Summary of §11 Partial Derivatives
Math 1321 (Qinghai Zhang)
2013-FEB-25
Definition 1. The open ball centered at P0 ∈ Rn with Theorem 12 (Clairaut’s). If f : Rn → R has continuous
radius r > 0 is the point set
second partial derivatives at P , then ∀i, j = 1, 2, · · · , n,
B(P0 , r) = P |P − P0 | < r .
(1)
∂ 2 f (P )
∂ 2 f (P )
=
.
(8)
∂xi ∂xj
∂xj ∂xi
It is an open interval in 1D and an open disk in 2D.
The open ball without the center is denoted by
Definition 13. The linear approximation of f (x, y) at
B0 (P0 , r) = B(P0 , r) \ {P0 }.
(2) P0 = (x0 , y0 ) is
Definition 2. P0 is a boundary point of an point set U
iff ∀r > 0, ∃P ∈ B(P0 , r) s.t. P 6∈ U.
∂f ∂f (x−x0 )+
(y−y0 ). (9)
f (x, y) ≈ f (x0 , y0 )+
∂x P0
∂y P0
Definition 3. A set is an open set if it contains none of
n
its boundary points. A set is an closed set if it contains Definition 14. The total differential of f : R → R is
defined in terms of the differential s dxi ’s:
all of its boundary points.
Definition 4. A is a subset of B, written as A ⊆ B,
iff x ∈ A ⇒ x ∈ B.
df =
n
X
∂f
dxi
∂x
i
i=1
(10)
Definition 5. A point set U ⊆ Rn is bounded iff
U ⊆ B(P0 , r) for some P0 ∈ Rn and r > 0.
Definition 15. The increment of a function z = f (x, y)
Definition 6. The limit of a function f : B (P , r) → R at a point P0 = (x0 , y0 ) is
0
0
exists as P approaches P0 , written as limP →P0 f (P ) = L,
iff
∆z = f (x, y) − f (x0 , y0 ).
(11)
∀ > 0, ∀ paths P → P0 , ∃δ > 0, s.t.
(3) Definition 16. f : B(P0 , r) → R is differentiable at Pn0
if there exists a linear function L(r) = c · r where c ∈ R
such that for all paths P → P0 ,
Formula 7. The limit of f does not exist at P0 if there
exists two different paths P → P0 and P
P0 s.t.
f (P ) − f (P0 ) − L(P − P0 )
= 0.
(12)
lim
P →P0
|P − P0 |
(4)
lim f (P ) = L 6= L = lim f (P ) .
∀P ∈ B(P0 , δ),
1
P →P0
|f (P ) − L| < .
2
P
P0
Theorem 17. f : Rn → R is differentiable at P iff
∂f
∀i ∈ 1, 2, · · · , n, ∂x
exists and is continuous at P .
i
Theorem 8 (The squeeze theorem). If ∃r > 0 s.t.
∀P ∈ B0 (P0 , r), f (P ) ≤ g(P ) ≤ h(P ), then
lim f (P ) = lim h(P ) = L ⇒ lim g(P ) = L.
P →P0
P →P0
Definition 18. A function f (x) is C 1 or continuously
differentiable if f 0 (x) exists and is itself continuous.
P →P0
(5)
Definition 19. A C 1 function f : R2 → R admits a
pair of tangent vectors along x- and y- directions for the
surface z = f (x, y):
n
Definition 9. f : R → R is continuous at Q iff
lim f (P ) = f (Q).
P →Q
(6)
u=
f is continuous on a point set U if (6) holds ∀Q ∈ U.
Theorem 10. A polynomial is continuous everywhere.
∂z
1, 0,
∂x
,
v=
∂z
0, 1,
∂y
.
(13)
The tangent plane of the surface at P0 is the plane that
contains P0 and the tangent vectors u(P0 ) and v(P0 ).
Definition 11. The partial derivative of f : Rn → R
with respect to the ith dimension at P0 is
∂f (P0 )
∂f f (P0 + hei ) − f (P0 )
=
= lim
, (7)
∂xi
∂xi P0 h→0
h
Lemma 20. 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 ∈ R and {ei | i = 1, 2, · · · , n} is the set of standard basis vectors of the Euclidean n-space, i.e. the ith
component of ei is 1 and all other components are 0.
z − z0 =
1
∂z ∂z (x
−
x
)
+
(y − y0 ).
0
∂x P0
∂y P0
(14)
Summary of §11 Partial Derivatives
Math 1321 (Qinghai Zhang)
2013-FEB-25
Theorem 21 (The chain rule). If f : Rn → R is dif- Definition 30. f : Rn → R has a local maximum at
ferentiable and each variable xi is a differentiable func- P0 ∈ Rn iff
tion of m variables t1 , t2 , · · · , tm , then f is a function of
∃r > 0, s.t. ∀P ∈ B(P0 , r),
f (P ) ≤ f (P0 ). (23)
t1 , t2 , · · · , tm and ∀i = 1, 2, · · · , m,
n
Changing ≤ to ≥ in (23) yields a local minimum.
(15) An extremum is either a maximum or minimum.
X ∂f ∂xj
∂f
=
.
∂ti
∂xj ∂ti
j=1
Definition 31. A critical point of f : Rn → R is a point
Definition 22. A level curve of F : R2 → R is the curve P0 ∈ Rn satisfying ∇f | = 0.
P0
with equation F (x, y) = c where c is a constant.
A level surface of F : R3 → R is the surface with Theorem 32 (Fermat’s). Every local extremum of
f : Rn → R is a critical point.
equation F (x, y, z) = c where c is a constant.
Theorem 23 (Implicit function theorem:
2D). Consider Formula 33 (Second derivatives test for critical points).
a critical point P0 = (a, b) of f : R2 → R. If
∂F 2
F : R → R at P0 = (x0 , y0 ). If ∂y 6= 0 and F is C 1 Consider
∂2f
∂2f
∂2f
P0
2 , ∂y 2 , and ∂x∂y are all continuous on B(P0 , r) for
∂x
on the open disk B(P0 , r) for some r > 0, then the level
some r > 0, then the discriminant
curve F (x, y) = c defines a C 1 function y = y(x) on the
2
∂ f
open interval B(x0 , r), and
∂2f ∂x2
∂x∂y (24)
D
=
det
∂F
2
2
dy
∂ f
∂ f ∂x
∂y∂x
= − ∂F
.
(16)
2
∂y
P0
dx
∂y
might determine the


