Qinghai Zhang
1
Summary of Math 1321-004
2013-MAR-18
Fundamentals
Definition 7. 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. X and Y are the domain and
range of f , respectively.
Nomenclature
• R, N: the sets of real and natural numbers.
• calligraphic uppercase letters S, U, P: sets,
• sans serif uppercase letters U, E: statements,
Definition 8. A function f : X → Y is said to be injective or one-to-one iff
• uppercase letters A, B, P : points,
• lowercase boldfaced letters a, v: vectors,
∀x1 ∈ X , ∀x2 ∈ X , x1 6= x2 ⇒ f (x1 ) 6= f (x2 ).
• lowercase letters x, y, c, d: scalars, or set elements,
It is surjective or onto iff
• lowercase letters m, n: natural numbers
Definition 1. A set S is a collection of distinct objects
x’s, often denoted with the following notation
S = {x | the conditions that x satisfies. }.
(9)
∀y ∈ Y, ∃x ∈ X , s.t. y = f (x).
(10)
It is bijective iff it is both injective and surjective.
(1)
Definition 2. A universal statement is a logic statement Definition 9. A set S is countable iff there exists an
injective function f : S → N that maps S to N.
of the form
U = (∀x ∈ S, A(x) holds).
An existential statement has the form
E = (∃x ∈ S, s.t. A(x) holds).
(2) Definition 10. A 3D coordinate system is a bijective
function whose domain and range are R3 and P, where
R3 = R × R × R and P is the set of all spatial locations.
(3)
Definition 11. The distance between two points
A = (a1 , a2 , a3 ) and B = (b1 , b2 , b3 ) is
A statement of implication/conditional has the form
A ⇒ B.
(4)
|AB| =
p
(a1 − b1 )2 + (a2 − b2 )2 + (a3 − b3 )2 .
(11)
Axiom 3 (First-order negation of logical statements).
The negations of the statements in Definition 2 are
¬U = (∃x ∈ S, s.t. A(x) does not hold),
¬E = (∀x ∈ S, A(x) does not hold).
Definition 12. Given two points A = (a1 , a2 , a3 ) and
B = (b1 , b2 , b3 ), the vector v = B − A is the displace(5) ment of B from A. The length of vector v equals the
(6) distance between A and B:
Definition 4. S is a subset of U, written as S ⊆ U,
iff x ∈ S ⇒ x ∈ U.
|v| = |AB|.
In particular, a = ha1 , a2 , a3 i can be regarded as a position vector that starts at the origin O = (0, 0, 0) and
ends at A.
Definition 5. 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:
X × Y = {(x, y) | x ∈ X , y ∈ Y}.
(12)
(7)
a = ha1 , a2 , a3 i ⇒ |a| =
q
a21 + a22 + a23 .
(13)
Axiom 6 (Fundamental principle of counting). A task
consists of a sequence of k steps. Let ni denote the num- Axiom 13 (Vector addition and scaling). If c ∈ R,
ber of different choices for the i-th step, the total number u = hxu , yu , zu i, v = hxv , yv , zv i, then
of distinct ways to complete the task is then
u + v = hxu + xv , yu + yv , zu + zv i ,
(14a)
k
Y
cv = hcvx , cvy , cvz i .
(14b)
ni = n1 n2 · · · nk .
(8)
i=1
1
Qinghai Zhang
Summary of Math 1321-004
2013-MAR-18
Axiom 14 (Vector algebra). If c ∈ R and u, v, w ∈ Rn , Definition 24. A set is an open set if it contains none
then
of its boundary points.
u + v = v + u,
u + (v + w) = (v + u) + w,
(15a) Definition 25. A set is a closed set if it contains all of
(15b) its boundary points.
Definition 26. A point set U ⊆ Rn is bounded iff
n
u + (−u) = 0
(15d) U ⊆ B(P0 , r) for some P0 ∈ R and r > 0.
c(u + v) = cv + cu,
(15e) Definition 27 (Limit of a scalar function with multiple
variables). The limit of a function f : B0 (P0 , r) → R ex(c + d)u = cu + du
(15f)
ists as P approaches P0 , written as limP →P0 f (P ) = L,
(cd)u = c(du),
(15g) iff
1u = u.
(15h)
∀ > 0, ∀ paths P → P0 , ∃δ > 0, s.t.
(22)
Definition 15. A scalar function is a function whose
∀P ∈ B(P0 , δ),
|f (P ) − L| < .
range is a subset of R.
Formula 28. The limit of f : B0 (P0 , r) → R does not
Definition 16. A vector function is a function whose exist at P0 if there exists two different paths P → P0 and
range is a subset of Rn with n > 1.
P
P0 s.t.
Definition 17. A vector field is a vector function
(23)
lim f (P ) = L1 6= L2 = lim f (P ) .
m
m
P P0
P →P0
F:R →R .
u+0=u
(15c)
Definition 18 (Limit of a scalar function with one vari- Theorem 29 (The squeeze theorem). If ∃r > 0 s.t.
able). Consider a function f : I → R with I(a, r) = ∀P ∈ B0 (P0 , r), f (P ) ≤ g(P ) ≤ h(P ), then
(a − r, a) ∪ (a, a + r). The limit of f (x) exists as x aplim f (P ) = lim h(P ) = L ⇒ lim g(P ) = L.
proaches a, written as limx→a f (x) = L, iff
P →P0
∀ > 0, ∃δ, s.t. ∀x ∈ I(a, δ), |f (x) − L| < .
P →P0
P →P0
(16)
(24)
Definition 19 (Limit of a scalar function with one vari- Definition 30. f : R → R is continuous at a iff
able at infinity). Consider a function f : (a, ∞) → R.
lim f (x) = f (a).
