EVALUATION OF THE CONTENT OF BALL-BASED OBJECTS IN ℝ� Title: Evaluation of the Content of Ball-based Objects in ℝ� (by: Fareed Saiepour) Abstract In this text, the contents of n-dimensional ball-based objects are derived avoiding the usage of Gamma function and using an alternative method of double-factorial as well as integration. The focus is to consider nballs, n-cylinders and n-cones as n-manifolds in finite-dimensional Euclidean spaces. The wording: “ballbased” will refer to objects with an n-ball as a cross-section. Key words: n-ball, n-cylinder, n-cone, double-factorial, content, boundary set, n-manifold, hyper-coordinates, objects in Euclidean spaces. Fareed Saiepour, PI: W3583376 Department of Mathematics & Statistics The Open University frd_spr@hotmail.com (October 2012) [Subject Classification: 57N05,10,12,13,15,16,25] Content: 1. Preliminaries (Page 2) 2. The content of n-balls and their boundaries (Page 6) 3. The content of ball-based objects: elliptic-balls, cylinders & cones (Page 15) Bibliography (Page 22) 1 Copyright©2012Fareed Saiepour. EVALUATION OF THE CONTENT OF BALL-BASED OBJECTS IN ℝ� 1. Preliminaries Throughout this text, ℕ denotes the set of Natural numbers (Positive integers), ℝ and ℝ+ denote the sets of Real and Positive Real numbers, respectively and for each � ∈ ℕ, the n-dimensional Euclidean space [1] is denoted by ℝ� . Let � ∈ ℕ and suppose that {� , … , �� } is an orthonormal basis [2] for the space ℝ� . Any point � ∈ ℝ� , can be expressed by the n-tuple � , … , �� such that for each � ∈ { , … , �}, �� ∈ ℝ �� where each ℝ �� is a 1-dimensional subspace of the space ℝ� corresponding to the unit vector �� . In such a case, we write: ℝ� ≡ ℝ � , … , �� whenever we wish to express the space ℝ� in terms of its orthogonal [2] subspaces: ℝ � , … , ℝ �� . Consequently, the expression: ∈ ℝ� , defines the origin of the space ℝ� indicating that: , … , ∈ ℝ � , … , �� , where , … , is the n-tuple representing the origin 0. The space ℝ� is regarded as a normed space and for this purpose, for � ∈ ℝ� as introduced above, the usual ⁄ has been associated. Euclidean norm [3] ‖�‖ = ∑��= �� Definition 1.1 Let � ∈ ℕ. We call the set Ω an object in the space ℝ� if and only if Ω is a compact and pathwise connected subset of ℝ� . In such a case, the notation Ω ⊏ ℝ� will be used to serve the purpose. Remarks Referring to Heine-Borel Theorem [4] as discussed in Topology of Metric spaces, it must be clear that since ℝ� is a Euclidean space, it is a necessary and sufficient condition for a compact subset of ℝ� to be closed and bounded. Notation 1.2 Let Ω ⊏ ℝ� , for some � ∈ ℕ. From this point onwards, �Ω and dimΩ will be used to denote the boundary set and the topological dimension of the object Ω, respectively. Notation 1.3 Let � ∈ ℕ and ∈ ℝ+ . We denote the n-ball with radius r and centred at the origin by � . That is: � = {� ∈ ℝ� |‖�‖ ≤ }. Moreover, since � satisfies Definition 1.1 as an object in ℝ� , using Notation 1.2, we can now write: � ⊏ ℝ� . Considering the definition and notations above, it becomes consistent to write down the following expressions for the boundary set of � : � � = �{� ∈ ℝ� ∍ ‖�‖ ≤ } = {� ∈ ℝ� ∍ ‖�‖ = } ⊏ ℝ� , dim� � = dim � − =�− . Remarks When considering the space ℝ� , it is important to notice that ℝ�− is not a unique subspace of ℝ� . For example, if � = , then ℝ� = ℝ ≡ ℝ � , � , � . Consequently, the orthogonal basis {� , � , � } gives rise to three 2-dimensional subspaces ℝ � , � , ℝ � , � and ℝ � , � . Therefore ℝ is not unique and to avoid loss of generality, unless the basis is specified, the use of the term ℝ must be avoided. Precisely speaking, if ℝ ≡ ℝ � , � , � , then either ℝ ≡ ℝ � , � or ℝ ≡ ℝ � , � , or even ℝ ≡ ℝ � , � . Moreover, when ℝ is defined by its basis {� , � , � } and ⊏ ℝ , then although dim ∂ = 2, it is is not an object in any space of the type ℝ as