Consider the three Frenet-Serret formulas
dT
N
ds
dN
T + B
ds
dB
N
ds
By definition of unit tangent vector,
r
T
r
r r T
r sT
s is arc length parameter.
Here
Differentiate r sT with respect to t both sides,
r sT + sT
The Frenet-Serret formula
dT
N can
a)
First method:
By definition of curvature
dT
dT
k so
will have the same magnitude as kN (because
ds
ds
N is a unit normal vector).
Recall that the unit normal vector is defined in parameter of time as
N t
T t
T t
Now if parameter is arc length s , t
a)
Let the orbital period of the planet isT , then from Keplers second law
dA 1
c area
dt 2
swept in one rotation is given by
T
1
A c dt
2 0
1
cT
2
However since we know from Keplers first law, the orbit of each planet is an ellipse and
area of ellipse i
a)
The key to Keplers second law is the fact that the following cross product is a constant
vector even though both r t and r t is changing with time:
c r t r t
The vector c is constant that is
dc
0
dt
By the product rule for cross products
dc d r t r t
If r t is position vector of particle with respect to time t , then r t represents velocity
of the particle.
Similarly if r s is the position vector of the particle with respect to arc length then r s
represents unit tangential vector T .
Now r t aT T aN
Consider the position of particle at certain instant of time t0 , when after releasing it hits
the target located at a, b
B a, b
r2
r1
r
A
O
Let t0 be the time of release, and then position vector r t R cos ti R sin tj at that time
will given by
r t0 R
a)
Since B 1 is a constant, the condition on B implies that
B B B
2
1
Differentiating both sides the equation B B 1 , with respect to s , we obtain,
d
B B
ds
dB dB
B +
B
ds ds
dB
2
B
ds
dB
B
ds
d
1
ds
Product rule
0
0
dB dB
B = ds
ds
B
0
Since the do
a)
The given vector function is r t
1 2
t 4t 8 i 1 3t 2 6t 4 j
5
5
0t 3
The parametric equations for the curve are
x
1 2
t 4t 8
5
y
1 2
3t 6t 4
5
By choosing the arbitrary real values of t , ordered pairs x, y can be created, as shown in
the table be
Using 8 bits to store a normalized floating point number with the leftmost bit as the sign
bit, the next 3 bits as the exponent (expressed in excess notation), followed by a 4-bit
mantissa, the pattern 11101110 represents
A.1.5
B.3.625
C.7.9375
D.3.25
E.3
Week 5 Lab
Name_
Part I: Separable Differential Equations
1. What is meant by separable? What should you look for in a differential equation to decide if it is
separable or not?
The equation
form
dy
f x, y
dx
f x dx g y dy
is said to be variables separa