(25)
x→a
limx→∞ f (x) = L iff
Definition 31. f : Rn → R is continuous at Q iff
∀ > 0, ∃M ∈ (a, ∞), s.t. ∀x > M, |f (x) − L| < .
(17)
lim f (P ) = f (Q).
(26)
P →Q
Definition 20 (Limit of a vector function with one variable). If r : R → Rn , r(t) = hr1 (t), r2 (t), . . . , rn (t)i , f is continuous on a point set U if (26) holds ∀Q ∈ U.
then
Definition 32. The derivative of a function f : R → R
D
E
lim r(t) = lim r1 (t), lim r2 (t), · · · , lim rn (t) . (18) at a is the limit
t→a
t→a
t→a
t→a
f (a + h) − f (a)
f 0 (a) = lim
.
(27)
Definition 21. The open ball centered at P0 ∈ Rn with
h→0
h
radius r > 0 is the point set
If the limit exists, f is differentiable at a.
B(P0 , r) = P |P − P0 | < r .
(19)
Definition 33. A function f (x) is C k or k times conIt is an open interval in 1D and an open disk in 2D.
tinuously differentiable if f (k) (x) exists and is itself conThe open ball without the center is denoted by
tinuous.
B0 (P0 , r) = B(P0 , r) \ {P0 }.
(20) Theorem 34. A polynomial is continuous everywhere.
Definition 22. f : Rn → R has a local maximum at Definition 35. A sequence is simply a countable set
P0 ∈ Rn iff
{an | n ∈ N+ } or a function defined on N+ .
∃r > 0, s.t. ∀P ∈ B(P0 , r),
f (P ) ≤ f (P0 ).
Changing ≤ to ≥ in (21) yields a local minimum.
An extremum is either a maximum or minimum.
(21) Definition 36 (Limit of a sequence). A sequence
{an } has the limit L, written as limn→∞ an = L, or
an → L as n → ∞, iff
∀ > 0, ∃N, s.t. ∀n > N, |an − L| < .
Definition 23. P0 is a boundary point of a point set U
iff ∀r > 0, ∃P ∈ B(P0 , r) s.t. P 6∈ U.
(28)
If such a limit L exists, we say that {an } converges to L.
2
Qinghai Zhang
2
Summary of Math 1321-004
2013-MAR-18
Infinite sequences and Series
Definition 37. A series is the sum of all terms in a
sequence {an }.
P∞
n−1
Theorem 38. The geometric series
with
n=1 ar
a 6= 0 is convergentPif |r| < 1 and divergent otherwise.
∞
a
.
In the former case, n=1 arn−1 = 1−r
P∞ 1
Theorem 39. The p-series
n=1 np is convergent if
p > 1 and divergent if p ≤ 1.
P∞
Theorem 40.
n=1 an converges ⇒ limn→∞ an = 0.
Theorem 50 (Remainder estimate for integral test).
Let f be a function that satisfies the conditions in Theorem 42, then ∀n > M ,
Z ∞
Z ∞
f (x)dx.
(31)
f (x)dx ≤ Rn ≤
n
n+1
Theorem 51 (Alternating series estimation). A convergent alternating series defined in Theorem 45 satisfies
|Rn | ≤ bn+1 .
Definition 52. A power series centered at a is a series
of the form
∞
Theorem 41 (Test for divergence).
X
P∞If limn→∞ an does
p(x)
=
cn (x − a)n ,
(32)
not exist or limn→∞ an 6= 0, then n=1 an diverges.
n=0
Theorem 42 (Integral test). Let f : R → R be a con- where cn ’s are the coefficients. The interval of convertinuous bounded function satisfying f (n) = an .
gence is the set of x values for which the series converges:
0
If ∃MP∈ N+ , s.t. ∀x ∈ [M, +∞),
f
(x)
>
0
and
f
(x)
<
0,
R∞
∞
Ic (p) = {x | p(x) converges}.
(33)
then n=1 an converges iff M f (x)dx is convergent.
Theorem 53 (Interval of convergence of power series).
Theorem
43P(Comparison test). Consider two series
P∞
There are only three possibilities for Ic (p) of the power
∞
+
n=1 an and
n=1 bn satisfying ∀n ∈ N , 0 < an ≤ bn .
series (32): (i) Ic = {a}, (ii) Ic = (−∞, +∞),
P∞
P∞
(iii) Ic = (a − R, a + R) or [a − R, a + R) or (a − R, a + R]
• If n=1 bn converges, then n=1 an converges.
or [a − R, a + R].
P∞
P∞
• If n=1 an diverges, then n=1 bn diverges.
R is called the radius of convergence. R = 0, +∞ for
cases (i) & (ii), respectively.
Theorem 44P(Limit comparison test). If two series
P
∞
∞
+
Theorem 54 (Term-by-term
differentiation and integraP∞
n=1 an and
n=1 bn satisfy ∀n ∈ N , an > 0, bn > 0,
tion).
If
f
(x)
=
c
(x
−
a)n has radius of convern
and
n=0
an
= c ∈ (0, ∞),
(29) gence R > 0, then f (x) is differentiable on (a − R, a + R).
lim
n→∞ bn
∞
X
then either both series converge or both diverge.
f 0 (x) =
ncn (x − a)n−1 ,
(34a)
n=0
Theorem 45 (AlternatingP
series test). An alternating
∞
series is a series of the form n=1 (−1)n−1 bn with bn > 0
+
∀n ∈ N . If it satisfies
(i) ∀n ≥ 1, bn+1 ≤ bn , and (ii)
P∞
limn→∞ bn = 0, then n=1 (−1)n−1 bn converges.
P∞
Definition 46. A series
P∞ n=1 an is called absolutely
convergent if the series n=1 |an | converges.
P∞
Theorem 47P(Absolute convergence). If n=1 |an | con∞
verges, then n=1 an converges.
