arXiv:0704.0747v1  [math.DG]  5 Apr 2007
Univ. Beograd. Publ. Elektrotehn. Fak.
Ser. Mat. 7 (1996), 105–109.
A NOTE ON HIGHER-ORDER
DIFFERENTIAL OPERATIONS
Branko J. Malešević
In this paper we consider successive iterations of the first-order differential
operations in space R
1. INTRODUCTION
Let C∞(R3) be the set of scalar functions f = f(x1, x2, x3) : R
3 7→ R which
have the continuous partial derivatives of the arbitrary order on coordinates xi (i =
1, 2, 3). Let ~C∞(R3) be the set vector functions ~f =
f1(x1, x2, x3), f2(x1, x2, x3),
f3(x1, x2, x3)
: R3 7→ R3 which have the coordinately continuous partial deriva-
tives of the arbitrary order on coordinates xi (i = 1, 2, 3). First-order differential
operations of the vector analysis of the space R3 are defined on the following set
of functions:
f : R3 7→ R | f ∈ C∞(R3)
and ~F =
~f : R3 7→ R3 | ~f ∈ ~C∞(R3)
First-order differential operations of the vector analysis of the space R3 are
defined as the following three linear operations [1], denoted here by ∇1,∇2 and ∇3
for a convenience:
(1) grad f = ∇1f =
~e1 +
~e2 +
~e3 : F 7→ ~F ,
(2) curl ~f = ∇2 ~f =
~e1 +
~e2 +
~e3 : ~F 7→ ~F ,
(3) div ~f = ∇3 ~f =
: ~F 7→ F.
Let Ω = {∇1,∇2,∇3} be the set of above defined operations and let Σ =
F ∪ ~F . Then the first-order differential operations can be considered as partial
operations Σ 7→ Σ, i.e. as operations whose domain (and codomain) are subsets F or
01991 Mathematics Subject Classification: 26B12
http://arxiv.org/abs/0704.0747v1
106 Branko J. Malešević
~F of Σ. Second and higher-order differential operations are then defined as products
of operations in Ω in the sense of composition of operations. Some of these products
might be meaningful, like ∇3 ◦ ∇1, while the others are meaningless, like ∇1 ◦ ∇1.
To all meaningless products for any argument we associate the value of nowhere
defined function ϑ (Dom (ϑ) = ∅ and Ran (ϑ) = ∅). Nowhere defined function ϑ(f∅)
is a concept from the recursive function theory [2]. We do not consider the function
ϑ as the starting argument for calculating the value of the higher-order differential
operations. In that way we increase set Σ into set Σ = F ∪ ~F ∪ {ϑ}.
All meaningful second-order differential operations are:
(4) ∆f = div grad f = (∇3 ◦ ∇1) (f),
(5) curl curl ~f = (∇2 ◦ ∇2) (~f),
(6) graddiv ~f = (∇1 ◦ ∇3)(~f),
(7) div curl ~f = (∇3 ◦ ∇2) (~f) = 0,
(8) curl grad f = (∇2 ◦ ∇1) (f) = ~0, f, ~f ∈ Σ \ {ϑ}.
In this paper we consider higher-order differential operations, search for mean-
ingful ones and present some applications.
2. HIGHER-ORDER DIFFERENTIAL OPERATIONS
Theorem 1. For arbitrary operations ∇i,∇j ,∇k ∈ Ω (i, j, k ∈ {1, 2, 3}) and
argument ξ ∈ Σ \ {ϑ} the associative law holds:
(9) ∇i ◦ (∇j ◦ ∇k)(ξ) = (∇i ◦ ∇j) ◦ ∇k(ξ).
Proof. Choosing the ∇i,∇j ,∇k from Ω and argument ξ from Σ\{ϑ}, (9) appears
in 54 possible cases. It is directly verified that whenever the left side of the equality
is meaningless, the right side is also meaningless. Than, all meaningless products
have the same value of the nowhere defined function ϑ, so that (9) is true in the
following form: ϑ = ϑ. Also, whenever the left side of equality is meaningful,
the right side is also meaningful. Then, according to the associative law of the
meaningful functions, we conclude that (9) is true.
