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Triangular integrals for 2-, 3- and 4-variable functions
Advances in Difference Equations volume 2014, Article number: 89 (2014)
Abstract
The triangular integrals for 2-, 3- and 4-variable functions are respectively and precisely defined as the single limits of double, triple, and quadruple sums in detail. A corollary of the divergence theorem in each dimension is useful to determine the triangular integral value. The indices of the sequence of the integrand must coincide with those of the corresponding integral variable to calculate the correct triangular integral value. In a triangular triple integral, one kind of two sets of increments is inappropriate for the convergence of numerical values, but the other kind is able to calculate numerical values by a computer algebra system.
1 Introduction
The primary theme of this article is a double integral for a 2-variable function in a domain D on the 2D plane. A double integral is usually regarded as a rectangular double integral. The calculation process of the rectangular double integral [1, 2] is conventionally defined as the double limits at infinity of double independent sums, for and for , of rectangularly divided areas by
where and . On the other hand, a triangle mesh or triangular mesh is widely used in the computer graphics. In addition to introducing triangular elements in the finite element method [3], a combination of a triangular area method and double dependent series was applied to sweep all of the area [4]. Proenca̧ and Filipe showed the advantage of a triangular region in comparison with rectangular one for a finite area in real-time face detection. They only investigated a finite sum of finite triangular areas, but our theory of the triangular integral [5, 6] treats infinite sum of infinitesimal triangular areas. Moreover, it involves the total differential and the antisymmetric property [7]. The calculation process of the triangular double integral on the 2D plane, where triangular double integral is expressed as (2.36), has not been defined even in the previous article [6]. A corollary of the divergence theorem on the 2D plane is useful to determine the triangular double integral value. The indices of the sequence of the integrand must coincide with those of the corresponding integral variable to calculate the correct triangular integral value. The calculation process of the triangular double integral for a 2-variable function on the 2D plane is precisely defined as the single limit of double dependent sums by (2.30) in Definition 3. Applying Definition 3, it is able to calculate numerical values by a computer algebra system in Example 1.
The secondary theme of this article is a triple integral for a 3-variable function in a domain D in the 3D space. A triple integral is usually regarded as a rectangular triple integral. The calculation process of the rectangular triple integral is conventionally defined as the triple limits at infinities of triple independent sums, for and for and for , of rectangularly divided volumes by
where , , and . As shown in the previous article [6], a triangular triple integral can be expressed as (3.65). A corollary of the divergence theorem in the 3D space is useful to determine the triangular triple integral value. In this calculation process of the triangular triple integral, new difficulty has arisen. For the integrand of the divergence theorem in the 3D space, there two alternative ways of decomposition of two kinds of double sequences and for and and . One way is used in the previous article [6], the other way is used as (3.39) and (3.43) in this article. One kind of the two sets of increments and for and and used in the previous article [6] is inappropriate for convergence of integral values since it is unable to calculate numerical values by a computer algebra system in Example 2. However, the other kind of two sets of increments
-
1.
, derived from (3.39),
-
2.
, derived from (3.43)
for and and and used in this article is able to calculate numerical values by a computer algebra system in Example 2. We formulate the divergence theorem in the 3D space and related corollary based on the appropriate two sets of increments in this article. The calculation process of the triangular triple integral for a 3-variable function in the 3D space is precisely defined as the single limit of triple dependent sums by (3.52) in Definition 6.
The tertiary theme of this article is quadruple integral for a 4-variable function in a domain D in the 4D time-space. Quadruple integral is usually regarded as the rectangular quadruple integral. The calculation process of the rectangular quadruple integral is conventionally defined as the quadruple limits at infinities of quadruple independent sums, for and for and for and for , of rectangularly divided hyper-volumes by
where , , and . The triangular quadruple integral is expressed as (4.110). A corollary of the divergence theorem in the 4D time-space is useful to determine the triangular quadruple integral value. Increments in the 4D time-space are replaced for the convergence of the triangular integral value. One kind of six sets of increments , , , , and for and and and used in the previous article [6] is inappropriate since they are the extension of the inappropriate increments and for and and to in the 4D time-space. For the integrand of the divergence theorem in the 4D time-space, there are six alternative ways of decomposition of six kinds of triple sequences , , , , , and for and and and . Extending the appropriate set of the increments and for and and and in the 3D space to in the 4D time-space, another kind of six sets of increments
-
1.