P∞
Theorem 48 (Ratio test). Consider a series Pn=1 an
∞
an+1
satisfying limn→∞ | an | = L. If L < 1, then n=1 an
P∞
converges; if L > 1, then n=1 an diverges; if L = 1, no
conclusion can be drawn.
P∞
Definition 49. The nth remainder of the series i=1 ai
is defined as
Rn =
∞
X
i=1
ai −
n
X
i=1
ai =
∞
X
ai .
Z
f (x)dx = C +
∞
X
cn
n=0
(x − a)n+1
.
n+1
(34b)
Furthermore,
the radii of convergence of both f 0 (x) and
R
f (x)dx are R.
Definition 55. If f (n) (x) exists for a function f : R → R
at x = a, then
Tn (x) =
n
X
f (k) (a)
k=0
k!
(x − a)k
(35)
is called the nth Taylor polynomial for f (x) at a.
In particular, the linear approximation for f (x) at a is
T1 (x) = f (a) + f 0 (a)(x − a).
(36)
Definition 56. The Taylor series (or Taylor expansion)
for f (x) at a is
(30)
lim Tn (x) =
n→∞
i=n+1
3
∞
X
f (k) (a)
k=0
k!
(x − a)k .
(37)
Qinghai Zhang
Summary of Math 1321-004
2013-MAR-18
Definition 57. The remainder of the nth Taylor poly- Definition 62. ∀k, n ∈ N, the binomial coefficients are
nomial in approximating f (x) is
( k!
if k ≥ n,
k
Rn (x) = f (x) − Tn (x).
(38)
(42)
= n!(k−n)!
n
0
if k < n.
Theorem 58. Let Tn be the nth Taylor polynomial for
f (x) at a.
∀k ∈ R, ∀n ∈ N, the binomial coefficients are
lim Rn (x) = 0 ⇔ f (x) = lim Tn (x).
(39)
( Qn−1 (k−i)
n→∞
n→∞
i=0
k
if n > 0,
n!
=
(43)
(m)
Lemma 59. ∀m = 0, 1, 2, . . . , n, Rn (a) = 0.
n
1
if n = 0.
Theorem 60 (Taylor’s theorem with Lagrangian form).
Consider a function f : R → R. If f (n+1) (x) exists on Theorem 63 (Binomial series). ∀k ∈ R, |x| < 1,
the interval I = (a − d, a + d), and f (n) (x) is continuous
∞ ∞ Qn−1
X
X
k n
i=0 (k − i) n
on [a − d, a + d], then ∀x ∈ I, ∃y ∈ I s.t.
(1 + x)k =
x (44)
x =1+
n!
n
(n+1)
n=0
n=1
f
(y)
∞
(x − a)n+1 .
(40)
Rn (x) =
X
k(k − 1) · · · (k − n + 1) n
(n + 1)!
x
=1+
n!
Theorem 61 (Taylor’s inequality). If ∃M < ∞, s.t.
n=1
∀x ∈ [a−d, a+d], f (n+1) (x) ≤ M , then ∀x ∈ (a−d, a+d),
|Rn (x)| ≤
M
|x − a|n+1 .
(n + 1)!
(41)
If ∀n ∈ N, (41) holds, then limn→∞ Tn = f (x).
4
Qinghai Zhang
3
Summary of Math 1321-004
2013-MAR-18
Vectors and Curves
Definition 73. The standard basis vectors in R3 are

 e1 = i = h1, 0, 0i ,
is a unit vector in the same direction
e2 = j = h0, 1, 0i ,
(54)

e3 = k = h0, 0, 1i .
Definition 64. v is a unit vector iff |v| = 1.
Formula 65.
of v.
v
|v|
Definition 66. A line is a set of points uniquely determined by a point P0 and a direction vector v:
{P | P (t) = P0 + tv, t ∈ (−∞, +∞)}.
(45)
a · b = |a||b| cos θ.
(48)
Definition 74 (Geometric definition of cross product).
The cross product of two vectors a, b ∈ R3 is
a × b = (|a||b| sin θ)n,
(55)
Definition 67 (Dot product: algebraic definition). The
where θ is the angle between a and b, and n is the unit
dot product of two vectors a, b ∈ R3 is a real number:
vector determined by the right-hand rule from a and b.
a · b = a 1 b1 + a 2 b2 + a 3 b3 .
(46)
Definition 75 (Algebraic definition of cross product).
Definition 68. The angle θ between two nonzero vec
tors a, b ∈ R3 satisfies
i
j k a × b = det a1 a2 a3 (56a)
a·b
b1 b2 b3 cos θ =
,
θ ∈ [0, π].
(47)
|a||b|
a2 a3 a1 a3 a1 a2 Theorem 69. The algebraic definition of the dot prod- = det b2 b3 i − det b1 b3 j + det b1 b2 k
uct is equivalent to its geometric definition:
(56b)
= (a2 b3 − a3 b2 )i − (a1 b3 − a3 b1 )j + (a1 b2 − a2 b1 )k
(56c)
Theorem 70 (Algebra of dot product). If u, v, w ∈ R3 ,
Theorem 76. The algebraic and geometric definitions
and c ∈ R, then
of cross product are equivalent.
u · u = |u|2
(49a)
u·v =v·u
(49b) Theorem 77 (Algebra of cross product). If c ∈ R and
3
u · (v + w) = u · v + u · w
(49c) u, v, w ∈ R , then
(cu) · v = c(u · v) = u · (cv)
0·u=0
(49d)
u × v = −v × u
(49e)
Definition 71. The scalar projection of b onto a is
compa b =
a·b
,
|a|
(57b)
u × (v + w) = u × v + u × w
(57c)
(u + v) × w = u × w + v × w
(57d)
(50)
Definition 78. Two nonzero vectors a, b are perpendicular or orthogonal, written as a ⊥ b, iff a · b = 0.
and the vector projection of b onto a is
proja b = ca = (compa b)
(57a)
(cu) × v = c(u × v) = u × (cv)
a
.
|a|
n
(51) Definition 79. Two nonzero vectors a, b ∈ R are parallel, written as a k b, iff ∃c 6= 0, s.t. a = cb.