D < 0,


2 D > 0, ∂∂xf2 > 0
P 0

2


D > 0, ∂∂xf2 < 0
Theorem 24 (Implicit function theorem: 3D).
Con
sider F : R3 → R at P0 = (x0 , y0 , z0 ). If ∂F
∂z P0 6= 0
and F is C 1 on the open ball B(P0 , r) for some r > 0,
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
∂z
=−
∂x
∂F
∂x
∂F
∂z
,
∂z
=−
∂y
∂F
∂y
∂F
∂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.
(25)
(17)
Theorem 34. If f : Rn → R is continuous on a closed,
n
Definition 25. The gradient vector of a function f : bounded set D ⊆ R , then f attains an absolute maximum value and an absolute minimum value on D.
Rn → R is a vector in Rn :
X
n
Theorem 35. Consider determining extremums of a C 1
∂f ∂f
∂f
∂f
∇f =
,
,··· ,
=
ei ,
(18) function f : Rn → R under the constraint F (P ) = c
∂x1 ∂x2
∂xn
∂xi
i=1
where F : Rn → R, F ∈ C 1 , c is a constant. If f attains
Definition 26. The directional derivative of a function an extremum at P0 and ∂F∂x(P0 ) 6= 0, then ∃λ ∈ R s.t.
n
f : Rn → R at P0 ∈ Rn in the direction of a vector
n
=
λ
∇F |P0 .
(26)
∇f
|
u ∈ R is the scalar
P0
f (P0 + hu) − f (P0 )
36. Consider determining extremums of a
.
(19) Definition
h
C 1 function f (x, y, z) under the constraint F (x, y, z) = c
1
Theorem 27. If f : Rn → R is differentiable at P , then where F ∈ C and c is a constant. By constructing the
Lagrangian function
Du f (P ) = u · ∇f (P ).
(20)
Λ(x, y, z, λ) = f (x, y, z) + λ c − F (x, y, z) ,
(27)
n
Theorem 28. If f : R → R is differentiable at P , then
max
Du f (P ) = |∇f (P )|.
(21) the method of Lagrangian multipliers first find the point
u∈Rn , |u|=1
set U = {(x, y, z) | ∇Λ(x, y, z, λ) = 0} by solving
Lemma 29. Let z = z(x, y) be the implicit function
∇f
= λ∇F,
(28)
defined by the level surface F (x, y, z) = c. The unit
F (x, y, z) =
c.
normal vector of the tangent plane to the level surface
and then evaluate f over U to obtain
F (x, y, z) = c is
∇F
n=
.
(22)
max f = max f (P ),
min f = min f (P ).
(29)
|∇F |
P ∈U
P ∈U
Du f (P0 ) = lim
h→0
2