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Transcript of Title: Multivariate Continuity and differentiation Md. Masum Murshed Lecturer Department of...
Title: Multivariate Continuity and differentiation
Md. Masum MurshedLecturer
Department of Mathematics, R.U.25 November, 2011, Friday
Time: 4:50 pm - 6 pm
1
Limit of a Function
2
Figure- 1
f(x)
L
L - ε
L + ε
a a - δ a + δ
Limit of a FunctionFind the value of delta () and Epsilon ()
3Figure- 2
4
q
q - r
q + r
p p - r p + r
(p , q)
Rectangular Neighborhood of a point in R2 Nr(p) = {x ϵ R : |x - p| < r} = (p – r, p + r).
Nr(q) = {y ϵ R : |y - q| < r} = (q – r, q + r).
5
q
p
Circular Neighborhood of a point in R2
Nr((p, q)) = {x, y ϵ R : (x – p)2 + (y – q) 2 < r2}
(p , q)
r
6
Circular Neighborhood of a point in R2
..
7
Norm on a Vector Space
V
Vector Space
R
k
Norm
8
Norm in Rn
Rn
Euclidean Norm is given by
0(0, 0)
Br(0)
r
+ x = Br(x)
x
Mathematically
Mathematically
In E = R2
Open Ball and Open sets
9
Continuous Function
10
A function f(x) is said to be continuous x = a, if for every neighborhood Nε(f(a)) there is some neighborhood Nδ(a) such that if x ϵ Nδ(a) then f(x) ϵ Nε(L).
A function f(x) is said to be continuous x = a, if for every neighborhood Nε(f(a)) there is some neighborhood Nδ(a) such that if x ϵ Nδ(a) then f(x) ϵ Nε(L).
A function f(x) is said to be continuous x = a, if
(i) f(x) is well defined at x = a, and
(ii)
A function f(x) is said to be continuous x = a, if
(i) f(x) is well defined at x = a, and
(ii)
A function f (x) is said to be continuous x = a, if for every number ε > 0 (however small) there is some number δ > 0 (usually depending on ε and a) such that
| f(x) – f(a)| < ε whenever |x-a| < δ
A function f (x) is said to be continuous x = a, if for every number ε > 0 (however small) there is some number δ > 0 (usually depending on ε and a) such that
| f(x) – f(a)| < ε whenever |x-a| < δ
Continuous Function
11
Figure- 3
f(x)
f(a)
f(a) - ε
f(a) + ε
a a - δ a + δ
CONTINUOUS FUNCTIONS
Observe that |x – x0| < δ means that x0 - δ < x < x0 + δ, or equivalently that x
belongs to the open interval (x0 - δ , x0 + δ).
Similarly, |f(x) - f(x0)| < ε means that f(x) belongs to the open interval (f(x0) - ε, f(x0) + ε).
Accordingly, the statement|x – x0| < δ implies |f(x) - f(x0)| < ε
is equivalent to the statement x ∈ (x0 - δ , x0 + δ) implies f(x) ∈ (f(x0) - ε, f(x0) + ε)
which is equivalent to the statement f [(x0 - δ , x0 + δ)] is contained in (f(x0) - ε, f(x0) + ε)
which is equivalent to the statement (x0 - δ , x0 + δ) is contained in f -1(f(x0) - ε, f(x0) + ε)
CONTINUOUS FUNCTIONS Hence we can restate the previous definition as follows.A function f : ℝ → ℝ is continuous at a point p ∈ ℝ if for any open set Vf(p), containing f(p) there exists an open set UP containing p such that f [UP] Vf(p) or UP f -1[Vf(p)]. ℝ ℝ
Continuous at p
f(p)p
f
Figure- 4
Continuous Function
14
A function f is continuous x = a, if and only if for every open set S containing f(a) , f -1(S) is an open set containing a.A function f is continuous x = a, if and only if for every open set S containing f(a) , f -1(S) is an open set containing a.
A function f is continuous x = a, if and only if for every ε > 0 there exists δ > 0 such that |x-a| < δ implies | f(x) – f(a)| < εA function f is continuous x = a, if and only if for every ε > 0 there exists δ > 0 such that |x-a| < δ implies | f(x) – f(a)| < ε
A function f is continuous x = a, if and only if for all {xn}, x ≠ a
xn → a implies f (xn ) → f (a)
A function f is continuous x = a, if and only if for all {xn}, x ≠ a
xn → a implies f (xn ) → f (a)
For f : ℝ → ℝ we know
Contiuity of Multivariate FunctionHow can we extend this definition for scalar/vector/ matrix/function valued functions of vector/matrix/function keeping important properties intact??
