Math 1321 (Qinghai Zhang)
Summary for §10.1 - §10.5
Theorem 10. For u, v : R → R3 , c ∈ R, f : R → R,
Definition 1. A universal statement is of the form
U = (∀x ∈ S, A(x) holds).
d
[u(t) + v(t)] = u0 (t) + v0 (t),
dt
d
[cu(t)] = cu0 (t),
dt
d
[f (t)u(t)] = f 0 (t)u(t) + f (t)u0 (t),
dt
d
[u(t) · v(t)] = u0 (t) · v(t) + u(t) · v0 (t),
dt
d
[u(t) × v(t)] = u0 (t) × v(t) + u(t) × v0 (t),
dt
d
[u(f (t))] = f 0 (t)u0 (f ).
dt
(1)
An existential statement is of the form
E = (∃x ∈ S, s.t. A(x) holds).
(2)
A statement of implication/conditional has the form
A ⇒ B.
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(3)
Axiom 2 (First-order negation of logical statements).
The negations of the statements in Definition 1 are
¬U = (∃x ∈ S, s.t. A(x) does not hold),
(4)
¬E = (∀x ∈ S, A(x) does not hold).
(5)
(10a)
(10b)
(10c)
(10d)
(10e)
(10f)
Definition 11. The unit tangent vector to a curve r(t)
at a point P (t) = O + r(t) is
Definition 3. A function f from X to Y , written as
f : X → Y or X 7→ Y , is a subset of the cartesian product X × Y satisfying that ∀x ∈ X, there is exactly one
y ∈ Y s.t. (x, y) ∈ X × Y .
T(t) =
r0 (t)
.
|r0 (t)|
(11)
The tangent line to r at P (t0 ) is the line determined by
Definition 4. A vector function is a function whose P (t ) and T: {P | P = P (t ) + tT, t ∈ R}.
0
0
range is a set of vectors in Rn . It is written as Rm 7→ Rn
m
n
+
or f : R → R (m, n ∈ N ).
Theorem 12. If |r(t)| = c where c is a constant, then
r(t) · r0 (t) = 0. Consequently r(t) · T(t) = 0.
Definition 5. A curve is (the image of) a vector function R 7→ R3 , or r(t) : R → R3 . The independent vaiable Definition 13. The arc length of a curve r : R → R3
t is its parametrization.
starting from P (a) = O + r(a) is a function R 7→ R
Definition 6. A surface is (the image of) a vector function R2 7→ R3 .
Z
s(t) =
t
|r0 (u)|du.
(12)
a
Definition 7 (Limit of a scalar function). Consider a
ds
function f : I → R with I(a, r) = (a − r, a) ∪ (a, a + r).
= |r0 (t)|.
Formula 14.
The limit of f (x) exists as x approaches a, written as
dt
limx→a f (x) = L, iff the following holds:
Definition 15. The curvature of a curve r(t) at the
point
P (t) = O + r(t) is
∀ > 0, ∃δ, s.t. ∀x ∈ I(a, δ), |f (x) − L| < . (6)
dT Definition 8 (Limit at infinity). Consider a function
.
κ(t) = (13)
ds f : (a, ∞) → R. limx→∞ f (x) = L iff
∀ > 0, ∃M ∈ (a, ∞), s.t. ∀x > M, |f (x) − L| < . (7) Formula 16.
|T0 (t)|
.
|r0 (t)|
(14)
Definition 9. The limit of a vector function r : R → R ,
r(t) = hf (t), g(t), h(t)i is
Theorem 17. The curvature of a curve r(t) at P (t) is
D
E
lim r(t) = lim f (t), lim g(t), lim h(t) .
(8)
t→a
t→a
t→a
t→a
|r0 (t) × r00 (t)|
κ(t) =
(15)
|r0 (t)|3
The derivative and integral of a vector function are also
defined component-wise, e.g.,
Corollary 18. The curvature of a 2D curve y = f (x) is
r(t
+
∆t)
−
r(t)
= hf 0 (t), g 0 (t), h0 (t)i .
r0 (t) = lim
|f 00 (x)|
∆t→0
∆t
κ(x) =
.
(16)
3/2
(9)
[1 + (f 0 (x))2 ]
κ(t) =
3
1
Summary for §10.1 - §10.5
Math 1321 (Qinghai Zhang)
Definition 19. The principal unit normal vector is
Theorem 21. The acceleration of a particle following
the curve r(t) consists of two parts
0
N(t) =
T (t)
,
|T0 (t)|
(17)
a = aT T + aN N,
and the binormal vector is
B(t) = T(t) × N(t).
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(19)
(18)
where aT is caused by the change of the speed, and aN
The normal plane of the curve at P = O + r(t) is the is caused by the change of the velocity direction:
plane determined by N(t) and B(t). The osculating
plane is that by T(t) and N(t).
r0 · r00
,
(20)
aT = v 0 = 0
Definition 20. Let t represent time and r(t) the trajec|r (t)|
tory of a moving particle. Then r0 (t) = v is called the
|r0 × r00 |
velocity of the particle, |r0 (t)| = |v| = v the speed of the
aN = κv 2 =
.
(21)
|r0 (t)|
particle, r00 (t) = a the acceleration of the particle.
2