 and 13.2 basically define and treat vectors in two- and three-dimensional spaces.
  - Uniform Circular Motion is well-understood using the concepts of vector geometry.
    - remember that ${x(t)=R\cos(t)}$ and ${y(t)=R\sin(t)}$
    - The three vectors that are (most) important are the vectors of
      - position,
      - velocity, and
      - acceleration.
    - For all of the gory details, a nice layout of all the physics, and a cool example, let's check out physclips -- the Australian dialect makes it more fun, too!
  - vector: a quantity with length and direction:
    
    If the vectors are in two-dimensions, then the lengths are just given using the Pythagorean theorem formula:
    
    in which case
    
    ${||v||=\sqrt{a^2+b^2+c^2}}$
    Obviously this can be generalized to vectors in four dimensions, five dimensions; even six dimensions. Maybe seven dimensions, too; or even n dimensions.
  - Vectors with length 1 are special, called unit vectors, and are important because they can be dedicated to showing direction:
    
    It's clear that we can turn any vector into a unit vector, by simply scaling it:
  - The vectors above are given with components:
  - Figure 11 illustrates the so-called parallelogram rule:
    
    Here it is in three-space:
    
    with the obvious changes to formulas because you now have three components, instead of two:
  - We use special components and the parallelogram rule to write each vector in terms of a basis of vectors (equal in number to the dimension of the space): for example,
    
    and, in three-space,
  - Theorems:
    
    This rule is familiar in the form of the old algebraic inequality
    
    ${\sqrt{a^2+b^2}\le{|a|+|b|}}$
- Let's look at some examples:
  - #5, p. 663
  - #9, p. 663
  - #17, p. 664

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