IR is continuous in

Chapter 6 DIFFERENTIABLE MAPPINGS In this chapter we will generalize the notion of derivability

  known from high school for several variable functions. The notion of differentiability which we will study here will allow us to approximate the value of a function in a certain point with the value of a polynomial of first degree in that point, improving the known results from the continuous functions .

  p

  We remind that if

  f : A  IR a  , A

  → IR is continuous in then there exists a neighbourhood  for which for

  U V a f   x  f (a )

    (approximation with a constant). Here we will show that,

  x U A

  in some conditions, there is a first-degree polynomial of

  p variables for which ,   . T f   x  T   x x U A

  1 ₁

1. PARTIAL DERIVATIVES

  p

  . A point 

  Definition 1. Let A  IR a A is named interior point

  of the set

  A if there exists a sphere with the centre in a included in A, i.e.

   such that r  S   a,r  . A The set of the interior points of the set

  A is named the interior of the set A and is denoted Å or Int A. p

  is open if and only if

  Remark 1. The set A  IR A=Å. Indeed,

  if z is open, then for any   so that

  a A there exists r S   a,r  , A

  hence  Å ; from definition

  a is an interior point and consequently A

  1 results that Å 

  

  A, hence A=Å. Conversely, if Å=A, then for any a A

  there exists  so that

  r S   a,r  , hence A is an open set. A Remark 2. The interior of the set A is the largest open set

  included in

   A; therefore if G is an open set included in A then G Å.

  3

  2

  2

  2 ≤

   Example 1. Let A={(x,y,z) IR  x,y,z ≥ 0, x +y +z 1}.

  3

  2

  2

  2 Then Å ={(x,y,z)  IR  x,y,z > 0, x +y +z < 1}. p

  then Å  Proposition 1. If A  IR A' .

  Proof. Let  a Å. According to definition 1 results that there

  exists , hence

  r>0 so that S   a,r  ; then A S   a,r  A\   a  Ø a is an

  accumulation point of the set A.

   then there exists so that Corollary. If a Å   x  A\   a _n

   . This affirmation results from proposition 1 and lim x a _{n n}   proposition 5 from chapter 5.

  p q

  be a function and

  Definition 2. Let f : A  IR IR

   respectively the canonical bases of the

  B e ,...,e B  e  ,..., e  _p     _{1 p q 1 q} p, q

  . We say that the function linear spaces IR respectively IR f is

  

partially derivable according to (where ) in the point

x i i  

  1 , 2 ,...,p  aÅ if exists:

  1

  q

     . (*) lim [f(a te ) f(a)] _{t  i}

  IR _{t  IR} t This limit is named the partial derivative of the function

  f

according to (with respect to, or shortly, w.r.t) in the point a and is

x i

  denoted 

  f or  .

    a f   a _{x i}

  

  x _i

  If f has a partial derivative according to every variable we say that

  f is partially derivable in

a. If A=Å and f is partially derivable according

  to x in any point a  A we say that f is partially derivable according to

  i (on x A); in this case the function: i

  

  f _q

     

  IR  f :A , a f (a) _{x x i i}

  

  x _i

  is named partial derivative of the function

  f according to x . If f is i

  continuous on in any point 

  A, partially derivable according to x i a A

  1 ¹

  we say that on

f is of class C A and we write f  C (A) .

   Å , then exists  so that

Remark 1. If a r S   a,r  ; in this A

  case for , then the problem of determining the

  t  r, a  te  A _i partial derivative from (*) has sense.

  

  f

  1     , or,

  Remark 2. In case p

  1 ,   a lim  f  a t  f   a  _t  

  x t

  denoting x=a+t,  

  f f   x f   a

   ;

  a  

  lim _{x a}   

  x x a

  then in the case of the functions with a single variable the partial derivative coincides with the derivative and we will keep the

  df

   notations from high-school:  .

  

f   a   a

dx

  

2

In case Å  : p=2, if   a,b 

  IR

  1 

     

  f   a,b f(     a,b t _{x lim  } 1 , ) f   a,b _t  t

   1 f   x,b f   a,b

      

  lim f  a t,b  f   a,b lim _{t  x  a  t x  a} 

  t x a

  and, analogously

  f   a,y  f   a,b f  a,b  . _{y lim}   _{y b}  y  b

   , and  , then according to

  Remark 3. If f f ,...,f a Å   _{1 q}

  proposition 1, , and from theorem 7 (chapter 5) it

  a  a ,...,a  A'   _{1 p}

  results that

  f is partially derivable if and only if the q scalar

  components have partial derivatives in

  a; also:

    f 

   f  f q ₁ .

