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Announcements 10/6/10. Prayer Exam goes until Saturday Correction to syllabus: on Saturdays, the Testing Center gives out last exam at 3 pm, closes at 4 pm. Homework survey—survey closes tonight. Please respond this afternoon/evening if you haven’t already. Taylor’s Series review:

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announcements 10 6 10
Announcements 10/6/10
  • Prayer
  • Exam goes until Saturday
    • Correction to syllabus: on Saturdays, the Testing Center gives out last exam at 3 pm, closes at 4 pm.
  • Homework survey—survey closes tonight. Please respond this afternoon/evening if you haven’t already.
  • Taylor’s Series review:
    • cos(x) = 1 – x2/2! + x4/4! – x6/6! + …
    • sin(x) = x – x3/3! + x5/5! – x7/7! + …
    • ex = 1 + x + x2/2! + x3/3! + x4/4! + …
reminder
Reminder
  • What is w?
  • What is k?
  • How do they relate to the velocity?
  • Relationship between w and T
  • Relationship between k and l
  • Consistency check: w/k = ?
reading quiz
Reading Quiz
  • What’s the complex conjugate of:
complex numbers a summary
Complex Numbers – A Summary

What is the square root of 1… 1 or -1?

  • What is “i”?
  • What is “-i”?
  • The complex plane
  • Complex conjugate
    • Graphically, complex conjugate = ?
  • Polar vs. rectangular coordinates
    • Angle notation, “Aq”
  • Euler’s equation…proof that eiq = cosq + isinq
    •  must be in radians
    • Where is 10ei(p/6) located on complex plane?
complex numbers cont
Complex Numbers, cont.
  • Adding
    • …on complex plane, graphically?
  • Multiplying
    • …on complex plane, graphically?
    • How many solutions are there to x2=1?
    • What are the solutions to x5=1? (xxxxx=1)
  • Subtracting and dividing
    • …on complex plane, graphically?
polar rectangular conversion
Polar/rectangular conversion
  • Warning about rectangular-to-polar conversion: tan-1(-1/2) = ?
    • Do you mean to find the angle for (2,-1) or (-2,1)?

Always draw a picture!!

using complex numbers to add sines cosines
Using complex numbers to add sines/cosines
  • Fact: when you add two sines or cosines having the same frequency (with possibly different amplitudes and phases), you get a sine wave with the same frequency! (but a still-different amplitude and phase)
    • “Proof” with Mathematica… (class make up numbers)
  • Worked problem: how do you find mathematically what the amplitude and phase are?
  • Another worked problem?
using complex numbers to solve equations
Using complex numbers to solve equations
  • Simple Harmonic Oscillator

(ex.: Newton 2nd Law for mass on spring)

  • Guess a solution like

what it means, really:

(and take Re{ … } of each side)

complex numbers traveling waves
Complex numbers & traveling waves
  • Traveling wave: A cos(kx – wt + f)
  • Write as:
  • Often:
  • …or
      • where “A-tilde” = a complex number, the phase of which represents the phase of the wave
      • often the tilde is even left off