Definition 72. A plane is a set of points uniquely de- Theorem 80. a, b ∈ R3 are parallel iff a × b = 0.
termined by a point P0 and a normal vector n = ha, b, ci:
{P | n · (P − P0 ) = 0}.
Definition 81 (scalar triple product). For a, b, c ∈ R3 ,
(52)
a1
a · (b × c) = (a × b) · c = det b1
c1
Equivalently, the scalar equation of a plane is
ax + by + cz + d = 0.
(53)
5
a2
b2
c2
a3
b3
c3
.
(58)
Qinghai Zhang
Summary of Math 1321-004
2013-MAR-18
Theorem 82. 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
Definition 90. The curvature of a curve r(t) at the
point P (t) = O + r(t) is
(59a)
dT .
(64)
κ(t) = ds
(59b)
Formula 91.
(59c)
κ(t) =
(59d)
(59e)
κ(t) =
(59f)
|r0 (t) × r00 (t)|
|r0 (t)|3
(66)
Corollary 93. The curvature of a 2D curve y = f (x) is
κ(x) =
Definition 84. A surface is (the image of) a vector function R2 7→ R3 .
|f 00 (x)|
[1 + (f 0 (x))2 ]
3/2
.
(67)
Definition 94. The principal unit normal vector is
N(t) =
Definition 85. The tangent vector to a curve
r(t) = hr1 (t), r2 (t), r3 (t)i at a point P (t) = O + r(t) is
T0 (t)
,
|T0 (t)|
(68)
and the binormal vector is
r(t + ∆t) − r(t)
= hr10 (t), r20 (t), r30 (t)i ,
∆t→0
∆t
(60)
r0 (t) = lim
B(t) = T(t) × N(t).
(69)
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).
the corresponding unit tangent vector is
r0 (t)
.
|r0 (t)|
(65)
Theorem 92. The curvature of a curve r(t) at P (t) is
Definition 83. 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) =
|T0 (t)|
.
|r0 (t)|
(61)
Definition 95. Let t represent time and r(t) the trajectory of a moving particle. Then r0 (t) = v is called the
Definition 86. The tangent line to r : R → R3 at velocity of the particle, |r0 (t)| = |v| = v the speed of the
P (t0 ) = O + r(t0 ) is the line determined by P (t0 ) and T: particle, r00 (t) = a the acceleration of the particle.
{P | P = P (t0 ) + tT, t ∈ R}.
(62) Theorem 96. The acceleration of a particle following
the curve r(t) is a vector a(t) consists of two parts:
Theorem 87. If |r(t)| = c where c is a constant, then
r(t) · r0 (t) = 0. Consequently r(t) · T(t) = 0.
a(t) = aT T + aN N,
(70)
Definition 88. The arc length of a curve r : R → R3
starting from P (a) = O + r(a) is a function s : R → R,
Z
s(t) =
where aT is caused by the change of the speed, and aN
is caused by the change of the velocity direction:
t
|r0 (u)|du.
r0 · r00
,
|r0 |
|r0 × r00 |
aN (t) = κv 2 =
.
|r0 |
(63)
aT (t) = v 0 =
a
Formula 89.
ds
= |r0 (t)|.
dt
6
(71a)
(71b)
Qinghai Zhang
4
Summary of Math 1321-004
2013-MAR-18
Partial Derivatives
Definition 97. The partial derivative of f : Rn → R Definition 102. The total differential of f : Rn → R is
with respect to the ith dimension at P0 is
defined in terms of the differential s dxi ’s:
n
f (P0 + hei ) − f (P0 )
∂f (P0 )
∂f X
∂f
= lim
=
, (72)
df
=
dxi
(79)
∂xi
∂xi P0 h→0
h
∂xi
i=1
where h ∈ R and {ei | i = 1, 2, · · · , n} is the set of stanDefinition 103. The increment of a function z =
dard basis vectors of the Euclidean n-space, i.e. the ith
f (x, y) at a point P0 = (x0 , y0 ) is
component of ei is 1 and all other components are 0.
∆z = f (x, y) − f (x0 , y0 ).
Theorem 98 (Clairaut’s). If f : Rn → R has continuous
second partial derivatives at P , then ∀i, j = 1, 2, · · · , n,
2
2
∂ f (P )
∂ f (P )
=
.
∂xi ∂xj
∂xj ∂xi
Definition 104. f : B(P0 , r) → R is differentiable at P0
if there exists a linear function L(r) = c · r where c ∈ Rn
(73) such that for all paths P → P ,
0
Definition 99. Big O notation describes the limiting behavior of a function in terms of another function. Given
f, g : R → R,
f (h)
= L 6= 0
g(h)
f (h)
f (h) = o(g(h)) ⇔ lim
=0
h→0 g(h)
f (h) = O(g(h)) ⇔ lim
h→0
(80)
lim
P →P0
f (P ) − f (P0 ) − L(P − P0 )
= 0.
|P − P0 |
(81)
Theorem 105. f : Rn → R is differentiable at P iff
∂f
∀i ∈ 1, 2, · · · , n, ∂x
exists and is continuous at P .
i
(74a)
Definition 106. A C 1 function f : R2 → R admits a
(74b) pair of tangent vectors along x- and y- directions for the
surface z = f (x, y):
Definition 100. The linear approximation of f (x, y) at
∂z
∂z
P0 = (x0 , y0 ) is
u = 1, 0,
,
v = 0, 1,
.