, derived from (4.92),
-
2.
, derived from (4.97),
-
3.
, derived from (4.98),
-
4.
, derived from (4.99),
-
5.
, derived from (4.100),
-
6.
, derived from (4.101)
for and and and and is derived in this article. This kind of six sets of increments is used in Definition 9. We formulate the divergence theorem in the 4D time-space and related corollary based on the appropriate 6 sets of increments in this article. The calculation process of the triangular quadruple integral for a 4-variable function in the 4D time-space is precisely defined as the single limit of quadruple dependent sums by (4.111) in Definition 9.
This article is basically about the calculation processes of the triangular double, triple and quadruple integrals for 2-, 3- and 4-variable functions. This article also includes revisions of the divergence theorems and the related corollaries based on the appropriate increments of the double and the triple sequences in the calculation processes of the triangular triple and quadruple integrals for 3- and 4-variable functions.
This article is structured as follows. In Section 2, the divergence theorem of the triangular integral and a related corollary on the 2D plane are reviewed. The calculation process of the triangular double integral for a 2-variable function is precisely defined in detail. In Section 3, the divergence theorem of the triangular integral and a related corollary in the 3D space are revised based on the appropriate increments of the double sequence. The calculation process of the triangular triple integral for a 3-variable function is precisely defined in detail. In Section 4, the divergence theorem of the triangular integral and a related corollary in the 4D time-space are revised based on the appropriate increments of the triple sequence. The calculation process of the triangular quadruple integral for a 4-variable function is precisely defined in detail.
2 Triangular double integral on the 2D plane
One kind of combined and antisymmetric finite line element vectors on the 2D plane is reviewed in Section 2.1. The triangular double integral for a 2-variable function is shown in Section 2.2. Component representation and an example of it for a 2-variable function are shown in Section 2.3. In the following, the Cartesian coordinates are denoted and .
2.1 One kind of finite line element vector on the 2D plane
For triangular double integral, the following increments of single sequence of points on the 2D plane are introduced.
The increments of single sequence of points are denoted as follows:
for and .
The finite line element vector for and is introduced as
The antisymmetric symbol on the 2D plane is
Using the antisymmetric symbol in (2.3), the antisymmetric finite line element vector for and is introduced as
and expressed as
where the index is summed over .
For example, we consider the case that the boundary of the domain is an ellipse:
where and . The following is shown in The curl theorem of a triangular integral [5].
The Cartesian coordinates of the sequence of points for , and for on the ellipse (2.6) are respectively expressed as
where angular arithmetic sequences and are respectively
2.2 Triangular double integral for a 2-variable function
Assume that D is a domain and ∂D is the boundary of the domain on the 2D plane, expressed in the Cartesian coordinates . Let and be partially differentiable functions with respect to and in D. There is only kind of single sequence
for and . There are sets of possible partial increments for a 2-variable function.
The total increments of for and are denoted
The increments of for and are denoted
The sets of possible partial increments of for and are denoted
Lemma 1 Let be partially differentiable functions with respect to for . The following holds:
for and , where the index is summed over .
Proof The proof of this lemma was shown in the previous article [6]. □
Our triangular single and double integrals and the divergence theorem on the 2D plane are shown as follows.
Definition 1 The triangular line single integral on 2D plane is defined as
where ∂D is the boundary of a domain, is the antisymmetric infinitesimal line element vector and the index is summed over .
Definition 2 The triangular double integral for integrands of which are partial differentials on the 2D plane is defined as
where D is a domain and the indices are summed over .
The following proposition is necessary for the condition (2.25) in Theorem 1.
Proposition 1 Denote constants as and , then
holds, where the index is summed over .
Proof In the case of and , Definition 1 is reduced to
In the case of and , Definition 1 is reduced to
A linear combination of (2.20) and (2.21) is
□
We show that (2.24) holds for a closed curve in the following.
The sum of in (2.5) over for is
where the index is summed over . In the case of a closed curve, i.e., and , it satisfies
The following is the refined version of the theorem shown in The divergence theorem of a triangular integral [6].