indicated above . In ⊏ ℝ . But ∂ true to state: ∂ such a case, without having to be specific about ℝ , it is consistent to state that: ∂ ⋢ ℝ . In other words, 2 Copyright©2012Fareed Saiepour. EVALUATION OF THE CONTENT OF BALL-BASED OBJECTS IN ℝ� ∂ is not an object in ℝ . Furthermore, if ⊏ ℝ , we must be specific when stating . For + example: ⊏ ℝ � , � . Generally speaking, for each � ∈ ℕ and ∈ ℝ , suppose that: ℝ� ≡ � ℝ � , … , �� . If ⊏ ℝ� , then although dim� � = � − and � � ⊏ ℝ� , but: ∀ℝ�− ⊂ ℝ� , � � ⋢ ℝ� , in other words: ℝ�− ⊂ ℝ� ∍ � � ⊏ ℝ� . The intention in the next part of the introduction is to introduce the idea of the content of an object in the space ℝ� for any � ∈ ℕ. Let � ∈ ℕ and ∈ ℝ+ and suppose that: ℝ ≡ ℝ � , � . Recalling from Differential Geometry, if ⊏ ℝ , then r is a 2-manifold [5] and � a 1-manifold. Similarly, if ℝ ≡ ℝ � , � , � and ⊏ ℝ , then is a 3-manifold and ∂ a 2-manifold. On the other hand, [6] recalling from Measure Theory, the Lebesgue measure of objects which are 1, 2 and 3-manifolds is the length, area and volume of each of the objects, respectively. Of-course, the objects involved in this text are smooth manifolds and we will be dealing with higher dimensions as well as the first three. Generally speaking, the considered objects are n-manifolds such that � ∈ ℕ. Moreover for the sake of simplicity, the word “content” will be used instead of “Lebesgue measure”. For this purpose, we will draw our attention to the following notation: Notation 1.4 Let � ∈ ℕ and ℝ� ≡ ℝ � , … , �� . Also, suppose that: Ω ⊏ ℝ� . We denote the content of the object Ω by: | Ω |. Referring to the notation above, since the object Ω is necessarily a subset of the space ℝ� , it follows that: Ω ∈ � ℝ� where � ℝ� is the power-set [7] of ℝ� . In other words, � ℝ� is the set of all subsets of ℝ� which is � sometimes denoted by ℝ . This enables us to regard the content of objects in ℝ� as a positive real-valued function in the form below: | . | ∶ � ℝ� ⟶ ℝ+ Moreover, if we wish to use formal language based on mathematical logic, it is convenient to start by a universal quantifier as well as considering the space ℝ� as the universe of discourse, as below: ∀Ω ⊏ ℝ� , | Ω | ∈ ℝ+ Before completing the introduction by introducing “hyper-coordinate systems”, although up to this point the concept of content has been made clear to the reader, a visual example as a completion to the description of the idea would give a better perspective. Taking the comments made so far into account, an arbitrary 3-ball centred at the origin as well as reduced cases will be considered by the following examples: Examples Let ∈ ℝ+ and ℝ ≡ ℝ � , � , � . Also suppose that: ⊏ ℝ . In this case, is a 3manifold and ∂ which is the well-known 2-sphere with radius r and centred at the origin, a 2-manifold as the boundary set of . The following expressions for the contents of and its boundary which both satisfy the conditions of Definition 1.1 as objects in the space ℝ are correct: ∴ | dim = , | = volume{� ∈ ℝ ∍ ‖�‖ ≤ } = � 3 ∍� = � . Copyright©2012Fareed Saiepour. EVALUATION OF THE CONTENT OF BALL-BASED OBJECTS IN ℝ� ∴ |� dim� = , � � | = surface area{� ∈ ℝ ∍ ‖�‖ = } = volume{� ∈ ℝ ∍ ‖�‖ ≤ } = � � � � = � ∍� = . Considering the 2-dimensional analogy, we would have had: ℝ ≡ ℝ � , � discourse, r ⊏ ℝ and ∂ r ⊏ ℝ . It follows that: ∴ | ∴ |� dim = , | = area{� ∈ ℝ ∍ ‖�‖ ≤ } = � dim� | = perimeter{� ∈ ℝ ∍ ‖�‖ = } = ∍� =�. = , as the universe of ∍� =�. � � area{� ∈ ℝ ∍ ‖�‖ ≤ } = � � � = � Finally, reducing one more dimension to consider the case: � = , without further explanation, we proceed as below: ∴ | ∴ |∂ dim = , | = length{� ∈ ℝ ∍ ‖�‖ ≤ } = � ∍� = . dim ∂ = , � � | = cardinal{� ∈ ℝ ∍ ‖�‖ = } = length{� ∈ ℝ ∍ ‖�‖ ≤ } = � � � □ =� ∍� = Remarks In the example above, when considering the latter case � = , the object has been simplified to which is the