(82)
∂x
∂y
∂f ∂f T1 (x, y) = f (x0 , y0 ) +
(x − x0 ) +
(y − y0 ), Definition 107. The tangent plane of the surface at P0
∂x P0
∂y P0
is the plane that contains P0 and the tangent vectors
(75)
u(P0 ) and v(P0 ).
which satisfies
Lemma 108. If f : R2 → R is C 1 at z0 = f (x0 , y0 ),
the scalar equation of the tangent plane of the surface
σ(x, y) = hx, y, f (x, y)i at (x0 , y0 , z0 ) is
where h = max (|x − x0 |, |y − y0 |).
∂z ∂z Definition 101. The quadratic approximation of f (x, y)
z − z0 =
(x
−
x
)
+
(y − y0 ).
(83)
0
∂x P0
∂y P0
at P0 = (x0 , y0 ) is
n
∂f ∂f T2 (x, y) =f (x0 , y0 ) +
(x
−
x
)
+
(y − y0 ) Theorem 109 (The chain rule). If f : R → R is dif0
∂x P0
∂y P0
ferentiable and each variable xi is a differentiable func
2 2 2
2
tion of m variables t1 , t2 , · · · , tm , then f is a function of
∂ f (y − y0 )
∂ f (x − x0 )
+
+
t
1 , t2 , · · · , tm and ∀i = 1, 2, · · · , m,
∂x2 P0
2
∂y 2 P0
2
n
X
∂ 2 f ∂f
∂f ∂xj
+
(x − x0 )(y − y0 ),
(77)
=
.
(84)
∂x∂y P0
∂ti
∂x
j ∂ti
j=1
f (x, y) = T1 (x, y) + O(h2 )
(76)
which satisfies
f (x, y) = T2 (x, y) + O(h3 )
Definition 110. A level curve of F : R2 → R is the
curve with equation F (x, y) = c where c is a constant.
A level surface of F : R3 → R is the surface with
equation F (x, y, z) = c where c is a constant.
(78)
where h = max (|x − x0 |, |y − y0 |).
7
Qinghai Zhang
Summary of Math 1321-004
Theorem 111 (Implicit function theorem:
2D). Con∂F 2
sider F : R → R at P0 = (x0 , y0 ). If ∂y 6= 0 and F
2013-MAR-18
Theorem 119 (Fermat’s). Every local extremum of
f : Rn → R is a critical point.
P0
is C 1 on the open disk B(P0 , r) for some r > 0, then the Formula 120 (Second derivatives test for critical
level curve F (x, y) = c defines a C 1 function y = y(x) on points). For a critical point P0 = (a, b) of f : R2 → R,
2
2
∂2f
the open interval B(x0 , r), and
if ∂∂xf2 , ∂∂yf2 , and ∂x∂y
are all continuous on B(P0 , r) for
some
r
>
0,
then
the
discriminant
∂F
dy
∂x
= − ∂F
.
(85)
2
dx
∂ f
∂2f ∂y
∂x2
∂x∂y (92)
D = det 2
2
∂
f
∂
f
Theorem 112 (Implicit function theorem: 3D).
Con
∂y∂x
2
∂y
P0
sider F : R3 → R at P0 = (x0 , y0 , z0 ). If ∂F 6= 0
∂z P0
and F is C 1 on the open ball B(P0 , r) for some r > 0, might determine the
then the level surface F (x, y, z) = c defines a C 1 function 
z = z(x, y) on the open disk B((x0 , y0 ), r), and

D < 0,



2 D > 0, ∂∂xf2 > 0
∂F
∂F
∂z
∂z
∂y
P 0
∂x
2
,
(86) 
= − ∂F
= − ∂F .


D > 0, ∂∂xf2 < 0
∂x
∂y
∂z
∂z
P0
type of the critical point:
⇒ f (P0 ) is not a local extremum,
⇒ f (P0 ) is a local minimum,
⇒ f (P0 ) is a local maximum.
(93)
Definition 113. The gradient vector of a function f :
n
n
R → R is a vector in R :
Theorem 121. If f : Rn → R is continuous on a closed,
X
n
bounded set D ⊆ Rn , then f attains an absolute maxi∂f ∂f
∂f
∂f
∇f =
,
,··· ,
=
ei ,
(87) mum value and an absolute minimum value on D.
∂x1 ∂x2
∂xn
∂xi
i=1
Theorem 122. Consider determining extrema of a C 1
function f : Rn → R under the constraint F (P ) = c
where F : Rn → R, F ∈ C 1 , c is a constant. If f attains
an extremum at P0 and ∂F∂x(Pn0 ) 6= 0, then ∃λ ∈ R s.t.
Definition 114. The directional derivative of a function f : Rn → R at P0 ∈ Rn in the direction of a vector
u ∈ Rn is the scalar
Du f (P0 ) = lim
h→0
f (P0 + hu) − f (P0 )
.
h
(88)
∇f |P0 = λ ∇F |P0 .
(94)
Theorem 115. If f : Rn → R is differentiable at P , Definition 123. Consider determining extrema of a C 1
then
function f (x, y, z) under the constraint F (x, y, z) = c
Du f (P ) = u · ∇f (P ).
(89) where F ∈ C 1 and c is a constant. By constructing the
Lagrangian function
Theorem 116. If f : Rn → R is differentiable at P ,
then
Λ(x, y, z, λ) = f (x, y, z) + λ c − F (x, y, z) ,
(95)
max
Du f (P ) = |∇f (P )|.
(90)
n
u∈R , |u|=1
the method of Lagrange multipliers first find the point
Lemma 117. Let z = z(x, y) be the implicit function set U = {(x, y, z) | ∇Λ(x, y, z, λ) = 0} by solving
defined by the level surface F (x, y, z) = c. The unit
normal vector of the tangent plane to the level surface
∇f
= λ∇F,
(96)
F (x, y, z) = c is
F (x, y, z) =
c.