Theorem 1 (The divergence theorem of the triangular integral on the 2D plane)
Assume that ∂D is a piecewise smooth curve of the equation and D is the region inside and on ∂D on the 2D plane, expressed in the Cartesian coordinates . Let for be a set of partially differentiable functions with respect to for in D, where . In the case of a closed integral path which satisfies
the divergence theorem of the triangular integral on the 2D plane holds:
where the indices are summed over .
Proof Combining (2.16) with (2.4) for the sum of , we obtain
where the indices are summed over . Using Proposition 1, (2.25) is rewritten as
where the index is summed over . The limit at infinity of (2.27) is expressed as (2.26) by Definitions 1 and 2 under the condition of a closed curve (2.28). □
The triangular double integral for a 2-variable function on the 2D plane by the infinitesimal area element of the triangular double integral on the 2D plane is given as
where D is a domain and the index is summed over .
The calculation process of the triangular double integral on the 2D plane is precisely defined as follows.
Let be a piecewise smooth function on the 2D plane, expressed in the Cartesian coordinates .
Definition 3 The triangular double integral for a 2-variable function on the 2D plane is defined as
for and as the indices of variables of function , where D is a domain and the index is summed over .
The following is a corollary of the divergence theorem on the 2D plane.
Corollary 1 (A corollary of Theorem 1)
Assume that ∂D is a piecewise smooth curve of the equation and D is the region inside and on ∂D on the 2D plane, expressed in the Cartesian coordinates . Let for be a set of partially differentiable functions with respect to for in D, where . In the case of
where is a 2-variable function and the index is summed over , the following holds:
Proof Substituting (2.26) and (2.29) into (2.31), we obtain
where the indices are summed over . Substituting (2.18) and (2.30) into (2.33), it is expressed as
where the indices are summed over . In order for (2.34) to hold for any value of integral variables, the following kind of formula for and and is required:
where the index is summed over .
On the 2D plane, holds for , where the index is summed over . We therefore obtain , where the indices are summed over . Multiplying by both sides of (2.35), it is reduced to the differential equation (2.32). □
2.3 Component representation and an example of it for a 2-variable function
In component representation, (2.29) is expressed as
Using (2.30) in Definition 3, each component of (2.36) is
In Corollary 1, (2.31) is expressed as
and (2.32) is expressed as
We show an example of Corollary 1 in the following.
Example 1 In the case of
Substituting (2.41) into (2.40), we obtain
The boundary of the domain is an ellipse (2.6).
1. The value of the left-hand side of (2.39) is
See (A.1) and (A.2) in Appendix 1 for calculations in detail.
2. The value of the right-hand side of (2.39) is
See (A.3), (A.4), (A.5), and (A.6) in Appendix 1 for calculations in detail.
We thus see the coincidence of the value of (2.43) and that of (2.44).
Equation (2.39) is verified also in the case of and .
We consider the following approximation formula of (2.39) for ,
In the case of , (2.45) coincides with (2.39). It is verified by (2.43) and (2.44) in Example 1. An approximation formula, see (2.39), for Example 1 is
The left- and the right-hand sides of (2.46), respectively expressed as L and R, are shown in Table 1 and plotted in Figure 1, where .
3 Triangular triple integral in the 3D space
Two kinds of combined and antisymmetric finite area element vectors in the 3D space are reviewed in Section 3.1. The triple integral for a 3-variable function is shown in Section 3.2. Component representation and an example of it for a 3-variable function are shown in Section 3.3. In the following, the Cartesian coordinates are denoted , , and .
3.1 Two kinds of finite area element vectors in the 3D space
1. For a triangular triple integral, the following increments of the double sequence of points in the 3D space are introduced.
The increments of the double sequence of points at are denoted as follows:
for and and .
The first combined finite area element vector for and and is introduced as
2. The increments of the double sequence of points at are denoted as follows:
for and and .
The second combined finite area element vector for and and is introduced as
The antisymmetric symbol in the 3D space is
Using the antisymmetric symbol in (3.11), the first antisymmetric finite area element vector and the second antisymmetric finite area element vector for and and are respectively introduced as
where the indices are summed over .
