Int. J. Anal. Appl. (2023), 21:44

Fractional Vector Cross Product in Euclidean 3-Space

Manisha M. Kankarej1,∗, Jai P. Singh2

1Rochester Institute of Technology, Dubai, UAE
2B. S. N. V. P. G. College, Lucknow University, India

∗Corresponding author: manisha.kankarej@gmail.com

Abstract. In this research a new definition of fractional vector cross product of two vectors in Euclidean

3-space is presented. Using this new definition the formulae for Euclidean norm, fractional triple vector

cross product, curl and divergence of fractional vector cross product of an electromagnetic vector field

is discussed. In particular cases, all the properties satisfy definition of standard vector cross product

for standard orthogonal basis of R3. The new definition has application in radiation characteristic in

micro-strip antenna and can be applied in other branches of mathematics and physics.

1. Introduction

In 1967, Crowe [1] presented vector analysis. In 2013, Das [2] defined fractional vector cross

product and derived fractional curl. He further defined the formulations that were useful in applications

in electrodynamics, elastodynamics, fluid flow etc. Mishra et. al. [4] gave a simulation approach and

applied fractional vector cross product in radiation characteristic in micro-strip antenna. Fractional

vector cross product was defined by Tripathi and Kim [5] in 2022. In this paper we present new

definition of fractional vector cross product in Euclidean 3-space different than in [5] and discussed

its properties further with respect to the new definition.

2. Fractional Vector Cross Product in Euclidean 3 - Space

Further to the study of fractional vector cross product in [5], we defined an alternative definition

below:

Received: Mar. 27, 2023.

2020 Mathematics Subject Classification. 15A30, 15A63, 26A99, 26B12.
Key words and phrases. vector cross product; fractional vector cross product; fractional vector triple cross product;

curl; divergence.

https://doi.org/10.28924/2291-8639-21-2023-44
ISSN: 2291-8639

© 2023 the author(s).

https://doi.org/10.28924/2291-8639-21-2023-44


2 Int. J. Anal. Appl. (2023), 21:44

Definition: Let R3 be the Euclidean 3-space equipped with standard inner product 〈., .〉. Let
(e1,e2,e3) be standard orthonormal basis of R3 and β ∈ [0,1] a real number. Then, for vectors
a = a1e1 +a2e2 +a3e3, b = b1e1 +b2e2 +b3e3 in R3, the β − f ractional vector cross product
is defined by

a×β b =
{
(a2b3 −a3b2)sin

(βπ
2

)
+(b2 +b3)a1cos

(βπ
2

)}
e1

+
{
(a3b1 −a1b3)sin

(βπ
2

)
+(b3 +b1)a2cos

(βπ
2

)}
e2

+
{
(a1b2 −a2b1)sin

(βπ
2

)
+(b1 +b2)a3cos

(βπ
2

)}
e3

(2.1)

From eqn (2.1) we have,

ei ×β ej =cos
(βπ
2

)
ei +sin

(βπ
2

)
ek (2.2)

ej ×β ei =cos
(βπ
2

)
ej − sin

(βπ
2

)
ek (2.3)

el ×β el =0 f or l = {1,2,3} (2.4)

where (i, j,k) is a cyclic permutation of (1, 2, 3). The equations (2.2), (2.3) and (2.4) are similar to

that in [4] and [5].

Matrix representation

By (2.2), (2.3), (2.4) and linearity we can write equation (2.1) as

a×β b =sin
(βπ
2

){
(a2b3 −a3b2)e1 +(a3b1 −a1b3)e2 +(a1b2 −a2b1)e3

}
+cos

(βπ
2

){
(b2 +b3)a1e1 +(b3 +b1)a2e2 +(b1 +b2)a3e3

} (2.5)
or

a×β b =sin
(βπ
2

){
(a2b3 −a3b2)e1 +(a3b1 −a1b3)e2 +(a1b2 −a2b1)e3

}
+cos

(βπ
2

)
(b1 +b2 +b3)a−cos

(βπ
2

)
(a1b1e1 +a2b2e2 +a3b3e3)