For examples, i) f1(x)=Ax, ii) f2(A)=A-1, iii) g(A)=|A|, iv) μ(f)=∫xf(x)dx, f3(x)=xTAx etc.
We need norm /metric in both domain and co-domain.
Norm is better. Why??
f is continuous at a if and only if for all ε > 0, there exists δ > 0 s.t. ||x-a||1 < δ implies ||f(x)-f(a)||2 < ε
f is continuous at a if and only if for all ε > 0, there exists δ > 0 s.t. d1(x,a) < δ implies d2( f(x), f(a)) < ε
f is continuous at a if and only if for all {xn}, xn ≠a
Our Well-known Derivative
For f : I→R , where I, an interval of R , x ε interior of I
How can we extend this definition for scalar/vector/ matrix/function valued functions of vector/matrix/function keeping important properties intact??
In co-domain there must exist the concept of subtraction. General vector spaces does have.
For examples, i) f1(x)=Ax, ii) f2(A)=A-1, iii) g(A)=|A|, iv) μ(f)=∫xf(x)dx, f3(x)=xTAx etc.
In vector/matrix/function space we can define limit with metric but not division. We cannot define division in general vector spaces
Math
ematical C
on
cepts
Figure- 5
A function f of n variables is a rule that assigns to each ordered n-tuple of real numbers (x1,…, xn) in a set D a unique real number denoted by f (x1,…, xn).
Study of Multivariate Functions
We often write z= f (x1,…, xn) to make explicit the value taken on by f at the general point (x1,…, xn).
The set D is the domain of f and its range is the set of values that f takes on, that is,
The variables x’s are independent variables and z is the dependent variable.
DEFINITION
Study of Multivariate Functions
0
D
y
(x, y)
(a, b)
0 zx
f (a, b)f
f (x, y)
Domain of f
y
0-1
-1
x =1
x + y + 1 =0
x
Figure- 6
Figure- 7
Study of Multivariate Functions2:f R R
Figure- 9
Figure- 8
DEFINITION
Graph of Multivariate Functions
Figure 11
If f is a function of two variables with domain D, then the graph of is the set of all points (x, y, z) in R3 such that z=f (x, y) and (x, y) is in D.
Figure- 10
Graph of Multivariate Functions
Figure- 12 Figure- 13
Graph of Multivariate Functions
Figure 14
Given a function f :Rn R and a real number c, we call the set, L = {(x1, x2,…., xn)| f(x1, x2,…., xn) = c} a level set of f at level c.
Graph of Multivariate FunctionsDEFINITION
We also call L a contour of f. When n = 2, we call L a level curve of f and when n = 3 we call L a level surface of f.
A plot displaying level sets for several different levels is called a contour plot.
Figure- 15
Figure 16: Level surface of f(x, y)=2x2 +y2
Figure 17: The parabloid f(x, y)=2x2 +y2
Figure- 18
Graph of Multivariate Functions
Figure- 19Figure- 20
The graph of h (x, y)=4x2+y2
is formed by lifting the level curves.
Graph of Multivariate Functions
Figure- 21
Standardard Ellipsoid
X2+2y2+3z2 =1
Very im
portant in Multivariate
Statistics
Figure- 22
29
Bivariate Gaussian
Its Importance in Statistics
30
Spherical, diagonal, full covariance
UMFigure- 23
April 20, 2023Md. Masum Murshed , Dept. of
Mathematics, RU31
Limits of functions f : R2→R
Note that in order for
to exist, it must have the same value regardless of the approach of (x, y) to (x0, y0).
It follows that if two different approaches give different values, the limit cannot exist.
For a limit in one variable to exist, the limits from both sides must yield the same value.
The analogous result for 2 variable limits is that the limit exists only if the limits along every curve through (x0, y0) are the same.