From Theorem 1 it follows (by induction) that the generalized associative
law also holds, so we may write the product ∇i1 ◦ ∇i2 ◦ · · · ◦ ∇in without brackets
(ij ∈ {1, 2, 3} : j = 1, 2, ..., n).
For higher-order differential operations, given as meaningful products, we
say that they are the trivial products if they are trivially anullated, i.e. if they are
identically the same as the anullating functions 0, ~0 from Σ. Otherwise, we refer to
the higher-order differential operations, given as meaningful products, as nontrivial
products (if they are nontrivially anullated).
Next, we prove the statement:
A note on higher-order differential operations 107
Theorem 2. Higher-order differential operations appear as nontrivial products in
the following three forms:
(grad) div . . . graddiv grad f = (∇1◦)∇3 ◦ · · · ◦ ∇1 ◦ ∇3 ◦ ∇1f,
curl curl . . . curl curl curl ~f = ∇2 ◦ ∇2 ◦ · · · ◦ ∇2 ◦ ∇2 ◦ ∇2 ~f,
(div) grad . . . div graddiv ~f = (∇3◦)∇1 ◦ · · · ◦ ∇3 ◦ ∇1 ◦ ∇3 ~f,
for arbitrary functions f, ~f ∈ Σ \ {ϑ}, where terms in brackets are included for odd
number of terms and are left out otherwise. All other meaningful operations are
identically zero in their domain.
Proof. Meaningful third-order differential operations appear in the form of eight
compositions as follows:
(10) graddiv gradf = ∇1 ◦ ∇3 ◦ ∇1f,
(11) curl curl curl ~f = ∇2 ◦ ∇2 ◦ ∇2 ~f,
(12) div graddiv ~f = ∇3 ◦ ∇1 ◦ ∇3 ~f,
(13) div curl curl ~f = ∇3 ◦ ∇2 ◦ ∇2 ~f = 0,
(14) div curl gradf = ∇3 ◦ ∇2 ◦ ∇1f = 0,
(15) curl curl grad f = ∇2 ◦ ∇2 ◦ ∇1f = ~0,
(16) curl graddiv ~f = ∇2 ◦ ∇1 ◦ ∇3 ~f = ~0,
(17) graddiv curl ~f = ∇1 ◦ ∇3 ◦ ∇2 ~f = ~0, f, ~f ∈ Σ \ {ϑ}.
Anullations of the operations (13)–(17) follow directly from the anullations (4)–(5).
The statement follows directly from the principle of mathematical induction by
means of using the general associative law and formulas (10)–(17).
For a given sequence of operations ∇i1 ,∇i2 , . . . ,∇in from the set Ω of func-
tions, let define the concept of the collection of functions as a subset of functions
Θ ⊆ Σ \ {ϑ} such that all functions ξ from Θ anullate the nontrivial product
∇i1 ◦ ∇i2 ◦ · · · ◦ ∇in (ξ).
Let us form some collections. Scalar functions f from Σ, such that ∆nf = 0
is true, define harmonic collection Hn of order n, as the form of the polyharmonic
functions. Let us notice that in the case of two dimensions there is a general form
of polyharmonic functions f as a solution of the equation ∆nf = 0, [3]. Vector
functions ~f from Σ, such that curln ~f = ~0 is true, define curling collection Cn of
order n.
We can remark that besides the total scalar operation ∆ : F 7→ F (partial
scalar operation ∆ : Σ 7→ Σ) we can also consider the total vector operation
~∆ : ~F 7→ ~F (partial vector operation ~∆ : Σ 7→ Σ) defined by:
(18) ~∆~f = (∆f1,∆f2,∆f3) = ∆f1 · ~e1 +∆f2 · ~e2 +∆f3 · ~e3.
108 Branko J. Malešević
Let set ~Hn be the sign for the vector functions ~f from Σ such that ~∆
n(~f) = ~0,
where ~∆n is iteration of order n of the vector operation ~∆ given by (18). The set
of vector harmonic functions ~Hn of order n, which is defined in such a way, is not
in the list of collections which appear in the previous theorem because it is not
obtained through the compositions of operations (1)–(3). For the set ~Hn we shall
keep the term collection.