(2.6)

From [3] we have

a×β b =


0 a1cos
(
βπ
2

)
−a3 sin

(
βπ
2

)
a1cos

(
βπ
2

)
+a2 sin

(
βπ
2

)
a2cos

(
βπ
2

)
+a3 sin

(
βπ
2

)
0 a2cos

(
βπ
2

)
−a1 sin

(
βπ
2

)
a3cos

(
βπ
2

)
−a2 sin

(
βπ
2

)
a3cos

(
βπ
2

)
+a1 sin

(
βπ
2

)
0







b1

b2

b3




(2.7)



Int. J. Anal. Appl. (2023), 21:44 3

or

a×β b =




(b2 +b3)cos
(
βπ
2

)
b3 sin

(
βπ
2

)
−b2 sin

(
βπ
2

)
−b3 sin

(
βπ
2

)
(b3 +b1)cos

(
βπ
2

)
b1 sin

(
βπ
2

)
b2 sin

(
βπ
2

)
−b1 sin

(
βπ
2

)
(b1 +b2)cos

(
βπ
2

)







a1

a2

a3




(2.8)

Thus considering a ∈ R3 as a fixed column vector in R3, we see that a ×β b : R3 → R3 is a linear
function as given in eqn (2.7) and (2.8).

Particular case:

Case 1: The 0-fractional vector cross product is defined by:

a×0 b =(b2 +b3)a1e1 +(b3 +b1)a2e2 +(b1 +b2)a3e3

By eqn (2.7) we have, a×0 b =



0 a1 a1

a2 0 a2

a3 a3 0





b1

b2

b3




Example 1: Using eqn (2.2) we have

ei ×0 ej = ei,ej ×0 ei = ej,el ×0 el =0.

Case 2: The 1-fractional vector cross product is defined by:

a×b =(a2b3 −a3b2)e1 +(a3b1 −a1b3)e2 +(a1b2 −a2b1)e3.

By eqn (2.7) we have

a×b =

∣∣∣∣∣∣∣∣
e1 e2 e3

a1 a2 a3

b1 b2 b3

∣∣∣∣∣∣∣∣ =


0 −a3 a2
a3 0 −a1
−a2 a1 0





b1

b2

b3




Example 2: Using eqn (2.2) we have

ei ×ej = ek,ej ×ei =−ek,el ×el =0.

where (i, j,k) is a cyclic permutation of (1,2,3). Thus, 1-fractional vector cross product is simply

the standard vector cross product.



4 Int. J. Anal. Appl. (2023), 21:44

3. Properties of Fractional Vector Cross product

Theorem 3.1. The β - fractional vector cross product satisfies

a×β (b+c)= a×β b+a×β c (3.1)

(a+b)×β c = a×β c +b×β c (3.2)

(λa+µb)×β c = λa×β c +µb×β c (3.3)

Proof. From eqn (2.5), equations (3.1), (3.2) and (3.3) can be proved.

�

Particular case:

Case 1: The 0-fractional vector cross product satisfies following properties:

a×0 (b+c)= a×0 b+a×0 c

(a+b)×0 c = a×0 c +b×0 c

(λa+µb)×0 c = λa×0 c +µb×0 c

Case 2: The 1-fractional vector cross product behaves as standard cross product.

a× (b+c)= a×b+a×c

(a+b)×c = a×c +b×c

(λa+µb)×c = λa×c +µb×c

Theorem 3.2. The β - fractional vector cross product satisfies

〈a,a×β b〉=cos
(βπ
2

){
a21(b2 +b3)+a

2
2(b1 +b3)+a

2
3(b1 +b2)

}
(3.4)

〈a,a×β b〉=cos
(βπ
2

)
(b1 +b2 +b3)‖a‖2 −cos

(βπ
2

)
(a21b1 +a

2
2b2 +a

2
3b3) (3.5)

Proof. From eqns (2.5) and (2.6), equations (3.4) and (3.5) can be proved.