Continuity Checking
Figure- 24
Continuity Checking
Figure- 25
Line in Vector Space
y + x
y - x
y
x
-x
Λ(y-x)
x+Λ(y-x) = (1- Λ)x+ Λy
Figure- 26
x
y
Line through x and yLine through x and y (1-λ) (1-λ) xx + λ + λ yy
λ = 0
λ = 1
λ = 1/2
λ = -0.3
λ = 1.3
Figure- 27
Line in Vector Space
X
Y
a
U||u||=1
a+hu
a+h(0,1)
a+h(1,0)
Figure- 28
Study of Multivariate FunctionsGiven this function, we can move in the direction of the
red, black or green vector and can measure rate change of the
but how??.
function in the direction,
Figure- 29
2x2 + y2 = 3(zero slope)
(x,y)=(1,1)
direction (-1,-1)steep rate of ascent
direction (1,1)steep rate of descent
In order to talk about the slopeof a function we must specify direction
Study of Multivariate Functions
Let F(x,y) = 4 – 2x2 –y2
Figure- 30
Study of Multivariate FunctionsIf we can measure rate change of the function in the direction or in other direction, can we measure the rate of change of the function all around from rates of range in all directions?
Figure- 31
Study of Multivariate FunctionsFor example, suppose a bug moves in the direction of v = (-1, -1). If we let
Note: f(x, y) = 4 – 2x2 –y2
Directional Derivative
(black arrow).
Figure- 32
Directional Derivative
0
( ) ( )( ) limu
h
f c hu f cD f c
h
Where ||u||=1.
The definition is valid even if f is vector valued.
Given a point c and a unit vector u, then the directional derivative of f in the direction of u at c is given by
Definition
Partial DerivativesIf we consider f as a function of x = (x1,x2,….,xn) and let ek be the k-th standard basis vector, k=1,2,…, then we call the partial derivative of f with respect to xk at c, which is also denoted as
Now suppose f : ℝn → ℝ and define g : ℝ → ℝ by g(t)=f(t, x2,….,xn).
1
1 2 1 1 21 2
0
1 2 1 2
0
1 2 1 2
0
1 1
0
(( , ,......, ) ) ( , ,......, )' ( , ,......, ) lim
(( , ,......, ) ( ,0,0,...0)) ( , ,......, )lim
( , ,......, ) ( , ,......, )lim
( ) ( )lim
n nx n
h
n n
h
n n
h
h
f x x x he f x x xf x x x
hf x x x h f x x x
hf x h x x f x x x
hg x h g x
h
1'( )g x
Partial Derivatives
In other words, we may compute the partial derivative by treating x2,…..xn as constants and differentiating with respect to x1 as we would in single-variable calculus
If f is a function of two variables, its partial derivatives are the functions fx and fy defined by
h
yxfyhxfyxf
hx
),(),(lim),(
0
h
yxfhyxfyxf
hy
),(),(lim),(
0
RULE FOR FINDING PARTIAL DERIVATIVES OF z=f (x, y)1. To find fx, regard y as a constant and differentiate f (x, y) with respect to x.2. To find fy, regard x as a constant and differentiate f (x, y) with
respect to y.
NOTATIONS FOR PARTIAL DERIVATIVES
fDfDfx
zyxf
xx
ffyxf xxx
11),(),(
fDfDfy
zyxf
yy
ffyxf yyy
22),(),(
If Z=f (x, y) , we write
The definition of PARTIAL DERIVATIVES is valid even if f is vector valued.
Partial Derivatives: Emamples
For examplef(x, y)=3x2 – 4xy2
Treating y as a constant and differentiating with respect to x
2( , ) 6 4x
ff x y x y
x
Treating x as a constant and differentiating with respect to y
( , ) 8y
ff x y xy
y
2 2
8ff
yx y y x
The partial derivatives of f at (a, b) arethe slopes of the tangents to C1 and C2.
Geometric Meanings of Partial Derivatives
Figure- 33
The Gradient
That is, we may view the gradient as a function which takes an n-dimensional vector for input and returns another n-dimensional vector. We call a function of this type a vector field.
Each partial derivative may be regarded as a multivariate function of x.
The n-vector of these partial derivatives is then a vector function of x, termed the gradient vector of F, and will be denoted by ∇ F(x) or g(x):
For example, if F(x) is defined by F(x)=x1x2
2+x2cos(x1), then
Notice that if f : ℝn → ℝ then ∇f : ℝn → ℝn
The Gradient of a Linear Function
• If the gradient of F is a constant vector, F is said to be a linear function of x:
( ) TF x x ( )
constant
F x
1 2
1 2
( ) 5 3 10
5, 3, 10
F x x x
F F
x x
Example
Gradients and Lines and PlanesWe recall that a one-dimensional derivative defines the slope of the
tangent line to the curve defined by F(x).
