Announcements 10/6/10

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# Announcements 10/6/10 - PowerPoint PPT Presentation

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
• 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
• 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 = ?
• What’s the complex conjugate of:
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.
• …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
• 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
• 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
• 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
• 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