Math 1321-004 (Qinghai Zhang)
Memorization sheet for Exam 2
Definition 1. The Cartesian product X × Y between
two sets X and Y is the set of all possible ordered pairs
with first element from X and second element from Y :
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Definition 9 (Dot product: algebraic definition). The
dot product of two vectors a, b ∈ R3 is a real number:
a · b = a1 b1 + a2 b2 + a3 b3 .
X × Y = {(x, y) | x ∈ X, y ∈ Y }.
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(1)
Definition 10. The angle θ between two nonzero vecDefinition 2. A function f from X to Y , written as tors a, b ∈ R3 satisfies
f : X → Y or X 7→ Y , is a subset of the Cartesian proda·b
uct X × Y satisfying that ∀x ∈ X, there is exactly one
,
θ ∈ [0, π].
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cos θ =
|a||b|
y ∈ Y s.t. (x, y) ∈ X × Y . X and Y are the domain and
range of f , respectively.
Theorem 11. The algebraic definition of the dot prodDefinition 3 (Limit of a scalar function). Consider a uct is equivalent to its geometric definition:
function f : I → R with I(a, r) = (a − r, a) ∪ (a, a + r).
a · b = |a||b| cos θ.
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The limit of f (x) exists as x approaches a, written as
(2) Definition 12. The scalar projection of b onto a is
lim f (x) = L,
x→a
iff the following holds:
compa b =
∀ > 0, ∃δ, s.t. ∀x ∈ I(a, δ), |f (x) − L| < .
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a·b
,
|a|
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and the vector projection of b onto a is
Definition 4. A vector function is a function whose
range is a set of vectors in Rn . It is written as Rm 7→ Rn
or f : Rm → Rn (m, n ∈ N+ ).
proja b = ca = (compa b)
a
.
|a|
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Definition 5. The limit of a vector function r : R → R3 , Definition 13. A plane is a set of points uniquely determined by a point P0 and a normal vector n:
r(t) = hf (t), g(t), h(t)i is
D
E
{P | n · (P − P0 ) = 0}.
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lim r(t) = lim f (t), lim g(t), lim h(t) .
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t→a
t→a
t→a
t→a
Equivalently, the scalar equation of a plane is
The derivative and integral of a vector function are also
defined component-wise, e.g.,
ax + by + cz + d = 0.
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r(t + ∆t) − r(t)
Definition 14. The standard basis vectors in R3 are
= hf 0 (t), g 0 (t), h0 (t)i .
∆t→0
∆t
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i = h1, 0, 0i ,
j = h0, 1, 0i ,
k = h0, 0, 1i . (16)
r0 (t) = lim
Definition 6. Given two points A = (a1 , a2 , a3 ) and
B = (b1 , b2 , b3 ), v = B − A is the vector that starts at A
and ends at B. The length of vector v equals the distance
between A and B:
p
|v| = |AB| = (a1 − b1 )2 + (a2 − b2 )2 + (a3 − b3 )2 .
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In particular, a = ha1 , a2 , a3 i can be regarded as a vector
that starts at the origin O = (0, 0, 0) and ends at A.
q
a = ha1 , a2 , a3 i ⇒ |a| = a21 + a22 + a23 .
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Definition 7. v is a unit vector iff |v| = 1.
The unit vector in the same direction of v is
Definition 15 (Geometric definition of cross product).
The cross product of two vectors a, b ∈ R3 is
a × b = (|a||b| sin θ)n,
where θ is the angle between a and b, and n is the unit
vector determined by the right-hand rule from a and b.
Definition 16 (Algebraic
i
j k
a × b = det a1 a2 a3
b1 b2 b3
a a3 i − det = det 2
b2 b3
v
|v| .
Definition 8. A line is a set of points uniquely determined by a point P0 and a direction vector v:
{P | P (t) = P0 + tv, t ∈ (−∞, +∞)}.
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definition of cross product).
(18a)
a1 a2 a1 a3 j
+
det
b1 b2 k
b1 b3 (18b)
= (a2 b3 − a3 b2 )i − (a1 b3 − a3 b1 )j + (a1 b2 − a2 b1 )k
(18c)
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Math 1321-004 (Qinghai Zhang)
Memorization sheet for Exam 2
Theorem 17. The algebraic and geometric definitions
of cross product are equivalent.
Formula 29.
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ds
= |r0 (t)|.
dt
Definition 18. Two nonzero vectors a, b are perpendic- Definition 30. The curvature of a curve r(t) at the
point P (t) = O + r(t) is
ular or orthogonal, written as a ⊥ b, iff a · b = 0.
Definition 19. Two nonzero vectors a, b ∈ Rn are par dT .
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κ(t) = allel, written as a k b, iff ∃c 6= 0, s.t. a = cb.
ds 3
Theorem 20. a, b ∈ R are parallel iff a × b = 0.
Formula 31.
Definition 21 (scalar triple product). For a, b, c ∈ R3 ,
|T0 (t)|
κ(t) = 0
.
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a1 a2 a3 |r (t)|
a · (b × c) = (a × b) · c = det b1 b2 b3 . (19)
Theorem 32. The curvature of a curve r(t) at P (t) is
c1 c2 c3 Theorem 22. For u, v : R → R3 , c ∈ R, f : R → R,
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
κ(t) =
Corollary 33. The curvature of a 2D curve y = f (x) is
(20b)
κ(x) =
(20c)
|f 00 (x)|
[1 + (f 0 (x))2 ]
3/2
.
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(20d) Definition 34. The principal unit normal vector is
(20e)
N(t) =
(20f)
T0 (t)
,
|T0 (t)|
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and the binormal vector is
B(t) = T(t) × N(t).
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The normal plane of the curve at P = O + r(t) is the
plane determined by N(t) and B(t). The osculating
plane is that by T(t) and N(t).
Definition 24. A surface is (the image of) a vector function R2 7→ R3 .
Definition 25. The tangent vector to a curve r(t) at a
point P (t) = O + r(t) is r0 (t);
the corresponding unit tangent vector is
r0 (t)
.
|r0 (t)|
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(20a)
Definition 23. A curve is (the image of) a vector function R 7→ R3 , or r(t) : R → R3 . The independent variable t is its parametrization.
T(t) =
|r0 (t) × r00 (t)|
|r0 (t)|3
Definition 35. Let t represent time and r(t) the trajectory of a moving particle. Then r0 (t) = v is called the
velocity of the particle, |r0 (t)| = |v| = v the speed of the
particle, r00 (t) = a the acceleration of the particle.
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Theorem 36. The acceleration of a particle following
Definition 26. The tangent line to r : R → R3 at the curve r(t) is a vector a(t) consists of two parts:
P (t0 ) = O + r(t0 ) is the line determined by P (t0 ) and T:
a(t) = aT T + aN N,
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{P | P = P (t0 ) + tT, t ∈ R}.
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Theorem 27. If |r(t)| = c where c is a constant, then where aT is caused by the change of the speed, and aN
is caused by the change of the velocity direction:
r(t) · r0 (t) = 0. Consequently r(t) · T(t) = 0.
Definition 28. The arc length of a curve r : R → R3
starting from P (a) = O + r(a) is a function s : R → R,
Z t
s(t) =
|r0 (u)|du.
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r0 · r00
,
|r0 |
|r0 × r00 |
aN (t) = κv 2 =
.
|r0 |
aT (t) = v 0 =
a
2
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