�



Int. J. Anal. Appl. (2023), 21:44 5

Particular case:

Case 1: The 0-fractional vector cross product satisfies following properties:

〈a,a×0 b〉=
{
a21(b2 +b3)+a

2
2(b1 +b3)+a

2
3(b1 +b2)

}
〈a,a×0 b〉=(b1 +b2 +b3)‖a‖2 − (a21b1 +a

2
2b2 +a

2
3b3)

Case 2: The 1-fractional vector cross product satisfies following properties:

〈a,a×b〉=0

Example 3: Using eqn (2.2) we have

〈ei,ei ×β ej〉=cos
(
βπ
2

)
For 0-fractional vector cross product 〈ei,ei ×0 ej〉=1

For 1-fractional vector product 〈ei,ei ×ej〉=0.

Theorem 3.3. The β - fractional vector cross product satisfies

〈b,a×β b〉=cos
(βπ
2

){
a1b1(b2 +b3)+a2b2(b1 +b3)+a3b3(b1 +b2)

}
(3.6)

〈b,a×β b〉=cos
(βπ
2

)
(b1 +b2 +b3)〈a,b〉−cos

(βπ
2

)
(a1b

2
1 +a2b

2
2 +a3b

2
3) (3.7)

Proof. From eqns (2.5) and (2.6), equations (3.6) and (3.7) can be proved.

�

Particular case:

Case 1: The 0-fractional vector cross product satisfies following properties:

〈b,a×0 b〉=
{
a1b1(b2 +b3)+a2b2(b1 +b3)+a3b3(b1 +b2)

}
〈a,a×0 b〉=(b1 +b2 +b3)〈a,b〉− (a1b21 +a2b

2
2 +a3b

2
3)



6 Int. J. Anal. Appl. (2023), 21:44

Case 2: The 1-fractional vector cross product satisfies following properties:

〈a,a×b〉=0

Example 4: Using eqn (2.2) we have

〈ej,ei ×β ej〉=0

Theorem 3.4. The β - fractional vector cross product satisfies

〈a+b,a×β b〉=cos
(βπ
2

){
(a1 +b1)(b2 +b3)a1

+(a2 +b2)(b1 +b3)a2 +(a3 +b3)(b1 +b2)a3
} (3.8)

〈a+b,b×β a〉=cos
(βπ
2

){
(a1 +b1)(a2 +a3)b1

+(a2 +b2)(a1 +a3)b2 +(a3 +b3)(a1 +a2)b3
} (3.9)

〈a+b,a×β b〉−〈a+b,b×β a〉=cos
(βπ
2

){
∣∣∣∣∣∣∣∣
a2 +b2 a3 +b3 a1 +b1

a1 a2 a3

b1 b2 b3

∣∣∣∣∣∣∣∣
−

∣∣∣∣∣∣∣∣
a3 +b3 a1 +b1 a2 +b2

a1 a2 a3

b1 b2 b3

∣∣∣∣∣∣∣∣
}

(3.10)

Proof. This theorem can be proved from eqn (3.6).

�

Particular case:

Case 1: The 0-fractional vector cross product satisfies following properties:

〈a+b,a×0 b〉=
{
(a1 +b1)(b2 +b3)+(a2 +b2)(b1 +b3)+(a3 +b3)(b1 +b2)

}
〈a+b,b×0 a〉=

{
(a1 +b1)(a2 +a3)b1 +(a2 +b2)(a1 +a3)b2 +(a3 +b3)(a1 +a2)b3

}
〈a+b,a×0 b〉−〈a+b,b×0 a〉

=
{
∣∣∣∣∣∣∣∣
a2 +b2 a3 +b3 a1 +b1

a1 a2 a3

b1 b2 b3

∣∣∣∣∣∣∣∣−
∣∣∣∣∣∣∣∣
a3 +b3 a1 +b1 a2 +b2

a1 a2 a3

b1 b2 b3

∣∣∣∣∣∣∣∣
}



Int. J. Anal. Appl. (2023), 21:44 7

Case 2: The 1-fractional vector cross product satisfies following properties:

〈a+b,a×b〉=0,〈a+b,b×a〉=0

〈a+b,a×b〉−〈a+b,b×a〉=0

Example 5: Using eqn (2) we have

〈ei +ej,ei ×β ej〉−〈ei +ej,ej ×β ei〉=(ei −ej)cos
(
βπ
2

)
For β =0,〈ei +ej,ei ×0 ej〉−〈ei +ej,ej ×0 ei〉= ei −ej

For β =1,〈ei +ej,ei ×ej〉−〈ei +ej,ej ×ei〉=0

Theorem 3.5. The β - fractional vector cross product satisfies

(a×β b)+(b×β a)= cos
(βπ
2

){
(b1 +b2 +b3)(a1e1 +a2e2 +a3e3)

+(a1 +a2 +a3)(b1e1 +b2e2 +b3e3)
}
−2cos

(βπ
2

)
(a1b1e1 +a2b2e2 +a3b3e3)

(3.11)

Proof. Using eqn (2.6) we can prove the theorem.

�

Particular case:

Case 1: The 0-fractional vector cross product satisfies following property:

(a×0 b)+(b×0 a)=
{
(b1 +b2 +b3)(a1e1 +a2e2 +a3e3)

+(a1 +a2 +a3)(b1e1 +b2e2 +b3e3)
}
−2(a1b1e1 +a2b2e2 +a3b3e3)

(3.12)

Case 2: The 1-fractional vector cross product satisfies following property:

(a×b)+(b×a)=0 (3.13)

Example 6: Using eqn (2.2) and (2.3) we have

(ei ×β ej)+(ej ×β ei)= cos
(
βπ
2

)
(ei +ej)

For β =0, (ei ×0 ej)+(ej ×0 ei)= ei +ej



8 Int. J. Anal. Appl. (2023), 21:44

For β =1, (ei ×ej)+(ej ×ei)=0

Theorem 3.6. The β - fractional vector cross product satisfies

‖a×β b‖2 =
{
‖a‖2‖b‖2 −〈a,b〉2

}
+
{
2b1b2(a1a2 +a

2
3)+2b1b3(a1a3 +a

2
2)+2b2b3(a2a3 +a

2
1)
}
cos
(βπ
2

)2

+

∣∣∣∣∣∣∣∣
a1(b3 +b3) a2(b1 +b3) a3(b1 +b2)

a1 a2 a3

b1 b2 b3

∣∣∣∣∣∣∣∣sin
(
βπ
) (3.14)

‖b×β a‖2 =
{
‖a‖2‖b‖2 −〈a,b〉2

}
+
{
2a1a2(b1b2 +b

2
3)+2a1a3(b1b3 +b

2
2)+2a2a3(b2b3 +b

2
1)
}
cos

(βπ
2

)2
+

∣∣∣∣∣∣∣∣
b1(a3 +a3) b2(a1 +a3) b3(a1 +a2)

a1 a2 a3

b1 b2 b3

∣∣∣∣∣∣∣∣sin
(
βπ
) (3.15)

Proof. Using theorem 3.2 we can prove the theorem. �

Particular case:

Case 1: The 0-fractional vector cross product satisfies following property:

‖a×0 b‖2 =
{{
‖a‖2‖b‖2 −〈a,b〉2

}
+2b1b2(a1a2 +a

2
3)+2b1b3(a1a3 +a

2
2)+2b2b3(a2a3 +a

2
1)
}

‖b×0 a‖2 =
{{
‖a‖2‖b‖2 −〈a,b〉2

}
+2a1a2(b1b2 +b

2
3)+2a1a3(b1b3 +b

2
2)+2a2a3(b2b3 +b

2
1)
}

Case 2: The 1-fractional vector cross product satisfies following property:

‖a×b‖2 =
{
‖a‖2‖b‖2 −〈a,b〉2

}
,‖b×a‖2 =

{
‖a‖2‖b‖2 −〈a,b〉2

}
Example 7: Using eqns ( 2.2) and (2.3) we have



Int. J. Anal. Appl. (2023), 21:44 9

‖ei ×β ej‖2 =cos
(
βπ
2

)2
,‖ej ×β ei‖2 = cos

(
βπ
2

)2
For β =0, ‖ei ×0 ej‖2 =1, ‖ej ×0 ei‖2 =1

For β =1, ‖ei ×ej‖2 =0, ‖ej ×ei‖2 =0

Theorem 3.7. The β - fractional vector cross product satisfies

〈a×β b,c〉=sin
(βπ
2

)
∣∣∣∣∣∣∣∣
c1 c2 c3

a1 a2 a3

b1 b2 b3

∣∣∣∣∣∣∣∣+cos
(βπ
2

)
(b1 +b2 +b3)〈a,c〉

−cos
(βπ
2

)
(a1b1c1 +a2b2c2 +a3b3c3)

(3.16)

Proof. Using eqn (3.4) we can prove the theorem. �

Particular case:

Case 1: The 0-fractional vector cross product satisfies following property:

〈a×0 b,c〉=(b1 +b2 +b3)〈a,c〉− (a1b1c1 +a2b2c2 +a3b3c3)

Case 2: The 1-fractional vector cross product satisfies following property:

〈a×b,c〉=

∣∣∣∣∣∣∣∣
c1 c2 c3

a1 a2 a3

b1 b2 b3

∣∣∣∣∣∣∣∣
Example 8: Using eqn (2.2) and (2.3) we have

〈ei ×β ej,ek〉=sin
(
βπ
2

)
For β =0, 〈ei ×0 ej,ek〉=0

For β =1, 〈ei ×ej,ek〉=1

Theorem 3.8. The β - fractional vector cross product satisfies



10 Int. J. Anal. Appl. (2023), 21:44

(a×β b)×β c =sin
(βπ
2

)2
(〈a,c〉b−〈b,c〉a)

+cos
(βπ
2

){
sin
(βπ
2
(b1 +b2 +b3)(a×c)

−sin
(βπ
2

)
(a1b1e1 +a2b2e2 +a3b3e3)×c +sin

(βπ
2

)
∣∣∣∣∣∣∣∣
1 1 1

a1 a2 a3

b1 b2 b3

∣∣∣∣∣∣∣∣c

+cos
(βπ
2

)(
(b2 +b3)a1 +(b1 +b3)a2 +(b1 +b2)a3

)
c − sin

(βπ
2

)
∣∣∣∣∣∣∣∣
c1e1 c2e2 c3e3

a1 a2 a3

b1 b2 b3

∣∣∣∣∣∣∣∣
−cos

(βπ
2

)(
(b2 +b3)a1c1e1 +(b1 +b3)a2c2e2 +(b1 +b2)a3c3e3

)}

(3.17)

Proof. For vectors c = c1e1+c2e2+c3e3 and d = d1e1+d2e2+d3e3 in R3 in view of eqn (2.6), we

have

d ×β c =sin
(βπ
2

){
(d2c3 −d3c2)e1 +(d3c1 −d1c3)e2 +(d1c2 −d2c1)e3

}
+cos

(βπ
2

){
(c2 +c3)d1e1 +(c3 +c1)d2e2 +(c1 +c2)d3e3

}
=sin

(βπ
2

){
(d2c3 −d3c2)e1 +(d3c1 −d1c3)e2 +(d1c2 −d2c1)e3

}
+cos

(βπ
2

)
(c1 +c2 +b3)d −cos

(βπ
2

)
(d1c1e1 +d2c2e2 +d3c3e3)