1. The first antisymmetric finite area element vector for and and is expressed as
where the indices are summed over . In detail, (3.14) for are respectively written as
2. The second antisymmetric finite area element vector for and and is expressed as
where the indices are summed over . In detail, the equations in (3.18) for are respectively written as
For example, we consider the case that the boundary of the domain is a sphere:
where . The following is shown in The divergence theorem of a triangular integral [6].
1. The Cartesian coordinates of the antisymmetric first finite area element vector for and and on the surface of the sphere (3.22) are respectively expressed as
where angular arithmetic sequences and are respectively
2. The Cartesian coordinates of the antisymmetric second finite area element vector for and and on the surface of the sphere (3.22) are respectively expressed as
where angular arithmetic sequences and are respectively
3.2 Triangular triple integral for a 3-variable function
Assume that D is a domain and ∂D is the boundary of the domain in the 3D space, expressed in the Cartesian coordinates . Let , , and be partially differentiable functions with respect to , , and in D. There are kinds of the double sequences
for and and . There are two alternatives for decomposition of two kinds of double sequences and for and .
As mentioned in the Introduction, the inappropriate two sets of increments, used in the previous article [6], are replaced by the appropriate kind of two sets of increments and for and and and to calculate the numerical values in Example 2.
The following formulae have been revised based on the appropriate set of increments. In order to prove Theorem 2, and for and and are respectively modified in Lemmata 2 and 3.
There are sets of possible partial increments for a 3-variable function.
1. The total increments of for and are denoted
The increments of for and are denoted
The sets of possible partial increments of for and are denoted
2. The total increments of for and and are denoted
The increments of for and and are denoted
The sets of possible partial increments of for and and are denoted
Lemma 2 In the case of for and and , the following holds:
where the index is summed over .
Proof Using (3.29) and (3.34), for and and is split into
1. Substituting (3.31), (3.32), and (3.33) into (3.29) for and , we obtain
where the index is summed over .
2. Substituting (3.36), (3.37), and (3.38) into (3.34) for and and , we obtain
where the index is summed over .
Substituting (3.41) and (3.42) into (3.40), we obtain (3.39). □
Lemma 3 In the case of for and and , the following holds:
where the index is summed over .
Proof In a similar manner as Lemma 2, we obtain (3.43). □
Our triangular double and triple integrals and the divergence theorem in the 3D space are shown as follows.
Definition 4 The triangular area double integral in the 3D space is defined as
where ∂D is the boundary of a domain, is the antisymmetric infinitesimal area element vector and the index is summed over .
Definition 5 The triangular triple integral for integrands of which are partial differentials in the 3D space is defined as
where D is a domain and the indices are summed over .
The following proposition is necessary for the condition (3.47) in Theorem 2.
Proposition 2 Denote constants as , , and , then
holds, where the index is summed over .
Proof The proof of this proposition is shown in The divergence theorem of a triangular integral [6]. □
The following is the revised version of the theorem shown in The divergence theorem of a triangular integral [6].
Theorem 2 (The divergence theorem of the triangular integral in the 3D space)
Assume that D is a domain and ∂D is the boundary of the domain in the 3D space, expressed in the Cartesian coordinates . Let for be a set of partially differentiable functions with respect to for in D, where . In the case of a closed 2D surface which satisfies
the divergence theorem of the triangular integral in the 3D space holds:
where the indices are summed over and are constants.
Proof Combining (3.39) with (3.12) and combining (3.43) with (3.13) for and , we obtain
where the indices are summed over . Using Proposition 2, (3.47) is rewritten as
where the index is summed over . The limit at infinity of (3.49) is expressed as (3.48) by Definitions 4 and 5 under the condition of a closed surface (3.50). □
The triangular triple integral for a 3-variable function in the 3D space by the infinitesimal volume element is given by
where the index is summed over .
The calculation process of a triangular triple integral in the 3D space is precisely defined as follows.
Let be a piecewise smooth function in the 3D plane, expressed in the Cartesian coordinates .
Definition 6 The triangular triple integral for a 3-variable function in the 3D space is defined as
for , , , and as the indices of variable of function , where D is a domain and the index is summed over .
The revised corollary shown below derived from Theorem 2 is the 3D version of Corollary 1.