closed interval [− , ] as a 1-manifold. It is important to note that since dim ∂ = , so ∂ is not an object. In fact, another reason for ∂ not being an object in ℝ (the real line), is that ∂ consists of a pair of antipodal points which are naturally distinct. That is:{− , }. Consequently, although ∂ is a compact subset of the real line, it is not pathwise connected and therefore does not meet all conditions of Definition 1.1. Referring to transfinite set theory, the term “cardinal” has been used to mean the number of elements in the given set (the size of the set). Precisely speaking, we have: cardinal{� ∈ ℝ ∍ ‖�‖ = } = cardinal{− , } = |{− , }| = . The introduction part of this text will be finalised by defining the hyper-coordinate system as a generalisation to the usual planar coordinate system. Using this system. We will evaluate the content of a number of significant objects in the space ℝ� . Definition 1.5 Let ℝ� ≡ ℝ � , … , �� for some � ∈ ℕ⁄{ } and suppose that �̅ represents the � − tuple � , … , �� . A hyper-coordinate system for the space ℝ� is a unique planar representation for this space which consists of a horizontal axis corresponding to the � − -dimensional subspace ℝ �̅ , namely the 4 Copyright©2012Fareed Saiepour. EVALUATION OF THE CONTENT OF BALL-BASED OBJECTS IN ℝ� hyper-axis and denoted by �̅ and a vertical axis which corresponds to the 1-dimensional subspace ℝ �� as an ordinary axis denoted by �� . Notation 1.6 Let ℝ� ≡ ℝ � , … , �� for some � ∈ ℕ⁄{ }. The expression ℝ� ≡ ℝ �̅, �� will be used whenever we need to express the space ℝ� in terms of its associated hyper-coordinated system. In such a case, we may also use ℝ� ≡ ℝ �̅ ×ℝ �� as a Cartesian product corresponding to the universe of discourse ℝ� . Examples Figure 1.1 describes the universe of discourse as ℝ ≡ ℝ �̅, � which indicates that the hyper-axis represents the subspace ℝ �̅ , horizontally and the ordinary axis the subspace ℝ � , vertically. The disc in this hyper-coordinate system represents the 3-ball of radius r and is centred at the origin, satisfying the expression: ⊏ ℝ . Furthermore, the horizontal closed interval [− , ] represents the 2-ball ⊏ ≤ }. On the other hand, the 1-ball ⊏ ℝ �̅ . That is the set: { � , � ∈ ℝ � , � ∍ � + � ≤ ℝ � has been represented by the vertical closed interval [− , ] indicated the set: { � ∈ ℝ � ∍ � }□ � � � �̅ − � − �̅ − − ������ . ������ . Figure 2.2 uses the same hyper-coordinate system. The boundary ∂ ⊏ ℝ ≡ ℝ �̅, � has been investigated. Following Notation 1.7, it becomes clear why for ∂ , the set {− , } has been used instead of the horizontal closed interval [− , ]: Notation 1.7 Let � ⊏ ℝ� ≡ ℝ �̅, �� , for some � ∈ ℕ⁄{ } and product” forms will be used for � and � � , respectively: � � � ≡ ≡� �− �− ⊚ ⊚ ∍( �− ∍ (� 5 �− ⊏ ℝ �̅ ⊏ ℝ �̅ ∈ ℝ+ . The following “rotational ⊏ ℝ �� ) , ⊏ ℝ �� ) . Copyright©2012Fareed Saiepour. EVALUATION OF THE CONTENT OF BALL-BASED OBJECTS IN ℝ� 2. The content of n-balls and their boundaries Let � ∈ ℕ. From this point onwards, unless specified, the space ℝ� will be regarded as the universe of discourse. In such a case, we assume that: ℝ� ≡ ℝ � , … , �� and whenever a hyper-coordinate system is involved, we have: ℝ� ≡ ℝ �̅, �� . The examples given after Notation 1.4 describe the contents of arbitrary n-balls for the first three dimensions, as below: � � ⊏ ℝ� . ∀� ∈ { , , }, ∀ ∈ ℝ+ , | � | = �� � ∍ � , � , � = ( , �, ) We also showed that: ∀� ∈ { , , }, ∀ ∈ ℝ+ , |� � |= � | � � | = ��� �− ∍ � ,� ,� = ( , �, � ). The aim in this chapter is to find formulas for | � | and |� � | for any � ∈ ℕ and ∈ ℝ+ . For this purpose, we will assume that: | � | = �� � and evaluate �� for all � ∈ ℕ. We start this procedure by introducing the following notation: Notation 2.1 For each � ∈ ℕ, � �⁄ denotes the Riemann integral: ∫−�⁄ cos � ���. Using the notation above, the following recursion