Similarly, for a differentiable multivariate function the tangent hyperplane at the point is defined by the gradient vector.
Figure- 34 Figure- 35
Figure- 36
Geometric Description of Gradient
For example, consider the quadratic form f(x) = ½ xTAx – bTx + c
This quadratic form is simply a scalar, quadratic function of a vector. 3 2 2
02 6 8
A b c
Figure- 37Figure- 38
Geometric Description of GradientFor example, consider the quadratic form
f(x) = ½ xTAx – bTx + c
2 2
3 2 2 0
2 6 8
1( , ) (3 6 4 ) 2 8
2
A b c
F x y x y xy x y
3 2 2 3 2 2( , )
6 2 8 2 6 8
x y xF x y
y x y
. .,
0(2, 2)
0
2(0,0)
8
e g
F
F
Geometric Description of Gradient
Note that the gradient vectors in the middle figure are orthogonal to the level curves of the contour plot (remember that the gradient is defined in the xy-plane).
Show that where ( ) 2f x Ax ( ) Tf x x Ax
Figure- 39Figure- 40Figure- 41
Geometric Description of GradientThis generalizes to higher dimensions. The gradient is normal to the level surface at that point.
For example, in this figure we see the level surface x2+y2+z2=1 (of the function F(x,y,z)= x2+y2+z2, which we obviously cannot plot in three dimensions).
Since at the level surface F(x,y,z)=constant, the function values don't change with change in x, y or z, thus the directional derivative is zero for all directions u that are tangential to the level surface:
All these tangential u's also define a tangent plane. Since this implies that the gradient is orthogonal to u and thus to the tangent plane.
Next slide
Geometric Description of Gradientgradient: orthogonal tosurface
Directional derivativein direction of u (tangent to surface) iszero
Figure- 42
Relation between the Gradient and the Directional Derivative
Later we shall show that
By C-S inequality
Relation between gradient and directional derivative
The gradient vector points in the direction of the maximum rate of increase of the function, and the negative of the gradient vector points in the direction of the maximum rate
of decrease of the function. The length of the gradient vector tells us the rate of increase/decrease
The gradient at a point has the following properties:
•The gradient direction at a point is in the direction of steepest ascent at that point
•The gradient magnitude is the steepness of that ascent
•The gradient direction is the normal to the level curve (or surface) at that point
•The gradient defines the tangent curve (or plane) at that point
•The gradient can be used to calculate the first derivative (i.e., directional derivative) in any direction.
Relation between Gradient and Directional Derivative
Relation between Gradient and Directional Derivative
( , ) ( 4 , 2 )f x y x y .
Thus
.
(1,1) ( 4, 2)f
Hence, if a bug standing above (1,1) in the xy-plane on the graph of f wants to head in the direction of most rapid ascent, it should move in the direction
of the unit vector (1,1) 1(2,1)
(1,1) 5
fu
f
.
If the bug wants to head in the direction of most rapid descent, it should move in the direction of the unit vector –u. Moreover,
2 2(1,1) (1,1) 4 2 20
(1,1) (1,1) 20
u
u
D f f
D f f
Relation between Gradient and Directional Derivative
Relation between the Gradient and the Directional Derivative
Scaled gradient vectors for f(x,y)=4-2x2-y2
f(x,y)=4-2x2-y2 with path of most rapid ascent from (1,1,1) y x
Figure- 43
Figure- 44
Recall that when we zoom in on a “sufficiently nice” function of two variables, we see a plane.
Relation between Continuity, Gradient and Directional Derivative
Figure- 45
What is meant by “sufficiently nice”?
Suppose we zoom in on the function z=f(x,y) centering our zoom on the point (a,b) and we see a plane. What can we say about the plane?
• The partial derivatives for the plane at the point must be the same as the partial derivatives for the function.
• Therefore, the equation for the tangent plane is
( , ) ( , ) ( , )( ) ( , )( )f f
L x y f a b a b x a a b y bx y
In particular. . .The Partial Derivatives Must Exist
If the partial derivatives don’t exist at the point (a,b), the function f cannot be locally planar at (a,b).
Example: A cone with vertex at the origin cannot be locally planar there, as it is clear that the x and y cross sections are not differentiable there.
Figure- 46
Not enough: A Puny Condition
Suppose we have a function
1 if 0 or 0( , )
0 if neither nor is 0
x yf x y
x y
Notice several things:
•Both partial derivatives exist at x=0. •The function is not locally planar at x=0.
