1. HW 1 (1) Do Exercise: 8, 16, 19, 21 (Section 16.2 in Stewart).
(2) Letr:I→R3be a regular parametrized curve (i.e. ris a curve withr0(t)6= 0 for allt∈I.).
The arc length function associated withris the functions: [a, b]→[0, L] defined by s(t) =
Z t
a
kr0(u)kdu.
The inverse function of s : [a, b] → [0, L] is denoted by t : [0, L] → [a, b]. Recall that the arc length parametrization of r is defined to be the parametrized curvec: [0, L]→R3 by c(s) =r(t(s)).The unit tangent vector fieldT(s) atsis given byT(s) =c0(s).
(a) LetC be the circlex2+y2= 4 parametrized by the function r(t) = (2 cost,2 sint), 0≤t≤2π.
Find the arc length function and the unit tangent vector fieldT(s) ofr.
(b) LetF:R2→R2 be the vector field
F(x, y) =xi+yj.
Evaluate Z
C
(F·T)dsand Z
C
F·dr.Are the two equal? HereCis the curve defined in 2(a).
(3) Let a, b >0 andC be the ellipse x2 a2 +y2
b2 = 1.parametrized by the function r(t) = (acost, bsint), 0≤t≤2π.
LetFbe the vector fieldF(x, y) =xi+yjas in 2(b). Evaluate Z
C
(F·T)ds.( Since it is not possible to find the arclength function directly in this example, you must find some other method to compute. Consider problem 2.)
(4) LetC1andC2be two different curves with the same initial points and with the same terminal points. Given a vector fieldF, is it always true that
Z
C1
F·dr= Z
C2
F·dr?
For example, consider the parametrized curves:
C1:r1(t) = (cost,sint), 0≤t≤π C2:r2(t) = (cost,−sint), 0≤t≤π.
Do C1 and C2 have the same initial points and have the same terminal points? Consider the vector field onR2− {(0,0)}
F(x, y) =− y
x2+y2i+ x x2+y2j.
Compute and compare Z
C1
F·drand Z
C2
F·dr.
(5) Due to the Newton’s second law of motion, moving a particle with massmby the forceFin a frictionless space gives us the relation between the accelerationaof the particle, its mass mand given forceFin the following mathematical expression
F=ma.
If the initial positionx0and the initial velocityv0of the particle are given, we can determine the positionr(t) of the particle at timetby solving the differential equation
md2r
dt2 =F, r(0) =x0, r0(0) =v0.
1
2
In class, we have seen that the work done in accelerating the particle by the forceF from timet= 0 to timet=b is given by
W =
Z
C
F·dr.
Here C is the path of the particle fromt= 0 to somet=b.From calculus I, we have seen the formula
d
dt(v(t)·v(t)) = 2v·dv dt.
Use the fundamental Theorem of Calculus and the above formula to show that W =1
2mkvbk2−1
2mkv0k2. Herevb is the velocity of the particle at timet=b.