    a    a ,...,   a

     x  x  x _{i i i}

    If we denote then  if and only if and:

  

x  a  te t x  a

_{i i i i}

   

  f f a ,...,a ,x ,a ,...,a f a ,...,a _{j j  1 i  1 i i  1 p   j 1 p }

   .

    a lim _{x a i i} 

   x x  a _{i i i} Therefore, the partial derivative of the function in

  f a is the derivative j

  of the partial function according to , namely the derivative of a

  x i

  single variable function. Consequently from the partially derivability of the function

  f in a results its partial continuity in a. The partially

  derivability, when 

  p

  1 , doesn’t imply the continuity, as in the following example.

  Example 2. The function: _{2 2} xy

  

    , x y

   ^{2 2}

  2 f : IR IR, f   x,y  x  y

  

    , x y

   is discontinuous in (0,0) (see example 26, chapter 5) but is partial derivable, because:

  

  f x, f ,    

    

  f   , lim _{x x } x

  and _{2 2 2 2 2}

  y x  y 

  2 x y y y  x

     

f  x,y   , for x,y  , .

_x       _{2 2 2 2 2 2} x  y x  y

     

  Then _{2 2 2 2} 

  

  y y x    

if x y

   

  f _{2 2} ²

   ,

    x,y

   

  x y  

  

  x

  

  

if  

x y

   and analogously _{2 2 2 2}  

  x x y  

  

,x  y 

  

  f  _{2 2} ²

  

    x,y 

   x y

   

  

  y

  

,x y .

  

 

  

  2 ¹

  2 Therefore  ) and consequently ). f C (IR f  C (IR

  Example 3. Let

  2 ð , ñ,  ñ   . f:A     ,   ,

  2  

  IR f     cos ,ñ sin  Certainly . Because  ñ,    and

  f  C   A f     cos , sin  _ñ

   ñ,  ñ   it results that   . Therefore

  f      sin ,ñ cos  f , f  C   A _{ 1 ñ } . f  C   A p q

  and  . If

  Definition 3. Let f : A IR IR a Å f is partially derivable according to on sphere (i.e. x i S   a,r  A ,

   and  f   x ,  x  S    a,r f  is partially derivable according to x j in a _{x x i i} we say

   and f is partially derivable two times according to x i x j , hence

  exists:   1  f

  q lim a  te  f a  .

   

  IR

   j 

    _t 

  t  x _i

    We denote this limit (if exists) with ₂

  

  f

   or

  f   a   a _{x x i j}

   

  x x _{i j}

  and we call it the partial derivative of second order (derivative II) of ₂ 

  f

the function and in  and

f according x i x j

  a. If i j then   a

    ₂

  x x _{i j}

  

  f

  are called mixed derivatives according to and

    a x i x j in a.

   

  x x _{j i}

  If and the partial

  i=j we call the derivative II according to x i x j

derivative of second order (or derivative II) according to x in a and

i

  we denote ₂ 

  f , or  . ₂   a f _{x i} ^{2   a}

  

  x _i

  Analogously we define the derivatives of higher order. If

  A=Å,

f is continuous and admits partial derivatives of any order h  , n

n

  continuous on on _n A we say f is of class C A and we write .

  f  C   A _n 

  If for any on

  f  C   A nIN we say f is of class C A and we  write . f  C   A

  The partial derivative of order (if exists) is

   n according to x i

  denoted _n 

  f   n

  , or n , _n f _{x i} 

  x _i

  and the partial derivative of order

   m according to x (if exists) for the j

    n

  function we denote:

  f n _{x i n m} 

   f 

   n m  or . _{n m} f n m _{x x i j}

   x  x _{i j}

  Remark. For the single variable function derivable n times it

  is known, from high school, the derivation formula of the product (Leibniz Formula): _n

    n  _{k  n k    k}

    ,

    u v C u v _n  _{k }

  (0) where, from derivative of order 0, i.e. u , of the function u (or v) we

  understand function

  u (respectively v). As the partial derivatives are,

  actually, the derivatives of the partial functions, therefore derivatives of the single variable functions, results that the Leibniz formula is valid for products of scalar functions of several variables as well. ₁₃

   f for the function

  Example 4. Let us compute _{7 6}

   x  y _{x y}  .

  f    x,y  x  y  e   ⁷ Calculate first f ⁷ _{7 x} using Leibniz’ formula: ₇

    k

   f  _{k x y} ^{7  k  } _{ k}

    x,y  C  x  y  ^{7 k e } _{7 x 7 x}   

   x _{7 x  y k  6 x  y x  y} .