•The function is not continuous at x=0.
Whoa! The existence of the
partial derivatives doesn’t even
guarantee continuity at the point!
Directional Derivatives?
A surprising fact is that a function can have direction derivative in every direction at a point without being continuous at that point
2 2 4( , ) 0
0( , )
xy x y x y
Otherwisef x y
0
( ) ( )( ) limu
h
f c hu f cD f c
h
1 2
2
1 2 1 22 2 4( , ) 0 01 2
( , )((0,0))
a a h h
f ha ha aaD f Lt Lt
h a ha
Our Well-known Derivative
(1) )()(
)( lim0 h
xfhxfxf
h
For f: I→R , where I, an interval of R , xε interior of I
0)()()(
0)(
where),()()()(
0
0
h
hxfxfhxfLt
h
hoLthohxfxfhxf
h
h
It can be easily shown, is a linear map. It can be shown,.
RRxf :)(
under usual norm every linear map between finite-dimensional spaces is always continuous ( shown later)
Our Well-known Derivative
0||
|)()()(|
0)()()(
0
0
h
hxfxfhxfLt
h
hxfxfhxfLt
h
h
It can be easily generalized to Banach space valued function, f: 2211 ,, BB
0||||
||)()()(||
1
2
0
h
hxfxfhxfLt
h
is a (continuous)linear map. 1 2
( ) :f x B B
It can be shown,.every linear map between infinite-dimensional spaces is not always continuous.
Called F
réchet derivative
Main Results of Fréchet Derivative
UA Banach Space
V,AnotherB-space
f,a nonlinear function
What is Differentiation?
Differentiation is nothing but local linearization
In differentiation we approximate a non-linear function locally by a (continuous) linear function
Linear Map and Matrices
Vn is a finite-dimensional vector space. Let v belongs to Vn
v=x1v1+x2v2+--------+xnvn
(x1, x2, --------- xn)nR
Every Vn is isomorphic to Rn
nn RV Its
Significance??
This isom
orphism is basis
dep
end
ent
Linear Map and Matrices
Every Vnm is isomorphic to
nmnmnm RRV
Its Significance??
This isom
orphism is basis
dep
end
ent
VnWm
k1v1+k2v2
k1L(v1)+k2L(v2))
L
L(k1v1+k2v2)
Vnm, the set of all such L’s is a vector space of dimension nm
nmR
=
Linear Map and Matrices
Linear mappings are almost omnipresent
If both domain and co-domain are both finite-dimensional vector space, each linear mapping can be uniquely represented by a matrix w.r.t. specific couple of bases
We intend to study properties of linear mapping from properties of its matrix
Normed spacesDefine the notion of the size of f, an element in F , a vector spaceNorm || f ||, || ||: F→[0,∞)1)||f||=0↔f=02) ||kf||=|k|||f||3) ||f+g|| <=||f||+||g||
0,f 0 0f f
1 2 1 2f f f f 1 2 1 2,d f f f f
Both R and Z are normed spaces are spaces with | |
Both Rn and are normed spaces are spaces with Euclidean norm,|| ||
n mR
Normed spaces become Banach spaces where every Cauchy sequence converges
A norm on the vector space is a mapping, ,
satisfying the following axioms:
Vector Norms
. : 0; 0 0
. : | | ;
. : ;
a Positivity v v v
b Linearity cv c v
c Triangleinequality v w v w
1v
n RR
Examples of Vector NormsnRvvvv ,),(|||| 21.Euclidean Norm
ni Cvnivv },,...,1:||{max||||
2.p Norm ),1[,,||||||/1
1
pCvvv npn
k
pkp
3,Max Norm
4nnCA nonsingular ppA Avv |||||||| ,
is a vector norm on nC for any ],1[ p
5.Show that ||f||= sup|f(x)| defines a norm on B(X).
Properties of Norms
|| || 0 ||| || || || | 0
|| || || || 0
|| || || ||
n n
n
n
v u v u
v u
v u
, ,v u V
( ): f v vf V R
thenTheorem 1 If
Proofs
| || || || || | || ||v u v u
|| || || || || ||,|| || || || || || (2)u u v v v v u u
(1)
(2) Is a continuous function
| || || || || | || ||n nv u v u
For any two norms and , defined in a finite
dimensional vector space, there exist positive constants
C≥c>0, such that:
Norms Equivalence
A B
BAB vCvvc,v
Proof (outline):
Due to property (b) of norms, it is enough to show (2) on the
unit sphere, . Due to property (c) the norm on
this sphere is a continuous function, thus it has the minimum
m and maximum M, which are both positive due to property
(a). Taking C=1/m and c = 1/M satisfies (2).
