Day 22 in Mat228

Today:

• Announcements
  • Homework for 11.6 due.
  • Homework for 11.7 due next time.
  • Exam revisions graded: come in to discuss it and get 100% on the redo. Email me to schedule a meeting (Please use subject line REDO).
  • Read Section 13.2 for next time responding to the worksheet for section 13.2.
  • Short quiz on 13.1/13.2 next time.
  • Today we start "Vector Geometry". When you're working with (and drawing vectors), it's sometimes convenient to have graph paper. I've put a link on our home page to some graph paper you can simply print off as you need it.
  • Just FYI: next week (Thursday) we'll have our second exam. It will cover the rest of chapter 11 (11.2-11.7).

• Any questions on old stuff?

• Section 11.7: Taylor Series
  • We've basically seen all this material before: we're now just summarizing:
    
    is the Taylor Series expansion of f(x) centered at x=c. The special case of c=0 is called the Maclaurin series. The factorials in the denominator sweep up all the powers that pop down during the differentiations.

    The coefficients a_kare "designed" so that after k differentiations of the series, when we evaluate the series we pick off only the k^th derivative evaluated at x=c:

    

    

    We've seen that, because of the radius of convergence, the function may not be represented by a power series at some values of x (even as it's perfectly represented at other values of x).

    If we can bound all derivatives of a function on an interval about x=c, then the Taylor series represents the function there:

    

    The expansion for the binomial series is a classic: it's another of the great achievements of Sir Isaac Newton, one of the founders of calculus:

    

    What about its radius of convergence? Let's look at #84, p. 599.

  • Examples:
    • #1, p. 596
    • #4, p. 597 (remember that there are several different ways to do these! You don't have to work from the definition....)
    • #10 p. 597
    • Let's take a look at some more interesting problems....
      • #48, p. 597
      • #49, p. 598 (Note: if you have the First Printing of our text, you don't have this problem -- and all the rest of your problems will be off by one!).
      • #77, p. 598 (homework)
      • #79, p. 598

• Sections 13.1 and 13.2 basically define and treat vectors in two- and three-dimensional spaces.
  • Uniform Circular Motion is best understood using the concepts of vector geometry.

    So are many other concepts from physics, where quantities have a direction and a "strength" -- a rocket in flight, for example. It is moving in a certain direction, and it has a speed, whose value can be represented by the length of a vector.

    • vector: a quantity with length and direction:
      

    • In uniform circular motion, we could represent the position of the bob by and
    • The three vectors that are (most) important in the physics of circular motion are the vectors of
      • position,
      • velocity, and
      • acceleration,
      each of which is a function of time.
    • Animations, demonstrations:
      • Uniform Circular Motion
      • For all of the gory details, a nice layout of all the physics, and a cool (but scary) example, let's check out physclips -- the Australian dialect makes it more fun, too!

  • If a vector is in two-dimensions, then the length is just given using the Pythagorean theorem formula:
    
    

    in which case

    

    Obviously this can be generalized to vectors in four dimensions, five dimensions; even six dimensions. Maybe seven dimensions, too; or even n dimensions.

  • The vectors above are given with components:
    

    examples:

    • #5, p. 663
    • #9, p. 663
    • #11, p. 663
    
    

  • Figure 11 illustrates the so-called parallelogram rule:
    

    example:

    • #17, p. 664

    Here it is in three-space:

    

    with the obvious changes to formulas because you now have three components, instead of two:

    

  • 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:

    

    example:

    • #38, p. 664

  • 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, in two-space, the usual basis is given by the two (perpendicular) vectors denoted i and j:
    

    and, in three-space,

    

  • Theorems:

    

    

    The Triangle inequality basically says that the diagonal of a parallelogram is shorter than or equal to the sum of the two sides:

    "The shortest distance between two points is a straight line."

    In the case of Figure 17 (for vectors at right angles to each other)

    this rule is familiar in the form of the old algebraic inequality
    

    and the only time you'll have equality is when either a or b is zero.

• Let's look at some more examples:
  • #19, p. 664
  • #29, p. 664
  • #47, p. 664
  • More physics:
    • #58, p. 665
    • #60, p. 665