∇F
n=
.
(91)
and then evaluate f over U to obtain
|∇F |
Definition 118. A critical point of f : Rn → R is a
point P0 ∈ Rn satisfying ∇f |P0 = 0.
max f = max f (P ),
P ∈U
8
min f = min f (P ).
P ∈U
(97)
Qinghai Zhang
5
Summary of Math 1321-004
2013-MAR-18
Multiple integral
Definition 132. The Riemann sum of f : R2 → R over
a partition Tm,n is
Definition 124. A partition of an interval I = [a, b] is
a finite ordered subset Tn ⊆ I of the form
Tn (a, b) = {a = x0 < x1 < · · · < xn = b}.
(98)
Sm,n (f ) =
m X
n
X
∗
f x∗ij , yij
∆Aij ,
(106)
i=1 j=1
The interval Ii = [xi−1 , xi ] is the ith subinterval of the
partition. The norm of the partition is the length of the
longest subinterval,
∗
∈ Rij
where ∆Aij = (xi − xi−1 )(yj − yj−1 ) and x∗ij , yij
is the sample point of the (i, j)th subrectangle.
hn = h(Tn ) = max(xi − xi−1 ),
i = 1, 2, . . . , n. (99) Definition 133. If f : R2 → R is integrable on R =
[a, b] × [c, d], then its limit is called the definite integral
Definition 125. The Riemann sum of f : R → R over of f on R:
a partition Tn is
ZZ
f (x, y)dA = lim Sm,n (f ).
(107)
n
X
m,n→∞
∗
R
Sn (f ) =
f (xi )(xi − xi−1 ),
(100)
i=1
Formula 134. The signed volume of the solid that lies
between a surface z = f (x, y) and a rectangle R inside
where x∗i ∈ Ii is a sample point of the ith subinterval.
the xy-plane is
ZZ
Definition 126. f : R → R is integrable (or more preV =
f (x, y) dA.
(108)
cisely Riemann integrable) on [a, b] iff
R
∃L ∈ R, s.t. ∀ > 0, ∃δ s.t.
Definition 135. The average value of f : R2 → R on
(101) R = [a, b] × [c, d] is
ZZ
1
Definition 127. If f : R → R is integrable on [a, b],
f (x, y) dA.
(109)
hf iR =
(b − a)(d − c)
then its limit is called the definite integral of f on [a, b]:
R
∀Tn (a, b) with h(Tn ) < δ, |Sn (f ) − L| < .
Theorem 136. Consider r ∈ R, R = [a, b] × [c, d], and
(102) continuous f, g : R → R.
n→∞
a
ZZ
ZZ
ZZ
Theorem 128. A continuous function f is integrable
(f + g) dA =
f dA +
g dA,
(110a)
R
R
over [a, b].
ZZ
ZR
Z
rf dA = r
f dA,
(110b)
Theorem 129. A monotonic function f is integrable
R
R
Z Z
ZZ
over [a, b].
(f ≥ g) ⇒
f dA ≥
g dA , (110c)
R
R
Definition 130. A closed rectangle R ⊂ R2 is the CarteZZ
sian product of two closed intervals [a, b] and [c, d]:
rdA = r(b − a)(d − c).
(110d)
R
R = (x, y) ∈ R2 | x ∈ [a, b], y ∈ [c, d] .
(103)
Theorem 137 (Fubini’s). If f : R2 → R is continuous
Formula 131. A 2D partition of a closed rectangle can on R = [a, b] × [c, d], then
Z
b
f (x)dx = lim Sn (f ).
be obtained by the Cartesian product of two 1D partitions:
ZZ
R
Tm,n (R) = Tm (a, b) × Tn (c, d).
Z
b
Z
f dA =
(104)
d
Z
d
Z
f dy dx =
a
c
b
f dx dy. (111)
c
a
Corollary 138. If f (x, y) = g(x)h(y) is continuous on
Any subrectangle is a smaller rectangle Ri,j = Ii × Ij . R = [a, b] × [c, d], then
! Z
!
The norm of the 2D partition is
ZZ
Z b
d
f (x, y) dA =
g(x) dx
h(y) dy . (112)
h(Tm,n ) = max h(Tm ), h(Tn ) .
(105)
R
a
c
9
Qinghai Zhang
Summary of Math 1321-004
2013-MAR-18
Formula 139 (Midpoint rule for a double integral).
If f : R2 → R is C 3 over a rectangle R, then
Theorem 146 (Change of variables in a double integral). Consider a C 1 , bijective function F : S → D that
maps a region S ⊂ R2 in the uv-plane to another region
ZZ
m X
n
X
D ⊂ R2 in the xy-plane. If both S and D are regular,
f (x, y) dA =
f (x̄i , ȳj ) ∆Aij + O h4m,n ,
then for a continuous f : D → R,
R
i=1 j=1
ZZ
ZZ
(113)
f
(x,
y)
dA
=
f x(u, v), y(u, v) |det(JF )| dudv.
where hm,n is the norm of the 2D partition (104), and
D
S
(119)
1
1
ȳj = (yj−1 + yj ).
(114) Definition 147. In the polar coordinate system, the pox̄i = (xi−1 + xi ),
2
2
sition of each point P on the Euclidean plane is deterDefinition 140. A planar region D is regular if it can mined by r = |P − O|, its distance from the origin O,
be described in either of these two ways
and θ, the angle between the two vectors P − O and i.
The relations between the polar coordinates (r, θ) and
Type I. D = (x, y) | x ∈ [a, b], y ∈ [g1 (x), g2 (x)] ;
the Cartesian coordinates (x, y) are
x = r cos θ,
Type II. D = (x, y) | y ∈ [c, d], x ∈ [h1 (y), h2 (y)] .