Let us notice that for scalar polyharmonic collections, vector polyharmonic
collections and curling collections, related to the index-order, the following inclu-
sions hold:
(19) H ⊂ H2 ⊂ · · · ⊂ Hn−1 ⊂ Hn ⊂ · · · ,
(20) ~H ⊂ ~H2 ⊂ · · · ⊂ ~Hn−1 ⊂ ~Hn ⊂ · · · ,
(21) C ⊂ C2 ⊂ · · · ⊂ Cn−1 ⊂ Cn ⊂ · · · .
Let emphasize that all previous considerations can be transformed in three-
dimensional orthogonal curvilinear coordinate system by introducing of correspond-
ing presumptions for functions from the sets F, ~F and Lamé’s coefficients.
Finally, let state a few examples where scalar and vector polyharmonic col-
lections appear.
Example 1. All meaningful products of third-and-higher-order differential opera-
tions for vector functions ~f ∈ ~H and scalar functions f ∈ H are anullated.
For vector functions ~f ∈ ~H the following equation holds:
(22) curl curl ~f = graddiv ~f.
Hence, for f ∈ H and ~f ∈ ~H, on the basis of formulas (22) and (10)–(17) the
following is true:
graddiv gradf = grad (∆f) = ~0,
curl curl curl ~f = curl (graddiv) ~f = ~0,
div graddiv ~f = div (curl curl) ~f = 0.
Thus, all eight meaningful products of third-order differential operations are anul-
lated, so that the statement is true.
Example 2. If f ∈ Hn−1, then x · f ∈ Hn, n ≥ 2.
Let us notice that if f ∈ F, then x · f ∈ F. For an arbitrary scalar function f ∈ F
the following equation is directly verified:
∆(x · f) = 2∂f/∂x+ x ·∆(f).
Inductive generalization is the following equation:
∆n(x · f) = 2n · ∂
∆n−1(f)
/∂x+ x ·∆n(f).
Thus, for (n− 1)-harmonic function f ∈ Hn−1 the conclusion x · f ∈ Hn is true.
A note on higher-order differential operations 109
Example 3. If f ∈ Hn−1, then (x
2 + y2 + z2) · f ∈ Hn, n ≥ 2.
Let us notice that if f ∈ F, then (x2 + y2 + z2) · f ∈ F. For the arbitrary scalar
function f ∈ F the following equations are directly verified:
∆(x2 · f) = 2 · f + 4x · ∂f/∂x+ x2 ·∆(f),
∆2(x2 · f) = 8 · ∂2f/∂x2 + 8x · ∂
/∂x+ 4 ·∆(f) + x2 ·∆2(f).
Inductive generalization is the equation as follows:
∆n(x2 · f) = 4n(n− 1) · ∂2
∆n−2(f)
+ 4nx · ∂
∆n−1(f)
/∂x+ 2n ·∆n−1(f) + x2 ·∆n(f).
Thus, if f ∈ Hn−1, then (x
2 + y2 + z2) · f ∈ Hn.
Two previous examples are the generalizations of the corresponding problems con-
tained in [4].
Acknowledgement. I wish to express my gratitude to Professors M. Merkle,
I. Lazarević and D. Tošić who examined the first version of paper and gave me
their suggestions and some very useful remarks.
REFERENCES
1. M. L. Krasnov, A. I. Kiselev, G. I. Makarenko: Vector Analysis. Moscow 1981.
2. N. Cutland: Computability. Cambridge University Press, London 1980.
3. D. S. Mitrinović, J. D. Kečkić: Jednačine matematičke fizike. Beograd 1985.
4. D. S. Mitrinović, in association with P. M. Vasić: Diferencijalne jednačine, Novi
zbornik problema 4. Beograd 1986.
5. M. J. Crowe: A History of Vector Analysis. University of Notre Dame Press, London
1967.
Faculty of Electrical Engineering, (Received May 6, 1996)
University of Belgrade,
P.O.B 816, 11001 Belgrade,
Yugoslavia
malesevic@kiklop.etf.bg.ac.yu

  In this paper we consider successive iterations of the first-order
differential operations in space ${\bf R}^3.$

arXiv:0704.0747v1  [math.DG]  5 Apr 2007
Univ. Beograd. Publ. Elektrotehn. Fak.