(2)
1vA
Norms Equivalence
|| || on S { :|| || 1}
|| |||| ||
B A A
B
A
m v M v v v
vm M
v
1|| ||
|| ||1 1 1
|| |||| ||
|| || || |||| ||
B
A
A
B
B BA
m v Mv
vM v mv v
vM m
xnxx1Show that
Matrix NormConsider the vector space M22 of 2 2 matrices. Let u and v defined as follows be arbitrary 2 2 matrices.
Prove that the following function is an inner product on M22.<u, v>= ae + bf + cg + dh
Determine the inner product of the matrices .
hg
fe
dc
bavu ,
Solution
Axiom 1:,<u, v>= ae + bf + cg + dh = ea + fb + gc + hd =<v, u>
Axiom 3: Let k be a scalar. Then
<ku, v>= kae + kbf + kcg + kdh = k(ae + bf + cg + dh) = k<u, v>
4)01()90()23()52( ,09
25
10
32
09
25 and
10
32
Consider the vector space M22 of 2 2 matrices. Let u and v defined as follows be arbitrary 2 2 matrices.
It is known that the function <u, v>= ae + bf + cg + dh is an inner product on M22 by Example 2.
The norm of the matrix is
hg
fe
dc
bavu ,
Matrix Norm
2 2 2 2, a b c d || u || u u
This norm is called Frobenius norm, denoted by || ||F.
Remember that it is generated from an inner product.We can define other norms in matrix spacesShow that
2|| || ( ) ( ) sum of eigen values of /T T T T
FA tr A A tr AA A A AA
Matrix Norms
Given ANY vector norm
nnC
on||||a CORRESPONDING matrix norm on
we can construct
}1||||:||||max{|||| vAvA
by
nC
Theorem .
0vProof. If
nnn CvCAvAAv ,||,||||||||||
then ||||||||0||0|||||| vAAv
If 0v then ||||||
||||||||||||
||||||||||
v
vAv
v
vvAAv
||||||||||||||||||
|| vAvv
vA
From this theorem you show that A is always continuous.
Matrix Norms
nnCBA ,
|||||||||||| BABA
thenTheorem 2 If
Proofs|||||||||||| BABA
|||||||||||||| BABA
||||||||||||max||||max||)(||max||||1||||1||||1||||
BABvAvvBABAvvv
)2(||||||||||||,|||||||||||| AABBBBAA
(1)
(2)
(3)
||||||||||)(||max||||1||||
BAvABABv
Examples of Induced Matrix Norms
}||{max||||11
n
j ijni aATheorem 2
i
Proof.
Choose
hencenjau ijj ,...,1),sgn(
then constructthat maximizes
n
j ija1
||
|| || 1v
by
}||{max||111
n
j ijni
n
j ij aa
1|||| unCu
}||{max||||11
n
j ijni aAv
hence }||{max||||11
n
j ijni aA
and n
j jiji uaAuAu1
|||)(|||||
pxp
p
xp
Axx
AxA
p 10supsup
For instance,
m
iij
ja
11
maxA
m
jij
ia
1maxA
(max column sum) (max row sum)
21
max2)]([ AAA T AAT of eigenvalue maximum
Examples of Induced Matrix Norms
There are bridges between matrix induced norm :
1111
2
2
2
21
1
)(
1)(
AAvAAA
AvAA
vvvAA
T
T
T
Proof
AAA12
Fréchet Derivative of f:Rn→Rm
Definition
Fréchet Derivative of f:Rn→Rm
Theorem 1
Fréchet Derivative of f:Rn→Rm
Theorem 2
Fréchet Derivative of f:Rn→Rm
Theorem 2
Fréchet Derivative of f:Rn→Rm
Theorem 3
Fréchet Derivative of f:Rn→Rm
Theorem 4
Fréchet Derivative of f:Rn→Rm
Theorem 4
Fréchet Derivative of f:Rn→Rm
Fréchet Derivative of f:Rn→Rm
Theorem 4 shows that
2
Theorem 5: Identification Theorem
Fréchet Derivative of f:Rn→Rm
Sufficient Condition for Differentiability
No.
No.
No.
No.
Theorem 6
Process of generalization of differentiation
100
Thank You All