(120)
y = r sin θ.
Theorem 141. If f is continuous on a regular region D,
then its double integral can be calculated by an iterated Corollary 148. If f (x, y) is continuous on the polar rectangle
integral :
#
ZZ
Z "Z
D = {(r, θ) | 0 ≤ a ≤ r ≤ b, 0 ≤ α ≤ θ ≤ β ≤ 2π},
b
g2 (x)
f (x, y) dA =
D
f (x, y) dy dx,
a
ZZ
Z
g1 (x)
d
"Z
f (x, y) dA =
D
#
h2 (y)
f (x, y) dx dy.
c
h1 (y)
(121)
then its double integral over D can be calculated as
ZZ
Z βZ b
(115b)
f (x, y) dA =
f (r cos θ, r sin θ) rdr dθ.
(115a)
D
α
a
(122)
Theorem 142. If (i) D1 ∩ D2 is empty or only contains
boundary points of D1 and D2 , (ii) f : R2 → R is con- Formula 149. If the density function ρ(x, y) is continuous for a lamina D, then the mass of the lamina is given
tinuous on D = D1 ∪ D2 , then
by
ZZ
ZZ
ZZ
ZZ
m=
ρ(x, y) dA,
(123)
f dA.
(116)
f dA +
f dA =
D
D
D2
D1
its moments about the x-axis and y-axis by
Definition 143. A function f : R → R is even iff
ZZ
ZZ
f (x) = f (−x); it is odd iff f (x) = −f (−x).
Mx =
yρ(x, y) dA, My =
xρ(x, y) dA, (124)
D
Formula 144. Let a > 0.
( Ra
Z +a
2 0 f (x)dx,
f (x)dx =
0,
−a
its center of mass (x̄, ȳ) by
if f (x) is even,
(117)
if f (x) is odd.
x̄ =
Definition 145. The Jacobian matrix of a function
F : R2 → R2 given by x = x(u, v), y = y(u, v) is

∂x
∂u
∂y
∂u
D
∂x
∂v
∂y
∂v
My
,
m
ȳ =
Mx
,
m
(125)
its moment of inertia about the x-axis and y-axis by
ZZ
ZZ
Ix =
y 2 ρ(x, y) dA, Iy =
x2 ρ(x, y) dA. (126)
D

D
(118) Formula 150. If σ(u, v) = hx(u, v), y(u, v), z(u, v)i is
C 1 and is injective on D, then the area of the surface
S = {P | P = σ(u, v), (u, v) ∈ D} is
and det(JF ) is the Jacobian (or more precisely Jacobian
ZZ ∂σ ∂σ determinant).
dA.
×
(127)
A(S) =
∂v D ∂u
JF = 
,
10
Qinghai Zhang
6
Summary of Math 1321-004
Vector calculus
2013-MAR-18
Formula 163. If C = {r(t) | t ∈ [a, b]}, then
Definition 151. A vector field F : Rm → Rm is C k if
each component is k times continuously differentiable.
Z
Z
f (x, y)ds =
C
b
q
2
2
f x(t), y(t) (x0 (t)) + (y 0 (t)) dt.
a
(136)
Definition 152. The vector differential operator ∇ is
More generally, for f : Rm → R and r : R → Rm ,
∂
∂
∂
∂
∂
∂
+j
+k
=
,
,
.
(128)
∇=i
Z
Z b
∂x
∂y
∂z
∂x ∂y ∂z
f (r(t))ds =
f r(t) r0 (t)dt.
(137)
C
a
Definition 153. The divergence operator of a vector
field F : Rm → Rm is
Formula 164. If C = {r(t) | t ∈ [a, b]}, then
div F = ∇ · F.
(129)
Z
Z b
f (x, y)dx =
f x(t), y(t) x0 (t)dt,
(138a)
Definition 154. A vector field F : Rm → Rm is said to
C
a
be incompressible if ∇ · F = 0 holds for all points.
Z
Z b
f x(t), y(t) y 0 (t)dt.
(138b)
f
(x,
y)dy
=
Definition 155. A point P is a source of a vector field
a
C
F if ∇ · F|P > 0; it is a sink of F if ∇ · F|P < 0.
m
m
Definition 156. The curl operator of a vector field Definition 165. Let a vector field F : R → R be continuous on an m-dimensional curve C = {r(t) | t ∈ [a, b]}.
F : R3 → R3 is
curl F = ∇ × F.
(130) The line integral of F along C is
Z
Z b
Z
Definition 157. A vector field F : Rm → Rm is said to
0
m
F · dr =
F r(t) · r (t)dt =
F · Tds. (139)
be conservative iff ∃f : R → R s.t. F = ∇f .
C
a
C
Then f is called a potential function for F.
Theorem 158. If a C 1 vector field F : R3 → R3 satisfies Formula 166. If F = hP (x, y, z), Q(x, y, z), R(x, y, z)i,
then
Z
Z
∇ × F = 0, then F is conservative.
F · dr =
P dx + Qdy + Rdz.
(140)
Theorem 159. If f : R3 → R has continuous secondC
C
order partial derivatives, then
Theorem 167 (Fundamental theorem of line integrals).
∇ × ∇f = 0.
(131) If f : Rm → R is differentiable and ∇f is continuous on
an m-dimensional curve C = {r(t) | t ∈ [a, b]}, then
Theorem 160. For a C 2 vector field F : R3 → R3 ,
Z
∇ · (∇ × F) = 0.
(132)
∇f · dr = f r(b) − f r(a) .
(141)
C
Definition 161. The line integral of f (x, y) along a
curve C (with respect to the arc length) is
Corollary 168. Line integrals of conservative vector
Z
fields are independent of path.
n
X
f (x, y)ds = lim
f (x∗i , yi∗ )∆si .