Ser. Mat. 7 (1996), 105–109.
A NOTE ON HIGHER-ORDER
DIFFERENTIAL OPERATIONS
Branko J. Malešević
In this paper we consider successive iterations of the first-order differential
operations in space R
1. INTRODUCTION
Let C∞(R3) be the set of scalar functions f = f(x1, x2, x3) : R
3 7→ R which
have the continuous partial derivatives of the arbitrary order on coordinates xi (i =
1, 2, 3). Let ~C∞(R3) be the set vector functions ~f =
f1(x1, x2, x3), f2(x1, x2, x3),
f3(x1, x2, x3)
: R3 7→ R3 which have the coordinately continuous partial deriva-
tives of the arbitrary order on coordinates xi (i = 1, 2, 3). First-order differential
operations of the vector analysis of the space R3 are defined on the following set
of functions:
f : R3 7→ R | f ∈ C∞(R3)
and ~F =
~f : R3 7→ R3 | ~f ∈ ~C∞(R3)
First-order differential operations of the vector analysis of the space R3 are
defined as the following three linear operations [1], denoted here by ∇1,∇2 and ∇3
for a convenience:
(1) grad f = ∇1f =
~e1 +
~e2 +
~e3 : F 7→ ~F ,
(2) curl ~f = ∇2 ~f =
~e1 +
~e2 +
~e3 : ~F 7→ ~F ,
(3) div ~f = ∇3 ~f =
: ~F 7→ F.
Let Ω = {∇1,∇2,∇3} be the set of above defined operations and let Σ =
F ∪ ~F . Then the first-order differential operations can be considered as partial
operations Σ 7→ Σ, i.e. as operations whose domain (and codomain) are subsets F or
01991 Mathematics Subject Classification: 26B12
http://arxiv.org/abs/0704.0747v1
106 Branko J. Malešević
~F of Σ. Second and higher-order differential operations are then defined as products
of operations in Ω in the sense of composition of operations. Some of these products
might be meaningful, like ∇3 ◦ ∇1, while the others are meaningless, like ∇1 ◦ ∇1.
To all meaningless products for any argument we associate the value of nowhere
defined function ϑ (Dom (ϑ) = ∅ and Ran (ϑ) = ∅). Nowhere defined function ϑ(f∅)
is a concept from the recursive function theory [2]. We do not consider the function
ϑ as the starting argument for calculating the value of the higher-order differential
operations. In that way we increase set Σ into set Σ = F ∪ ~F ∪ {ϑ}.
All meaningful second-order differential operations are:
(4) ∆f = div grad f = (∇3 ◦ ∇1) (f),
(5) curl curl ~f = (∇2 ◦ ∇2) (~f),
(6) graddiv ~f = (∇1 ◦ ∇3)(~f),
(7) div curl ~f = (∇3 ◦ ∇2) (~f) = 0,
(8) curl grad f = (∇2 ◦ ∇1) (f) = ~0, f, ~f ∈ Σ \ {ϑ}.
In this paper we consider higher-order differential operations, search for mean-
ingful ones and present some applications.
2. HIGHER-ORDER DIFFERENTIAL OPERATIONS
Theorem 1. For arbitrary operations ∇i,∇j ,∇k ∈ Ω (i, j, k ∈ {1, 2, 3}) and
argument ξ ∈ Σ \ {ϑ} the associative law holds:
(9) ∇i ◦ (∇j ◦ ∇k)(ξ) = (∇i ◦ ∇j) ◦ ∇k(ξ).
Proof. Choosing the ∇i,∇j ,∇k from Ω and argument ξ from Σ\{ϑ}, (9) appears
in 54 possible cases. It is directly verified that whenever the left side of the equality
is meaningless, the right side is also meaningless. Than, all meaningless products
have the same value of the nowhere defined function ϑ, so that (9) is true in the
following form: ϑ = ϑ. Also, whenever the left side of equality is meaningful,
the right side is also meaningful. Then, according to the associative law of the
meaningful functions, we conclude that (9) is true.