(3.18)

Using d = a×β b, then from eqn (2.1) we have,

d1 =
{
(a2b3 −a3b2)sin

(
βπ
2

)
+(b2 +b3)a1cos

(
βπ
2

)}
e1

d2 =
{
(a3b1 −a1b3)sin

(
βπ
2

)
+(b3 +b1)a2cos

(
βπ
2

)}
e2

d3 =
{
(a1b2 −a2b1)sin

(
βπ
2

)
+(b1 +b2)a3cos

(
βπ
2

)}
e3

Using above details with eqn (3.18), we can prove theorem 3.8. �

Particular case:

Case 1: The 0-fractional vector cross product satisfies following property:

(a×0 b)×0 c =
(
(b2 +b3)a1 +(b1 +b3)a2 +(b1 +b2)a3

)
c

−
(
(b2 +b3)a1c1e1 +(b1 +b3)a2c2e2 +(b1 +b2)a3c3e3

)}



Int. J. Anal. Appl. (2023), 21:44 11

Case 2: The 1-fractional vector cross product satisfies following property:

(a×b)×c = 〈a,c〉b−〈b,c〉a

Example 9: Using (2.2) we have

(ei ×β ej)×β ek =cos
(
βπ
2

)(
ei cos

(
βπ
2

)
− sin

(
βπ
2

)
ek
)

For β =0, (ei ×0 ej)×0 ek =
(
ei cos

(
βπ
2

)
For β =1, (ei ×ej)×ek =0

4. Divergence and Curl of Fractional Vector Cross Product

4.1. Curl of a Curl of Fractional Vector Cross Product. Using eqn (2.5) we have,

a×β b =sin
(βπ
2

)
∣∣∣∣∣∣∣∣
e1 e2 e3

a1 a2 a3

b1 b2 b3

∣∣∣∣∣∣∣∣+cos
(βπ
2

)


0 a1 a1

a2 0 a2

a3 a3 0





b1

b2

b3


 (4.1)

Alternatively

a×β b =sin
(βπ
2

)
∣∣∣∣∣∣∣∣
e1 e2 e3

a1 a2 a3

b1 b2 b3

∣∣∣∣∣∣∣∣
+cos

(βπ
2

)


(b2 +b3) 0 0

0 (b3 +b1) 0

0 0 (b1 +b2)





a1

a2

a3




(4.2)

From eqn (4.1) and [2] for a vector field F = F1e1 +F2e2 +F3e3 the fractional curl is:

∇×β F =sin
(βπ
2

)
∣∣∣∣∣∣∣∣∣∣∣∣∣

e1 e2 e3

∂β

∂x
∂β

∂y
∂β

∂z

F1 F2 F3

∣∣∣∣∣∣∣∣∣∣∣∣∣
+cos

(βπ
2

)



0 ∂
β

∂x
∂β

∂x

∂β

∂y
0 ∂

β

∂y

∂β

∂z
∂β

∂z
0







F1

F2

F3




(4.3)

Alternatively, from eqn (4.2) we have



12 Int. J. Anal. Appl. (2023), 21:44

a×β b =sin
(βπ
2

)
∣∣∣∣∣∣∣∣∣∣∣∣∣

e1 e2 e3

∂β

∂x
∂β

∂y
∂β

∂z

F1 F2 F3

∣∣∣∣∣∣∣∣∣∣∣∣∣
+cos

(βπ
2

)



(F2 +F3) 0 0

0 (F3 +F1) 0

0 0 (F1 +F2)







∂β

∂x

∂β

∂y

∂β

∂z




(4.4)

By theorem (3.8) we have,

a×β (b×β c)= sin
(βπ
2

)2
(b〈a,c〉−c〈a,b〉)+cos

(βπ
2

){
sin
(βπ
2
(c1 +c2 +c3)(a×b)