(133)
n→∞
Definition 169. A curve r(t), t ∈ [a, b] is simple iff r(t)
C
i=1
is injective on (a, b); it is closed iff r(a) = r(b). It is
More generally, if C is an m-dimensional curve given by simple closed or Jordan iff it is both simple and closed.
r(s), then for f : Rm → R,
Z
Definition 170. A point set P is connected (or more
n
X
f (r(s))ds = lim
f r(s∗i ) ∆si .
(134) precisely path-connected ) iff for any two points in P there
n→∞
C
exists a path in P that connects them.
i=1
Definition 162. The line integral of f (x, y) along a
curve C (with respect to x and y) is
Z
n
X
f (x, y)dx = lim
f (x∗i , yi∗ )∆xi ,
(135a)
C
n→∞
Z
f (x, y)dy = lim
C
n→∞
Theorem 171 (Jordan curve). The complement of a
simple closed curve in the plane consists of two components: the bounded one is called its interior and the unbounded its exterior ; both are open and path-connected.
i=1
n
X
i=1
f (x∗i , yi∗ )∆yi .
Definition 172. A point set P is simply-connected iff it
(135b) is connected and every simple closed curve in P encloses
only points that are in P.
11
Qinghai Zhang
Summary of Math 1321-004
2013-MAR-18
R
Theorem
173. F · dr is independent of path in D iff Definition 183. A surface is orientable if it has two
H
F · dr = 0 for all closed path C ⊂ D.
sides. The positive side is the side to which the unit norC
m
m
Theorem 174. Let F : R → R be a vector field that mal vector points. By convention, a simple closed surface
is oriented by its unit outward normal vector.
Ris continuous on an open and connected region D. If
F · dr is independent of path in D, then F is conser- Definition 184. For a continuous vector field F defined
C
vative on D.
on an oriented surface S with unit normal vector n, the
1
Theorem 175. If F = hP (x, y), Q(x, y)i is C and con- surface integral of F over S (or the flux of F across S)
∂Q
is
ZZ
ZZ
servative on D, then ∀(x, y) ∈ D, ∂P
∂y = ∂x .
F
·
dS
=
F · n dS.
(147)
Theorem 176. If F = hP (x, y), Q(x, y)i is C 1 on an
S
S
∂Q
open and simply-connected region D and ∂P
∂y = ∂x , then Formula 185. Consider a continuous vector field
F is conservative.
F = hP (x, y, z), Q(x, y, z), R(x, y, z)i and an oriented
Definition 177. The surface integral of f (x, y, z) over surface S given by z = g(x, y).
ZZ
ZZ a surface S given by r(u, v) = hxr (u, v), yr (u, v), zr (u, v)i
∂g
∂g
−Q
+ R dA, (148)
F · dS =
−P
is
∂x
∂y
S
D
ZZ
m
n
XX
f (Pij∗ )∆Sij , (142) where D is the projection of S onto the xy-plane.
f (x, y, z)dS = lim
m,n→∞
S
where
Pij∗
=
i=1 j=1
Theorem 186 (Gauss’s). Let V be the interior of a simple closed surface ∂V. If F : R3 → R3 is C 1 on an open
region that contains V, then
ZZ
ZZZ
F · dS =
∇ · F dV.
(149)
r(u∗i , vj∗ ).
Formula 178. If a smooth surface S is given by
r(u, v) = hxr (u, v), yr (u, v), zr (u, v)i, then
ZZ
ZZ
∂r
∂V
V
∂r f (x, y, z)dS =
f (r(u, v)) ×
dA, (143)
∂u ∂v
Theorem 187 (Green’s). Let D be the interior of
S
D
a positively-oriented, piecewise-smooth, simple closed
where D is the integral domain in the uv-plane.
curve ∂D in the plane. If F = hP (x, y), Q(x, y)i is C 1
Formula 179. If a smooth surface S is given by on an open region that contains D, then
z = g(x, y), then
I
ZZ ∂Q ∂P
ZZ
P dx + Qdy =
−
dA.
(150)
∂x
∂y
f (x, y, z)dS
(144)
∂D
D
S
s Equivalently, if G = hP (x, y), Q(x, y), 0i, then
2
ZZ
2
∂z
∂z
I
ZZ
=
f (x, y, g(x, y))
+
+ 1 dA,
∂x
∂y
G · dr =
(∇ × G) · kdA.
(151)
D
∂D
where D is the projection of S onto the xy-plane.
Definition 180. The unit outward normal vector of a
closed 2D curve r(t) = hx(t), y(t)i is
n(t) = −N(t),
D
Equivalently, if H = hQ(x, y), −P (x, y)i, then
I
ZZ
H · n ds =
∇ · H dA,
∂D
(145)
(152)
D
where n is defined in (145).
where N(t) is the principal unit normal vector defined in
Formula 188. The area of a region D bounded by a
Definition 94.
simple closed curve ∂D is given by
Definition 181. A simple closed curve is positively oriI
I
I
1
ented if an observer traversing the curve always has the
A(D) =
xdy = −
ydx =
xdy − ydx.
2 ∂D
interior of the curve on her left side.
∂D
∂D
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Definition 182. The unit normal vector of a surface
Theorem 189 (Stokes’). Let S be an oriented piecewiser(u, v) is
∂r
∂r
smooth surface with its boundary ∂S a simple, closed,
×
∂v n(u, v) = ∂u
.
(146) piecewise-smooth curve. If F : R3 → R3 is C 1 on an
∂r
∂r
× ∂u
∂v
open region that contains S, then
n(u, v) is called the unit outward normal vector if the
Z
ZZ
surface r(u, v) is simple closed and n(u, v) points from
F · dr =
(∇ × F) · dS.
(154)
its interior to its exterior.
∂S
S
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