TOTAL AND DIRECTIONAL FRACTIONAL DERIVATIVES

Abstract: Vector calculus is an important subject in mathematics with applications in all areas of applied sciences. Till now researchers deal with the partial fractional derivative as the fractional derivative with respect to x, y,... . In this paper we shall define total and directional fractional derivative of functions of several variables, we set some basics about fractional vector calculus then we use our definition to modify the definition of conformal fractional derivative obtained by R. Khalil et al [6].


Introduction
The subject of fractional derivative is as old as calculus.The most popular definitions of fractional derivative are: (i) Riemann-Liouville definition [6]: If n is a positive integer and α ∈ [n − 1, n), the α th derivative of f given is by (ii) Caputo definition [5].If n is a positive integer and α ∈ [n − 1, n) , the α th derivative of f is All definitions of fractional derivatives do not satisfy the known product rule, quotient rule, chain rule.In [6] Khalil etall gave a new definition of fractional derivative "conformable fractional derivative" of a function f .This definition seems to be a natural extension of the usual definition of derivative.Many theorems which are proved using the classical definitions are still valid using the new definition as product rule, quotient rule, chain rule.Many authors used the new definition to solve a fractional differential equations as in [3] and [4] , since the computation using the new definition is more easier than using Riemann-Liouville or Caputo definition of fractional derivative.Thabet Abdeljawd in [1] define the left and right conformable fractional derivative, so the connection can be mad between the conformable fractional derivative and the classical definition.
In this paper we shall define the concepts of directional fractional derivative and total fractional derivative of functions of several variables, these definitions announce the born of fractional vector calculus.Also we set some basics about fractional vector calculus, beside we shall give a simple modification of the definition given in [6] as an application of the new definition.

Basics
The concept of conformable fractional derivative is recently introduced by R. Khalil etall in 2014 by imitating the usual definition of derivative.Definition 1. [6].Let f : [0, ∞) −→ R, t > 0, and α ∈ (0, 1].Then the fractional derivative of f of order α, T α or f (α) is defined as: If f is α− differentiable in some (0, a) , a > 0, and lim It is clear that T α (t p ) = pt p−α .Further this definition coincides with the classical definition of Riemann-Liouville definition and Caputo definition on polynomials ( up to constant multiple), also if α = 1 we have the classical definition of derivative.Definition 2. [6].Let α ∈ (n, n + 1) , and f be an n− differentiable function at t > 0.Then T α (f ) (t) is defined by where ⌈α⌉ is the smallest integer greater than or equal to α.

If, in addition
dt .In the following example the fractional derivatives of well known functions are given.

T α
Using the new definition of conformable fractional derivative, most of the important Theorems in calculus as Roll's theorem, Mean Value theorem and differentiability implies continuity still valid.

Directional Fractional Derivative
So far researchers deal with the partial fractional derivative as the fractional derivative with respect to x or y, etc.In this section we shall generalize the definition given in [6] ,to cover the concepts of directional fractional derivative.
Let f be any function on R n , with domain D f and u ∈ R n be any vector.Put u ⊥ = {v ∈ R n : v is orthogonal to u} and u be the Euclidean norm of u.Finally, let c • u be the dot product of c and u.By A\B we mean the points in the set A not in the set B. Definition 3. Let f : D f ⊆ R n → R m be any function, α ∈ (0, 1] , u ∈ R n be a non-zero vector and c in the interior of 1.If c ∈ D • f \u ⊥ and there exists a vector L α u ∈ R m that satisfies for each ∈> 0,there is a δ (∈) > 0 such that for all t ∈ R satisfying 0 < |t| < δ (∈) ,we have u is called the fractional directional derivative of f of order α in the direction of u at c and denoted by D α u f (c).If there is no such a vector then we say that f is not α−differentiable at c in the direction of u.
the limit exist.If the limit does not exist we say that f not α−differentiable at c in the direction of u.

Using the above definition it is clear that if
, if the limit exists.And f not α−differentiable at c in the direction of u if the limit does not exist.
Note that, since f is continuous the limit is independent of the choice of c n .
As consequence of the above definitions we have.

If the directional derivative of
. Now one can easily prove the following theorem.