Corollary 2 (A corollary of Theorem 2)
Assume that D is a domain and ∂D is the boundary of the domain in the 3D space, expressed in the Cartesian coordinates . Let for be a set of partially differentiable functions with respect to for in D, where . In the case of
where is a 3-variable function and the index is summed over , the following holds:
Proof Substituting (3.48) and (3.51) into (3.53), it is rewritten as
where the indices are summed over . Substituting (3.45) and (3.52) into (3.55), it is expressed as
where the indices are summed over . In order for (3.56) to hold for any value of integral variables, the following kinds of formulae in two categories are required.
1. The formulae for and ,
where the index is summed over .
2. The formulae for and and ,
where the index is summed over .
In the 3D space, holds for , where the indices are summed over . We therefore obtain , where the indices are summed over . Multiplying by both sides of the four kinds of (3.57), (3.58), (3.59), and (3.60), they are reduced to the differential equation (3.54). □
3.3 Component representation and an example of it for a 3-variable function
The left-hand side of (3.44) is expressed as
where the index is summed over . Using (3.44) in Definition 4, each component of (3.61) is
In component representation, (3.51) is expressed as
Using (3.52) in Definition 6, each component of (3.65) is
In Corollary 2, (3.53) is expressed as
and (3.54) is expressed as
We show an example of Corollary 2 in the following.
Example 2 In the case of
Substituting (3.71) into (3.70), we obtain
The boundary of the domain is a sphere (3.22).
Since , , and , (3.71) satisfies the condition of a closed surface (3.50).
1. The value of the left-hand side of (3.69) is
See (B.3), (B.6), and (B.9) in Appendix 2 for calculations in detail.
The value of the right-hand side of (3.69) is
See (B.14), (B.19), and (B.24) in Appendix 2 for calculations in detail.
We thus see the coincidence of the value of (3.73) and that of (3.74).
We consider the following approximation formula of (3.69) for :
In the case of , (3.75) coincides with (3.69). It is verified by (3.73) and (3.74) in Example 2. An approximation formula, see (3.69), for Example 2 is
The left- and the right-hand sides of (3.76), respectively expressed as L and R, are shown in Table 2 and plotted in Figure 2, where .
4 Triangular integral in the 4D time-space
Six kinds of combined and antisymmetric finite hyper-surface element vectors in the 4D time-space are reviewed in Section 4.1. The triangular quadruple integral for a 4-variable function is shown in Section 4.2. In the following, the Cartesian coordinates are denoted , , , and .
4.1 Six kinds of finite hyper-surface element vectors in the 4D time-space
For the triangular quadruple integral, the following increments of a triple sequence of points in the 4D time-space are introduced.
1. The increments of a triple sequence of points at are denoted as follows:
for and and and .
The first combined finite hyper-surface element vector for and and and is introduced as
2. The increments of a triple sequence of points at are denoted as follows:
for and and and .
The second combined finite hyper-surface element vector for and and and is introduced as
3. The increments of a triple sequence of points at are denoted as follows:
for and and and .
The third combined finite hyper-surface element vector for and and and is introduced as
4. The increments of a triple sequence of points at are denoted as follows:
for and and and .
The fourth combined finite hyper-surface element vector for and and and is introduced as
5. The increments of a triple sequence of points at are denoted as follows:
for and and and .
The fifth combined finite hyper-surface element vector for and and and is introduced as
6. The increments of a triple sequence of points at are denoted as follows:
for and and and .
The sixth combined finite hyper-surface element vector for and and and is introduced as
The antisymmetric symbol in the 4D time-space is
Using the antisymmetric symbol in (4.61), the first antisymmetric finite hyper-surface element vector , the second antisymmetric finite hyper-surface element vector , the third antisymmetric finite hyper-surface element vector , the fourth antisymmetric finite hyper-surface element vector , the fifth antisymmetric finite hyper-surface element vector and the sixth antisymmetric finite hyper-surface element vector for and and and are respectively introduced as
where the indices are summed over .
4.2 Triangular quadruple integral for a 4-variable function
Assume that D is a domain and ∂D is the boundary of the domain in the 4D time-space, expressed in the Cartesian coordinates . Let , , , and be partially differentiable functions with respect to , , , and in D. There are kinds of triple sequences,
for and and and . There are six alternatives for decomposition of six kinds of triple sequences , , , , , and for and and and .