   C  x  y  e  C e  e  x  y  _{7 7} 7 

  Then: _{13 6 7 6}

    ^{6 k }

   f    f    k _{k x  y 6  k}

    x,y    x,y  C e   x  y 

  7  k  _{7 6 6 7  y 6 y}    

   x  y  y  x _{k }   _{x y}    _{1 x y x y}         .

  C e  x y ₆ 7  C e   ₆ 1 e  x y 1  Example 5. Let us calculate the mixed derivatives of second

  order for the function: _{3 2 2} 

  x y ,x  y 

  

  2 ²

²

. f : IR IR ,f   x,y   x y

  

  ,x  y 

   For we have:

      x,y  0, _{2 2 2 4 4 2 3}

  3 x y x  y  2 x y x y  3 x y

   

  

  f   x,y   _{x 2 2 2 2 2 2} x  y x  y     and _{/ 4 2 2 2 2 2 2 2 4 2 3}     

  x

  9 x y x y 4 y x y x y 3 x y

       

     

  

  f   x,y   f   x,y _{xy x y 2 2 4}

  

  x y _{6 4 2 2 4}   x 

  6 x y  3 x y ;

   _{2 2 3}

  x  y  

  analogously: _{' x  x  y   3 2 2 3 2 5 3 2} 2 x y x  x y

  f   x,y   _{y 2 2 2 2 2 2}  x  y   x  y 

  and _{4 2 2 2 2 2 2 2 5 3 2}      5 x

  3 x y x y 4 x x y x x y

       

   

    .

  f   x,y _{yx 2}

₂

₄ f   x,y _xy

  

  x y  

  For :

      x,y  0, f   x,  f   ,

  

  f   ,  lim  _{x x } x

  and  

  f   ,y  f   , _{x x}

  ,

  f    ,   _{xy lim y}  y f     , y  f , f    ,  lim  _{y y}  y

  and 

  

  f     x , f , _{y x}

      .

  f   , _{yx lim lim x  x }

  1

  x x

  Then: _{2 2}  f  f .

    x  y  y  x

  We remark that, if we pass to polar coordinates       , then:     x , y  ,  

  x cos , y sin  ₂

  and    ; hence , therefore

  f   x , y  lim f   x , y   f   , _{x , y} 

  2

  ); moreover:

  f  C ( IR _'

  2

  , hence  ) and      ( IR

  f  x , y  _{x x} 4 f C

  2

       , hence f   C ).

  f  x , y  _{y y} 2 ( IR

  But for mixed derivatives _{2 4} 1  6 m  3 m lim f    x , y  lim f   x , mx    lim f    x , y , _{xy xy yx 3 x x x}   ^{2 } _{y mx   y mx}

  

  1  m  hence doesn’t exist limit in origin, namely (0,0) is a point of _{2 2} discontinuity of the mixed derivatives; consequently .

  f  C (

  IR )

  2 As following, we will show that the functions of class have C

the mixed derivatives equal. For simplifying the notations we will

  show this property for  and  in general hypothesis.

  p

  2 q

  1

  2

  . If

  

Theorem 1. (Schwarz) Let f : A  IR  IR and  

  a, b  Å

  there exists an open neighbourhood 

  V V , there exists the partial (a,b)

  derivatives for any , and

  f    x , y , f    x , y , f    x , y   x , y  _{x y xy xy ''} V f  is

  continuous in , then there exists and

   

  a, b f   a , b _yx   . f   a , b  f   a , b _{xy yx}

• so that and

  Proof. Let

  

h, k IR  a  , h b  k  

  V

  (1)

  F   h , k  f  a  h , b  k    f a  h , b    f a , b  k     f a , b

  If we denote g x , y  f x , y  k  f x , y , then:

        .

  F   h , k  g  a  h , b     g a , b

  With respect to the definition of function g it results that:        (2)

  F   h , k hk f  a h , b k  _xy

  But

  f  is continuous in (a,b), hence the function _xy       (3)

    h , k  f  a  h , b  k   f   a , b _{xy xy}

  has, in the origin, the limit:

   . (4) _{h , k } lim   h , k 

  Let  . (5)

     h lim    h , k _k 

  Certainly, from (4) and (5) results that:

    . (6)

  lim   h _{h }

  1 Multiplying the equality (1) with , replacing F , h k from (2) and

   

k

        form (3) we obtain:

  f  a h , b k  _xy

  1        

  h f a , b h  h , k  f a h , b k f a h , b  _xy         k

  1    (7)

   f  a , b k  f   a , b  k

  Because on

  V there exists the partial derivative f  , passing to the _y

  limit when  in (7) and considering (5) we have:

  k

      ,

  h f   a , b  h   h  f  a  h , b   f   a , b _{xy y y}

  or:

  1        . (8)

  f a h , b f a , h f a , b  h        

   y y  xy h

  Passing at the limit in (8) for  , from (6) it results that

  h f    a , b _yx exists, and . f    a , b  f    a , b _{yx xy} p ²

  be an open set. If , then the

  Corollary. Let A  IR f  C   A

  mixed derivatives of second order are equal on ₂ A (their number being C for p2). _p

2. DIRECTIONAL DERIVATIVES

  We have noticed that the partial derivative of a function w.r.t. the variable assumes the existence of the limit (*) (definition

  x i

  2). In this context the vector

  p y

  tends to when  .

  a  te a IR t _i

  2 (a,b)

  For example, in , for the

  IR

  A

  derivative in w.r.t the variable

   

  a, b

  

  x,  a  t , b  tends (when t ) to x along a parallel line with Ox.

   

  a, b

  Let there be . Then the ( u , v )    , point of coordinates

   a  , tu b  tv  (a,b)

  tends to the point of coordinates (when  ) along the line

   

  a, b t (u,v) x

  parallel with the vector . We

  u  i v j

  will generalize the notion of partial derivative replacing the vector

  e _i

  from (*) with an arbitrary vector, introducing the concept of

  

(directional) derivative along a vector, concept that plays an

important role in the study of electromagnetic phenomena. p q

  a vectorial

  Definition 4. Let there be f : A  IR  IR _ p

  function, \{0} a vector. If there exists

  a  and A v  IR

  1 q   

  I R (**) lim f  a tv    f a

    _t  t _t  _{I R}

  we say that

  f is derivable at a along the vector v. The limit (**), if

  

  f

  exists, is denoted and is called the directional derivative of

    a

  

  v function f along the vector (with respect to (w.r.t.)) v at the point a.

  The directional derivative at a of the function

  f along the versor of

  1 the vector ) is also called the (directional)

  v (i.e. with respect to v

v

derivative of f in the direction v.

   

  f f   then  .

  Remark 1. If v e B   a   a _{i p}

   

  e x _{i i}

  or  1  , we

  

Remark 2. For p=q=1, denoting a  tv  x t  x a 

v

  remark that (**) becomes:  

  f f     x f a

     ,

  a lim v f a     _{x a} 

   

  v x a

  so, in this particular case, the derivability along a vector implies the derivability.

• Conversely, if , taking we

  f is derivable in a, and v IR vt  x  a

  have:  

  f   x f   a

  1

  1 1 f 

       ,

  

f   a lim lim f  a tv  f   a   a

  _{x a t}  

   

  x a t v v v _t  _{I R} so, derivability implies derivability along any non-zero vector.

  Example 6. The derivative along any non-zero vector v of a

  linear application is exactly the respective application calculated in

  p q p p

  ), and \{0} then

  v. Truly if fL(IR , IR a IR v IR

  

  f

  , for any  ,

  

f  a  tv   f   a  tf   v t IR. From (**) results   a f   v

  

  v p for any . a IR p  IR .

  Definition 5. Let there be x = (x 1 ,…, x p ), y = (y 1 ,…, y p )

  The real number _p     

  x , y x y x y ... x y _{k k 1}

_{1 p p}

 _k  ₁

  is named the scalar product of the vectors x and y.

  Remark. The scalar product has the following properties: ₂

p

  (P1)     and

  x , x x , x

  IR  x , x   x 

  p

  (P2) <   IR

  x,y>=<y,x>, x,y p

  (P3)      

  I R  

   x , y  x , y ,  , x , y

  IR

  p

  (P4)         

  x y , z x , z y , z , x , y , z

  IR

  p

  (P5)  x , y   x  y ,  x , y  (the Cauchy-Schwarz

  IR inequality).

  p Theorem 2 (Riesz). Let fL(IR , IR). Then there exists a p

  unique such that:

  a IR p

  (a) , for any

  f   x  a x ,  x IR

  (b) , where A  a A is the matrix of the linear application f.

  

p

    . Then:

  Proof. Let x x ,..., x

  IR _{p p}   _{1 p}  

  ,

  f   x  f x e  x f   e  x , a  _{i i i i}

      _i   _{1 i 1}  

  p p

  where   . If exists   such

  a f   e ,..., f e

  IR b b ,..., b

  IR

  

     

_{1 p 1 p}

p

  that: , for any , then , for any

  f   x  b x ,  x IR  e , b  e , a  _{i i}

   , so hence

  i

  1 , p b  a ,  i _{i i} 1 , p b  and the point (a) is proved. a

  

  Of course A =a, so (b) is, also, proved. We will give a set of sufficient conditions for derivability w.r.t. a vector and the formula of the derivative w.r.t. a vector.