−sin
(βπ
2

)
(b1c1e1 +b2c2e2 +b3c3e3)×c +sin

(βπ
2

)
∣∣∣∣∣∣∣∣
1 1 1

b1 b2 b3

c1 c2 c3

∣∣∣∣∣∣∣∣c

+cos
(βπ
2

)
a
(
(c2 +c3)b1 +(c1 +c3)b2 +(c1 +c2)b3

)
− sin

(βπ
2

)
∣∣∣∣∣∣∣∣
a1e1 a2e2 a3e3

b1 b2 b3

c1 c2 c3

∣∣∣∣∣∣∣∣
−cos

(βπ
2

)(
(c2 +c3)a1b1e1 +(c1 +c3)a2b2e2 +(c1 +c2)a3b3e3

)}

(4.5)

From (4.3) and (4.5) we have

∇×β (∇×β F)= sin
(βπ
2

)2
(∇〈∇,F〉−F〈∇,∇〉)

+cos
(βπ
2

){
sin
(βπ
2

)
(∇×∇)(F1 +F2 +F3)

−sin
(βπ
2

)
(
∂β

∂x
F1e1 +

∂β

∂y
F2e2 +

∂β

∂z
F3e3)×F +sin

(βπ
2

)
∣∣∣∣∣∣∣∣∣∣∣∣∣

1 1 1

∂β

∂x
∂β

∂y
∂β

∂z

F1 F2 F3

∣∣∣∣∣∣∣∣∣∣∣∣∣
c

+cos
(βπ
2

)
∇
(∂β
∂x
(F2 +F3)+

∂β

∂y
(F1 +F3)+

∂β

∂z
(F1 +F2)

)
− sin

(βπ
2

)
∣∣∣∣∣∣∣∣∣∣∣∣∣

∂β

∂x
e1

∂β

∂y
e2

∂β

∂z
e3

∂β

∂x
∂β

∂y
∂β

∂z

F1 F2 F3

∣∣∣∣∣∣∣∣∣∣∣∣∣
−cos

(βπ
2

)(
(
∂β

∂x
)2(F2 +F3)e1 +(

∂β

∂y
)2(F1 +F3)e2 +(

∂β

∂z
)2(F1 +F2)e3

)}

(4.6)



Int. J. Anal. Appl. (2023), 21:44 13

Particular case:

Case 1: The 0-fractional vector cross product satisfies following property:

∇×0 (∇×0 F)=
{
∇
(∂0
∂x
(F2 +F3)+

∂0

∂y
(F1 +F3)+

∂0

∂z
(F1 +F2)

)
−
(
(
∂0

∂x
)2(F2 +F3)e1 +(

∂0

∂y
)2(F1 +F3)e2 +(

∂0

∂z
)2(F1 +F2)e3

)} (4.7)
Case 2: The 1-fractional vector cross product satisfies following property:

∇× (∇×F)=∇〈∇,F〉−F〈∇,∇〉 (4.8)

4.2. Divergence of a Curl of Fractional Vector Cross Product. Using eqn (2.5) and (4.3) we have,

∇.(∇×β F)=
(∂β
∂x
+
∂β

∂y
+
∂β

∂z

)
sin
(βπ
2

)
{
(
∂β

∂y
F3 −

∂β

∂z
F2)e1 +(

∂β

∂z
F1 −

∂β

∂x
F3)e2 +(

∂β

∂x
F2 −

∂β

∂y
F1)e3

}
+cos

(βπ
2

){∂β
∂x
(F2 +F3)e1 +

∂β

∂y
(F3 +F1)e2 +

∂β

∂z
(F1 +F2)e3

}
(4.9)

∇.(∇×β F)= cos
(βπ
2

){
(
∂β

∂x
)2(F2 +F3)e1 +(

∂β

∂y
)2(F3 +F1)e2

+(
∂β

∂z
)2(F1 +F2)e3

} (4.10)
Particular case:

Case 1: The 0-fractional vector cross product satisfies following property:

∇.(∇×0 F)=
{
(
∂0

∂x
)2(F2 +F3)e1 +(

∂0

∂y
)2(F3 +F1)e2 +(

∂0

∂z
)2(F1 +F2)e3

}
(4.11)

Case 2: The 1-fractional vector cross product satisfies following property:

∇.(∇×F)=0 (4.12)

4.3. Curl of a Divergence of Fractional Vector Cross Product.

∇.F =
∂β

∂x
F1 +

∂β

∂y
F2 +

∂β

∂z
F3 (4.13)

∇×β (∇.F)=
∂β

∂x
F1 +

∂β

∂y
F2 +

∂β

∂z
F3 (4.14)

Using eqn (4.3) we have,



14 Int. J. Anal. Appl. (2023), 21:44

∇×β (∇.F)= sin
(βπ
2

)
∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣

e1 e2 e3

∂β

∂x
∂β

∂y
∂β

∂z

∂β

∂x
F1

∂β

∂y
F2

∂β

∂z
F3

∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣
+cos

(βπ
2

)



0 ∂
β

∂x
∂β

∂x

∂β

∂y
0 ∂

β

∂y

∂β

∂z
∂β

∂z
0







∂β

∂x
F1

∂β

∂y
F2

∂β

∂z
F3




(4.15)

Particular case:

Case 1: The 0-fractional vector cross product satisfies following property:

∇×0 (∇.F)=




0 ∂
0

∂x
∂0

∂x

∂0

∂y
0 ∂

0

∂y

∂0

∂z
∂0

∂z
0







∂0

∂x
F1

∂0

∂y
F2

∂0

∂z
F3



=0 (4.16)

Case 2: The 1-fractional vector cross product satisfies following property:

∇× (∇.F)=0 (4.17)

5. Conclusion

Fractional vector cross product is one of the important property in the study of fractional calculus. In

this research we have defined a new fractional vector cross product named as β fractional vector cross

product and further discussed properties of Euclidean norm, fractional triple vector cross product,

curl and divergence of fractional vector cross product. This definition is useful for applications in

electrodynamics, polarisation, impedance studies etc. where we can see real application of fractional

solution.

Conflicts of Interest: The authors declare that there are no conflicts of interest regarding the publi-

cation of this paper.

References

[1] M.J. Crowe, A History of Vector Analysis, the Evolution of the Idea of a Vectorial System, University of Notre

Dame Press, Notre Dame, 1967.

[2] S. Das, Geometrically Deriving Fractional Cross Product and Fractional Curl, Int. J. Math. Comput. 20 (2013),

1-29.

[3] R.A. Horn, C.R. Johnson, Matrix Analysis, Cambridge University Press, Cambridge, 1985.



Int. J. Anal. Appl. (2023), 21:44 15

[4] S. Mishra, R.K. Mishra, S. Patnaik, Fractional Cross Product Applied in Radiation Characteristic in Micro-Strip

Antenna: A Simulation Approach, Test Eng. Manage. 81 (2019), 1392-1401.

[5] M.M. Tripathi, J.R. Kim, Fractional Vector Cross Product, Pure Appl. Math. 29 (2022), 103-112. https://doi.

org/10.7468/JKSMEB.2022.29.1.103.

https://doi.org/10.7468/JKSMEB.2022.29.1.103
https://doi.org/10.7468/JKSMEB.2022.29.1.103

	1. Introduction
	2.  Fractional Vector Cross Product in Euclidean 3 - Space
	3. Properties of Fractional Vector Cross product
	4. Divergence and Curl of Fractional Vector Cross Product
	4.1. Curl of a Curl of Fractional Vector Cross Product
	4.2. Divergence of a Curl of Fractional Vector Cross Product
	4.3. Curl of a Divergence of Fractional Vector Cross Product

	5. Conclusion
	References