As mentioned in the Introduction, the inappropriate kind of six sets of increments, used in the previous article [6], is replaced by the appropriate kind of six sets of increments , , , , and for and and and and to calculate appropriate numerical values.
The following formulae have been revised based on the appropriate set of increments. In order to prove Theorem 3, , , , , , and for and and and are respectively modified in Lemmata 4, 5, 6, 7, 8, and 9.
There are sets of possible partial increments for a 4-variable function.
1. The total increments of for and are denoted
The increments of for and are denoted
The sets of possible partial increments of for and are denoted
2. The total increments of for and and are denoted
The increments of for and and are denoted
The sets of possible partial increments of for and and are denoted
3. The total increments of for and and and are denoted
The increments of for and and and are denoted
The sets of possible partial increments of for and and and are denoted
Lemma 4 In the case of for and and and , the following holds:
where the index is summed over .
Proof Using (4.74), (4.80), and (4.86), for and and and is split into
1. Substituting (4.76), (4.77), (4.78), and (4.79) into (4.74) for and , we obtain
where the index is summed over .
2. Substituting (4.82), (4.83), (4.84), and (4.85) into (4.80) for and and , we obtain
where the index is summed over .
3. Substituting (4.88), (4.89), (4.90), and (4.91) into (4.86) for and and and , we obtain
where the index is summed over .
Substituting (4.94), (4.95), and (4.96) into (4.93), we obtain (4.92). □
In a similar manner, we obtain the cases of , , , , and as follows.
Lemma 5 In the case of for and and and , the following holds:
where the index is summed over .
Lemma 6 In the case of for and and and , the following holds:
where the index is summed over .
Lemma 7 In the case of for and and and , the following holds:
where the index is summed over .
Lemma 8 In the case of for and and and , the following holds:
where the index is summed over .
Lemma 9 In the case of for and and and , the following holds:
where the index is summed over .
Our triangular triple and quadruple integrals and the divergence theorem in the 4D time-space are shown as follows.
Definition 7 The triangular hyper-surface triple integral in the 4D time-space is defined as
where ∂D is the boundary of a domain, is the antisymmetric infinitesimal hyper-surface element vector and the index is summed over .
Definition 8 The triangular quadruple integral for integrands of which are partial differentials in the 4D time-space is defined as
where D is a domain and the indices are summed over .
The following proposition is necessary for the condition (4.105) in Theorem 3.
Proposition 3 Denote constants as , , , and , then
holds, where the index is summed over .
Proof The proof of this proposition is shown in The divergence theorem of a triangular integral [6]. □
The following is the revised version of the theorem shown in The divergence theorem of a triangular integral [6].
Theorem 3 (The divergence theorem of the triangular integral in the 4D time-space)
Assume that D is a domain and ∂D is the boundary of the domain in the 4D time-space, expressed in the Cartesian coordinates . Let for be a set of partially differentiable functions with respect to for in D, where . In the case of a closed 3D hyper-surface which satisfies
the divergence theorem of the triangular integral in the 4D time-space holds:
where the indices are summed over and are constants.
Proof Combining (4.92) with (4.62), combining (4.97) with (4.63), combining (4.98) with (4.64), combining (4.99) with (4.65), combining (4.100) with (4.66) and combining (4.101) with (4.67) for and and , we obtain
where the indices are summed over . Using Proposition 3, (4.105) is rewritten as
where the index is summed over . The limit at infinity of (4.107) is expressed as (4.106) by Definitions 7 and 8 under the condition of a closed hyper-surface, (4.108). □
The triangular quadruple integral for a 4-variable function in the 4D time-space by the infinitesimal hyper-volume element is given as
where the index is summed over . In component representation, (4.109) is expressed as
The calculation process of the triangular quadruple integral in the 4D time-space is precisely defined as follows.
Let be a piecewise smooth function in the 4D time-space, expressed in the Cartesian coordinates .
Definition 9 The triangular quadruple integral for a 4-variable function in the 4D time-space is defined as
for , , , , , and as the indices of variable of function , where D is a domain and the index is summed over .
The revised corollary shown below derived from Theorem 3 is the 4D version of Corollary 2.
