MAT 241- Calculus 3- Prof.
Santilli
Toughloves Chapter 13
2
1.) Vector-value function: r (t) x(t)i y(t) j : plane curve in
3
r (t) x(t)i y(t) j z(t)k : Space curve in
Domain: Intersection of the vector function components domains.
2.) Tangent vector to the curve is T (t) r (t) x (t)i y (t) j z (t)k
T (t) r (t)
3.) Unit tangent vector to the curve is T (t)
T (t) r (t)
4.) r (t) x(t)i y(t) j z(t)k is smooth if x (t), y (t), z(t) are all continuous and r (t) 0 .
5.) Piecewise smooth = curve that is made up of a finite number of smooth pieces.
d u (t) v (t) du (t) dv (t)
6.) u (t) v (t)
dt dt dt
d cu (t) du (t)
7.) c c u (t)
dt dt
8.) Product rule (scalar funct)(vector)
d f (t)u (t) df (t) du (t)
u (t) f (t) f (t)u (t) f (t)u (t)
dt dt dt
9.) Product rule (vect)(vect)
d u (t) v (t) du (t) dv (t)
v (t) u (t) u (t) v (t) u (t) v (t)
dt dt dt
10.) Product rule (vect)x(vect) ORDER IS IMPORTANT!
d u (t) v (t) du (t) dv (t)
v (t) u (t) u (t) v (t) u (t) v (t)
dt dt dt
11.) Chain rule
d u ( f (t)) du ( f (t)) d f (t)
u ( f ) f (t)
dt df dt
12.) IMPORTANT!
If r (t) r (t) constant, then r (t) r (t) 0 ,
which means r (t) constant, then r (t)r (t) or r (t)T (t) .
NOTE:
For any UNIT VECTOR, r(t) r(t) r(t) 1 constant, and thus r(t) r (t) 0
which means r(t)r(t) . Any unit vector is perpendicular to its derivative vector.
13.) For r (t) x(t)i y(t) j z(t)k ,
r (t)dt x(t)dt i y(t)dt j z(t)dt k and
b b b b
r (t)dt x(t)dt i y(t)dt j z(t)dt k
a a a a
14.) Arc length function:
t t
ds(t)
s(t)
r (t) dt
T (t)dt , therefore
dt
r (t) T (t) or ds r (t)dt T (t) dt
a a
15.) r ( s) T ( s) 1 , therefore T ( s) T ( s)
16.) Differential notation: For r (t) x(t)i y(t) j z(t)k ,
dr dx i dy j dz k and dr r (t)dt T (t)dt , therefore ds dr
17.) Curvature- measures how sharply a curve bends
dT T (t) r (t) r (t)
a N f (x)
x y y x
T (s) r (s)
r (t) 3 v2
2 32
ds r (t) 32
1 f (x) x 2 y 2
1
18.) Curvature of a circle of radius a:
a
19.) Curvature of a line: 0
20.) TNB Frame:
T (t) r (t)
T (t) Unit tangent vector
T (t) r (t)
T (t)
N (t) Unit normal vector
T (t)
B(t) T(t) N (t) Unit binormal vector (order is important)
21.) Normal Plane contains N (t) and B(t)
22.) Osculating Plane contains T (t) and N (t) and osculating circle (circle of curvature at the
point of tangency on the curve)
23.) Rectifying plane contains T (t ) and B (t )
dr
24.) Velocity in space: v (t) r (t) r(t)
dt
dr ds
25.) Speed in space: v(t) v (t) r (t) r(t)
dt dt
2
dv d r
26.) Acceleration in space: a (t) v (t) v(t) 2 r (t)
r
(t)
dt dt
27.) Tangent and normal components of acceleration:
2
dv 2 d s ds 2 r r r r 1
a (t) T v N 2 T N T N v a, v a
dt dt dt r r v
Or
a (t) a T T a N N
where
dv d 2 s r r v a
aT a v
dt dt 2 r v
ds
2
r r v a
a N v
2
a v
dt r v
And
a T a N
2 2
a (t)