Total Fractional Derivative
In this section we shall define total fractional derivative of functions on R n .Using the new definition the most important theorems of functions of several variables are still valid.
Definition 5. .[2].Let f : D f ⊆ R n → R m be any function, and c an interior point ofD f .We say that f is differentiable at c if there exist a linear mapT c : To define total fractional derivative of f, we replace v by |c • v| 1−α v. Definition 6. f : D f ⊆ R n → R m be any function, α ∈ (0, 1] and c an interior point of D f .We say that the α fractional derivative of f at c exist, if there exist a linear map T α c : R n → R m such that where lim Obviously ( * ) can be expressed more compactly by writing Alternatively ( * ) can be rephrased as.For any ∈> 0, there exist δ (∈) > 0 such that if v ∈ R n and v ≤ δ (∈) , then ( * * ) Such a linear map T α c is called the fractional total derivative of f of order α at c and we shall denote it by D α f (c) and D α f (c) (v) for the value of the linear map From the above definition easily one can show.
z z, so T α c (z) ≤∈ z for all ∈> 0. Hence T α c (z) = 0. Now one can easily prove the following lemma.
any function, α ∈ (0, 1) and c an interior point of D f .1.If f is linear then the fractional total derivative of f of order α at c is the function f.That is If the total derivative of f at c, Df (c) exists then Moreover we have the following.
It is known that a differentiable function is continuos.The following lemma show that this is still true if D α f (c) exist for some α ∈ (0, 1) , c ∈ D In particular f is continuous at c.
Proof.Since every linear map on a finite dimensional normed space is bounded, there exists a positive constant M such that T α c (u) ≤ M u .Now by Definition 3, it follows that for ∈= 1, there exist 0 < δ (1) such that for v ∈ R n with v < δ (1) .Then u , then, We know that if f is differentiable at c, then the directional derivative of f in the direction of u at c, D u f (c) exist and D u f (c) = Df (c) (u) .So we shall end this section with the following Theorem.
f , and u ∈ R n be a non zero vector.Then the fractional directional derivative of f of order α in the direction of u at c, D α u f (c) exist and Proof.For c ∈ u ⊥ , the result follows.Suppose c / ∈ u ⊥ .Since the fractional derivative of order α of f at c exist, then given ∈> 0, there exist 0 < δ (∈) , such that Let u be a non zero vector, thus if 0 < |t| ≤ δ(∈) u , we have This shows that D u f (c) exist and Proof.Let u = e i , by Theorem 3 . From the theory of functions of several variables for f : In this case we can represent y = f (x) by a system of m functions of n variables In the following theorem we shall show that a similar result is obtained if we use total fractional derivative.
Proof.Let us present y = f (x) by the system, Now D α f (c) maps the point (u 1 , u 2 , ...u n ) of R n into the point w = (w 1 , w 2 , ...w m ) of R m given by So the fractional derivative D α f (c) determined by the n × m matrix whose elements are This matrix is as usual called the fractional Jacobi matrix and denoted by J α f (c).

Application
In his paper [5] U. Katugampola, wrote " one of the limitation of this version of the fractional derivative is that it assumes that the variable t > 0. So the question is, wether we can relax this condition on a special class of functions?, if so, what it is?."Using our definition of total fractional derivatives we shall give a simple modification of the definition of conformable fractional derivative of f , in this modification we don't assume t > 0. Also we shall show that all the results obtained in [6] , are still valid using the modified conformable fractional derivative of f .
It is known that if f is a function of several variable, the directional derivative of f in the direction of a vector u, at a point c is defined by , so if we let u = 1 and c = x 0 , then we obtain f ′ (x 0 ) .Using this idea we can modify the definition given in [6] .
) and c a non zero interior point of D f .Let u = 1 in 3. Then we have, Definition 8. Let f : R −→ R be any function,and t ∈ R\ {0} , α ∈ (0, 1) .Then define the modified conformable fractional derivative of f of order α, T α or f α by.
entiable in some (−a, a) \ {0} , a > 0, and lim Clearly, if α = 1 the modified definition coincide with the classical definition of derivative.
2. Similarly D α j f (c) , D α k f (c) the modified partial conformable fractional derivative of f with respect to y, z respectively.Remark 2. Let {e 1 , e 2 , ..., e n } be the standard basis of R n .Then D α e i f (c) = D α i f (c) is the modified conformable fractional partial derivative of f with respect to the variable x i .Definition 9. Let α ∈ (1, ∞) , f : R −→ R be (⌈α⌉ − 1)differentiable at t ∈ R\ {0} ,where ⌈α⌉ is the smallest integer greater than or equal to α.Then the modified conformable fractional derivative of f of order α, T α or f α is defined by Remark 3. As a consequence of Definition 9 it is easy to show that for α ∈ (1, ∞).
Theorem 5.If f : R −→ R, if the modified conformable the fractional derivative of f of order α exists at t 0 ∈ R\ {0} , α ∈ (0, 1] , then f is continuous at t 0 Proof.We want to show that lim It is clear that if α ∈ (1, ∞) and T α (f ) (t) exists then f is continuous.The following theorem is an easy consequence of the modified definition.Theorem 6.
6. If, in addition f is differentiable, then T α (f ) (t) = |t| 1−α df dt .In the following example the fractional derivative of a well known functions are given using the modified definition.(ii) f is α−differentiable for some α ∈ (0, 1] . (iii) f (a) = f (b) .