  Theorem 3 (Sufficient conditions for derivability). p q

  Let and  If exists a neighbourhood f : A  IR  IR a Å.

  V

  such that

  

V f has partial derivatives on V, continuous in a, then f

a

p

  is derivable w.r.t. any vector \{0} and: _p v IR  f  f

  a  a v ,     _i

  

   v   x _{i 1 i} where

  v = (v , … , v ).

  1 p

• such that . Then:

   

  Proof. Let t IR a tv

  V

f a  tv  f a  f a  tv , a  tv ,..., a  tv  f a , a  tv ,..., a  tv 

   

      _{1 1 2 2 p p 1 2 2 p p}

           f a , a tv , a tv ,..., a tv f a , a , a tv ,..., a tv ...

   ^{1 2 2 3 3 p p   1 2 3 3 p p }

     (1)

  f a , a ,..., a  , a tv f a , a ,..., a  , a  ^{1 2 p 1 p p   1 2 p 1 p }

  Knowing that f has partial derivatives on V, applying the theorem of Lagrange on the intervals , for the

  

a , a  tv , k 

  1 , p p differences

    _{k k k}

  from (1) it results the existence of the elements , such that:

  c    t  a , a  tv  , k  _{k k k k} 1 , p

  

  f

       

  f  a tv  f   a tv c   t , a tv ,..., a tv ₁  

₁

_{2 2 p p}

  

  x ₁ 

   f f

       

tv a tv , c   t , a tv ,..., a tv ... tv a , a , a ,..., a , c   t (2)

₂     _{1 1 2 3 3 p p p 1 2 3 p  1 p}

    x ₂ x ₂ From the hypothesis of the theorem we know that the partial

  derivatives are continuous in

  a; multiplying both members from (2)

  1 with and passing to limit with  , knowing that

  t t

   , it follows that:

  c   t a ,  k _{k k} 1 , p f

  1 f f    a  lim  f a  tv  f a   a v  ...  a v ,

  

         

_{1 p t} 

  v t x x    _{1 p} so

  f is derivable w.r.t. the vector v, and the formula of the directional derivative is proved.

p

  Definition 6. Let f : A IR  IR be a scalar function, partial

  derivable in

  aÅ. The vector: _p

      

  f f f

    

    a ,...,   a   a e _k 

      

  x x  x _{1 p k 1 k}

    is named the gradient of function , or

  f at a and is denoted grad f   a

   f   a (to be read “nabla”, or del applied to the function f in the point

  a”). If A=Å and f is partial derivable on A, the vectorial function: _p

   f p 

  grad f   f  e ,  f : A 

  IR , f (a ) _k a→

   _{k  1 x}  _k is called the gradient of function f, or, the nabla operator applied to f.

   f  f  f  f  f If  , , and if  , .

  p

  2  f  i  j p 3  f  i  j  k  x  y  x  y  z

  

p

Corollary 1. If f : A IR  IR verifies the hypothesis of

  

  f theorem 3, then    .

    a f ( a ), v

  

  v

p

₁

  and then

  Corollary 2. If A = Å 

  IR f  C ( A ) f is derivable 

  f w.r.t. any non-zero vector    . v from A and f , v

  

  v

  2 2 x  z x  z

  , . Let us

  Example 7. Let f : IR f   x , y  xe , ye

  → IR   calculate the derivative of

  f in (0,0) w.r.t. a versor v=(v 1 ,v 2 ) which

  forms an angle of 30  with the axis Ox.

  3

  1 Then    and    .

  v cos ₁ 30 v sin ₂

  30

  2

  2  

  3

  1 Therefore  . Moreover:

  v ,

   

  2

  2   _{x  y x  y x  y x  y}

     , 

  f    x , y

  1 x  e , ye f   x , y  xe ,  1  y  e _{x y}     and, according to theorem 3:

   f  f  f

  3

  1 ,  , v  , v  1 ,  , 1  v .

            _{1 2}

   v  x  y

  2

  2 Remark. The derivability of a function in a point w.r.t. any non-zero vector implies the continuity of it along any line which passes through the respective point. Nevertheless it does not imply the continuity for  .

  p

  1

2 Example 8. Let f : IR

  ₂ → IR, 

  , y  x  .

  f   x , y  y

   

  , y  ₂