I sit here, before you Rome, on a Friday, the 19th of September. I am at home, currently occupied by a large amount of nothing. What’s new, you idly wonder to yourself as you plan for your idle, meaningless date with your latest fling (yes, I’m feeling rather bitter). What’s new? Is anything ever truly new? Let’s see…
My senior year isn’t as bad as it could be. I’ve been doing fairly well in all my classes, including Math 270A over at Cal State. I feel confident about my ability to study Spanish. 2nd Year Calculus is fun, although problems that take up to half an hour of my free time (that even Li didn’t solve right) are definitely not fun.
To give an example:
“Find the vectors T, N, and B = T x N for the vector-valued function r(t) at the specified value of t. The vector B is called the unit binormal vector to the curve.
r(t)=ti+t^2j+t^3/3k, t = 1
Doesn’t look too bad, right?
OK, v(t)=r’(t)=<1,2t,t^2>
||v(t)||=sqrt(1+4t^2+t^4)
a(t)=v’(t)=<1,2,2t>
T = <1,2t,t^2>/sqrt(1+4t^2+t^4)
T(1) = <1,2,1>/sqrt(6)
= sqrt(6) * <1,2,1> / 6
N = T’/||T’||
T’ = (<0,2,2t>*sqrt(1+4t^2+t^4)-<1,2t,t^2>*(1/2)*(1+4t^2+t^4)^(-1/2)*(4t^3+8t))/(1+4t^2+t^4)
= (<0,2,2t>*(1+4t^2+t^4)-2<1,2t,t^2>(t^3+2t))/(1+4t^2+t^4)^(3/2)
= (<-2t^3-4t, 2+8t^2+2t^4-4t^4-8t^2, 2t+8t^3+2t^5-2t^5-4t^3>)/(1+4t^2+t^4)^(3/2)
= (<-2t^3-4t, 2-2t^4, 2t+4t^3>)/(1+4t^2+t^4)^(3/2)
||T’|| = sqrt((-2t^3-4t)^2+(2-2t^4)^2+(2t+4t^3)^2)/(1+4t^2+t^4)^(3/2)
N = (<-2t^3-4t, 2-2t^4, 2t+4t^3>)/(1+4t^2+t^4)^(3/2) / (sqrt((-2t^3-4t)^2+(2-2t^4)^2+(2t+4t^3)^2)/(1+4t^2+t^4)^(3/2))
= (<-2t^3-4t, 2-2t^4, 2t+4t^3>) / sqrt((-2t^3-4t)^2+(2-2t^4)^2+(2t+4t^3)^2)
N(1) = <-6, 0, 6> / sqrt(72)
= sqrt(72) * <-6, 0, 6> / 72
= 2 * sqrt(18) * <-6, 0, 6> / 72
= 6 * sqrt(2) * <-6, 0, 6> / 72
(at least I hope so)
Therefore, T x N / ((sqrt(6)/6) * (6*sqrt(2)/72)) =
| i j k |
| 1 2 1 |
| -6 0 6 |
12i-6j+12k-6j
T x N = ((sqrt(6)/6) * (6*sqrt(2)/72)) * <12,-12,12>
= sqrt(12)/72 * <12,-12,12>
= sqrt(3)/36 * <12,-12,12>
= sqrt(3)/3 * <1,-1,1>
No idea if that’s right, it’s quite a bit different from the answer I got last night. At least I proved that LiveJournal is no good for mathematics :).
So yeah, I’m bored. Talk to me, make me feel good, etc. That is, if you have any sympathy, you pigs ;).