Corollary 3 (A corollary of Theorem 3)
Assume that D is a domain and ∂D is the boundary of the domain in the 4D time-space, expressed in the Cartesian coordinates . Let for be a set of partially differentiable functions with respect to for in D, where . In the case of
where is a 4-variable function and the index is summed over , the following holds:
Proof Substituting (4.106) and (4.109) into (4.112), it is rewritten as
where the indices are summed over . Substituting (4.103) and (4.111) into (4.114), it is expressed as
where the indices are summed over . In order for (4.115) to hold for any value of integral variables, the following kinds of formulae in three categories are required.
1. The formulae for and ,
where the index is summed over .
2. The formulae for and and ,
where the index is summed over .
3. The formulae for and and and ,
where the index is summed over .
In the 4D time-space, holds for , where the indices are summed over . We therefore obtain , where the indices are summed over . Multiplying by both sides of the 18 kinds of formulae (4.116), (4.117), (4.118), (4.119), (4.120), (4.121), (4.122), (4.123), (4.124), (4.125), (4.126), (4.127), (4.128), (4.129), (4.130), (4.131), (4.132), and (4.133), they are reduced to the differential equation (4.113). □
Appendix 1: Calculations for Example 1 in detail
The value of in (2.43) is
The value of in (2.43) is
The value of in (2.44) is
The value of in (2.44) is
The value of in (2.44) is
The value of in (2.44) is
Appendix 2: Calculations for Example 2 in detail
1. Calculation of (3.62) for Example 2 in detail.
Since and , the value of is
Since and , the value of is
Substituting (B.1) and (B.2) into (3.62), we obtain
2. Calculation of (3.63) for Example 2 in detail.
Since and , the value of is
Since and , the value of is
Substituting (B.4) and (B.5) into (3.63), we obtain
3. Calculation of (3.64) for Example 2 in detail.
The value of is
The value of is
Substituting (B.7) and (B.8) into (3.64), we obtain
4. Calculation of (3.66) for Example 2 in detail.
Since , the value of is
Since and , the value of is
Since and , the value of is
Since and , the value of is
Substituting (B.10), (B.11), (B.12), and (B.13) into (3.66), we obtain
5. Calculation of (3.67) for Example 2 in detail.
Since , the value of is
Since and , the value of is
Since , the value of is
Since and , the value of is
Substituting (B.15), (B.16), (B.17), and (B.18) into (3.67), we obtain
6. Calculation of (3.68) for Example 2 in detail.
Since , the value of is
The value of is
The value of is
Since , the value of is
Substituting (B.20), (B.21), (B.22), and (B.23) into (3.68), we obtain
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References
Lebesgue H: Intégrale, longueur, aire. Ann. Mat. Pura Appl. 1902, 7: 231–359. Thèses présentées à la Faculté des sciences de Paris pour obtenir le grade de Docteur ès sciences mathématiques. Bernardoni de C. Rebeschini, Milano (1902)
Lebesgue H: Leçon sur L’intégration et la Recherche des Fonctions Primitives. Gauthier-Villars, Paris; 1904.
Bramble JH, Zlámal M: Triangular elements in the finite element method. Math. Comput. 1970, 24(112):809–820. 10.1090/S0025-5718-1970-0282540-0
Proenca̧ H, Filipe S: Combining rectangular and triangular image regions to perform real-time face detection. Signal Processing, 2008. ICSP 2008. 9th International Conference on 2008, 903–908.
Tokunaga K: The curl theorem of a triangular integral. Adv. Differ. Equ. 2012., 2012: Article ID 23 10.1186/1687-1847-2012-23
Tokunaga K: The divergence theorem of a triangular integral. Adv. Differ. Equ. 2012., 2012: Article ID 168 10.1186/1687-1847-2012-168
Cartan É: Sur l’intégration des systèmes d’équations aux différentielles totales. Ann. Sci. Éc. Norm. Super. 1901, 18: 241–311.
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Tokunaga, K. Triangular integrals for 2-, 3- and 4-variable functions. Adv Differ Equ 2014, 89 (2014). https://doi.org/10.1186/1687-1847-2014-89
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DOI: https://doi.org/10.1186/1687-1847-2014-89