From Theorem 1 it follows (by induction) that the generalized associative
law also holds, so we may write the product ∇i1 ◦ ∇i2 ◦ · · · ◦ ∇in without brackets
(ij ∈ {1, 2, 3} : j = 1, 2, ..., n).
For higher-order differential operations, given as meaningful products, we
say that they are the trivial products if they are trivially anullated, i.e. if they are
identically the same as the anullating functions 0, ~0 from Σ. Otherwise, we refer to
the higher-order differential operations, given as meaningful products, as nontrivial
products (if they are nontrivially anullated).
Next, we prove the statement:
A note on higher-order differential operations 107
Theorem 2. Higher-order differential operations appear as nontrivial products in
the following three forms:
(grad) div . . . graddiv grad f = (∇1◦)∇3 ◦ · · · ◦ ∇1 ◦ ∇3 ◦ ∇1f,
curl curl . . . curl curl curl ~f = ∇2 ◦ ∇2 ◦ · · · ◦ ∇2 ◦ ∇2 ◦ ∇2 ~f,
(div) grad . . . div graddiv ~f = (∇3◦)∇1 ◦ · · · ◦ ∇3 ◦ ∇1 ◦ ∇3 ~f,
for arbitrary functions f, ~f ∈ Σ \ {ϑ}, where terms in brackets are included for odd
number of terms and are left out otherwise. All other meaningful operations are
identically zero in their domain.
Proof. Meaningful third-order differential operations appear in the form of eight
compositions as follows:
(10) graddiv gradf = ∇1 ◦ ∇3 ◦ ∇1f,
(11) curl curl curl ~f = ∇2 ◦ ∇2 ◦ ∇2 ~f,
(12) div graddiv ~f = ∇3 ◦ ∇1 ◦ ∇3 ~f,
(13) div curl curl ~f = ∇3 ◦ ∇2 ◦ ∇2 ~f = 0,
(14) div curl gradf = ∇3 ◦ ∇2 ◦ ∇1f = 0,
(15) curl curl grad f = ∇2 ◦ ∇2 ◦ ∇1f = ~0,
(16) curl graddiv ~f = ∇2 ◦ ∇1 ◦ ∇3 ~f = ~0,
(17) graddiv curl ~f = ∇1 ◦ ∇3 ◦ ∇2 ~f = ~0, f, ~f ∈ Σ \ {ϑ}.
Anullations of the operations (13)–(17) follow directly from the anullations (4)–(5).
The statement follows directly from the principle of mathematical induction by
means of using the general associative law and formulas (10)–(17).
For a given sequence of operations ∇i1 ,∇i2 , . . . ,∇in from the set Ω of func-
tions, let define the concept of the collection of functions as a subset of functions
Θ ⊆ Σ \ {ϑ} such that all functions ξ from Θ anullate the nontrivial product
∇i1 ◦ ∇i2 ◦ · · · ◦ ∇in (ξ).
Let us form some collections. Scalar functions f from Σ, such that ∆nf = 0
is true, define harmonic collection Hn of order n, as the form of the polyharmonic
functions. Let us notice that in the case of two dimensions there is a general form
of polyharmonic functions f as a solution of the equation ∆nf = 0, [3]. Vector
functions ~f from Σ, such that curln ~f = ~0 is true, define curling collection Cn of
order n.
We can remark that besides the total scalar operation ∆ : F 7→ F (partial
scalar operation ∆ : Σ 7→ Σ) we can also consider the total vector operation
~∆ : ~F 7→ ~F (partial vector operation ~∆ : Σ 7→ Σ) defined by:
(18) ~∆~f = (∆f1,∆f2,∆f3) = ∆f1 · ~e1 +∆f2 · ~e2 +∆f3 · ~e3.
108 Branko J. Malešević
Let set ~Hn be the sign for the vector functions ~f from Σ such that ~∆
n(~f) = ~0,
where ~∆n is iteration of order n of the vector operation ~∆ given by (18). The set
of vector harmonic functions ~Hn of order n, which is defined in such a way, is not
in the list of collections which appear in the previous theorem because it is not
obtained through the compositions of operations (1)–(3). For the set ~Hn we shall
keep the term collection.
Let us notice that for scalar polyharmonic collections, vector polyharmonic
collections and curling collections, related to the index-order, the following inclu-
sions hold:
(19) H ⊂ H2 ⊂ · · · ⊂ Hn−1 ⊂ Hn ⊂ · · · ,
(20) ~H ⊂ ~H2 ⊂ · · · ⊂ ~Hn−1 ⊂ ~Hn ⊂ · · · ,
(21) C ⊂ C2 ⊂ · · · ⊂ Cn−1 ⊂ Cn ⊂ · · · .
Let emphasize that all previous considerations can be transformed in three-
dimensional orthogonal curvilinear coordinate system by introducing of correspond-
ing presumptions for functions from the sets F, ~F and Lamé’s coefficients.
Finally, let state a few examples where scalar and vector polyharmonic col-
lections appear.
Example 1. All meaningful products of third-and-higher-order differential opera-
tions for vector functions ~f ∈ ~H and scalar functions f ∈ H are anullated.
For vector functions ~f ∈ ~H the following equation holds:
(22) curl curl ~f = graddiv ~f.
Hence, for f ∈ H and ~f ∈ ~H, on the basis of formulas (22) and (10)–(17) the
following is true:
graddiv gradf = grad (∆f) = ~0,
curl curl curl ~f = curl (graddiv) ~f = ~0,
div graddiv ~f = div (curl curl) ~f = 0.
Thus, all eight meaningful products of third-order differential operations are anul-
lated, so that the statement is true.
Example 2. If f ∈ Hn−1, then x · f ∈ Hn, n ≥ 2.
Let us notice that if f ∈ F, then x · f ∈ F. For an arbitrary scalar function f ∈ F
the following equation is directly verified:
∆(x · f) = 2∂f/∂x+ x ·∆(f).
Inductive generalization is the following equation:
∆n(x · f) = 2n · ∂
∆n−1(f)
/∂x+ x ·∆n(f).
Thus, for (n− 1)-harmonic function f ∈ Hn−1 the conclusion x · f ∈ Hn is true.
A note on higher-order differential operations 109
Example 3. If f ∈ Hn−1, then (x
2 + y2 + z2) · f ∈ Hn, n ≥ 2.
Let us notice that if f ∈ F, then (x2 + y2 + z2) · f ∈ F. For the arbitrary scalar
function f ∈ F the following equations are directly verified:
∆(x2 · f) = 2 · f + 4x · ∂f/∂x+ x2 ·∆(f),
∆2(x2 · f) = 8 · ∂2f/∂x2 + 8x · ∂
/∂x+ 4 ·∆(f) + x2 ·∆2(f).
Inductive generalization is the equation as follows:
∆n(x2 · f) = 4n(n− 1) · ∂2
∆n−2(f)
+ 4nx · ∂
∆n−1(f)
/∂x+ 2n ·∆n−1(f) + x2 ·∆n(f).
Thus, if f ∈ Hn−1, then (x
2 + y2 + z2) · f ∈ Hn.
Two previous examples are the generalizations of the corresponding problems con-
tained in [4].
Acknowledgement. I wish to express my gratitude to Professors M. Merkle,
I. Lazarević and D. Tošić who examined the first version of paper and gave me
their suggestions and some very useful remarks.
REFERENCES
1. M. L. Krasnov, A. I. Kiselev, G. I. Makarenko: Vector Analysis. Moscow 1981.
2. N. Cutland: Computability. Cambridge University Press, London 1980.
3. D. S. Mitrinović, J. D. Kečkić: Jednačine matematičke fizike. Beograd 1985.
4. D. S. Mitrinović, in association with P. M. Vasić: Diferencijalne jednačine, Novi
zbornik problema 4. Beograd 1986.
5. M. J. Crowe: A History of Vector Analysis. University of Notre Dame Press, London
1967.
Faculty of Electrical Engineering, (Received May 6, 1996)
University of Belgrade,
P.O.B 816, 11001 Belgrade,
Yugoslavia
malesevic@kiklop.etf.bg.ac.yu