as a lemma can be obtained: Lemma 2.2 Let � ∈ ℕ. The recursion below, holds: � ; = , , �⁄ �⁄ verified. For the rest of the proof, starting from �⁄ = ∫ cos �+ � �� = −�⁄ = + �+ = �+ = �+ �+ ∙ � and ∫−�⁄ cos ��� = �⁄ , the seed-value of the recursion above can be Proof From ∫−�⁄ cos��� = �+ �+ �⁄ and integrating by parts, we proceed as below: �+ ∫ cos �+ � � sin � = [cos�+ �] �=−�⁄ �⁄ �⁄ ∫ sin � cos � � �� −�⁄ ∫ cos � � − cos �+ � = � + −�⁄ Rearranging the latter result, we obtain: �+ = �+ ⁄ �+ Using the result above, we prove a finer recursion as below: 6 � − ∙ �⁄ �=−�⁄ �+ �, − �⁄ ∫ sin � � cos �+ � �=−�⁄ . which completes the proof ■ Copyright©2012Fareed Saiepour. EVALUATION OF THE CONTENT OF BALL-BASED OBJECTS IN ℝ� Lemma 2.3 Let � ∈ ℕ. The following recursion holds: = ; = �+ � ∙ �+ − � Proof The seed-value is simply a result of the previous lemma. For the rest of the proof, it suffices to use mathematical induction by verifying the base-case as below: � : [ � ∙ �+ � − ] �= � = ∙ − =�∙ = =[ �+ ] �= . Next by selecting an arbitrary � ∈ ℕ⁄{ }, we hypothesise � � to proceed through the inductive step, as below: � ∙ � − . � � : �+ = �+ To complete this step, we return to Lemma 2.2 as well as using the hypothesis to show that the proposition � � + is also true: − � �+ � � ∙ �+ − = ∙( ∙ � − ) = ∙ � = �+ . � � + : �+ = �+ �+ �+ �+ By the principle of mathematical induction, we conclude that the proposition � � is true for all � ∈ ℕ and the proof completes ■ To proceed further, for the sake of simplicity, we introduce another notation as below: Notation 2.4 Let , ∈ ℝ . For each � ∈ ℕ, we write: � = + − − ∙ − �, � ={ which is defined as below: ; � is odd, ; � is even. The notations and the proven recursions so far as well as recalling “double-factorial” [8] from combinatorics, enable us to prove the following lemma: Lemma 2.5 Let � ∈ ℕ. The following statement is true: � = �− ‼ ∙ �‼ � � Proof It suffices to use the method of induction, verifying the base-case as below: 7 Copyright©2012Fareed Saiepour. EVALUATION OF THE CONTENT OF BALL-BASED OBJECTS IN ℝ� � : [ �− ‼ ∙ �‼ � �] �= = ‼ ∙ ‼ � = [ �] = × = �= . To proceed through the inductive step by hypothesising � � for some � ∈ ℕ⁄{ }, we continue as follows: � � : � = �− ‼ ∙ �‼ � � . Referring to Lemma 2.3 and the hypothesis as well as the property: because of � ∙ �+ = Notation 2.4, this step also completes as we show that the proposition � � + is also true, as below: − � � �− ‼ � �‼ − � � + : �+ = = ∙ � ∙ ∙ � � ∙ = �+ �+ �‼ � � �+ �− ‼ �‼ = ∙ � �+ . �+ ‼ Following the latter obtained result for � � + , by the principle of mathematical induction we conclude that the proposition � � is true for all � ∈ ℕ which completes the proof ■ The next notation which is followed by a lemma giving another recursion, plays a key role in the evaluation of the content of n-balls: Notation 2.6 Let � ∈ ℕ. We use �� to denote the finite product: ∏��= Lemma 2.7 Let � ∈ ℕ. The following recursion holds: � = = ; ��+ = �+ � . �� Proof Using the latter notation, to verify the seed-value we simply write: � = ∏�= Without further explanation, the rest of the proof completes as below: �+ ��+ = ∏ �= � = �+ � ∏ �= � = �+ �� � = = . ■ Examples Using the recursion of Lemma 2.7, it is now a convenient task to evaluate �� for � ∈ { , , }. Starting from � = , we obtain values for � and � : � = � = � =( � =( ‼ ∙ ‼ ‼ ∙ ‼ 8 � )∙ � )∙� = = � ∙ =�, ∙� = � . Copyright©2012Fareed Saiepour. EVALUATION OF THE CONTENT OF BALL-BASED OBJECTS IN ℝ� It follows that: � , � , � = , �, �⁄ = � , � , � . Consequently: ∀� ∈ { , , }, ∀ ∈ ℝ+ , | � | = �� � . The following theorem which is fundamental to the theory, extends the consequence above to � ∈ ℕ ∶ Theorem 2.8 Let � ∈ ℕ, ∈ ℝ+ and suppose that: | � � | = �� � ⊏ ℝ� . The following statement is true: . Proof We use the method of induction by considering the space ℝ�+ ≡ ℝ �̅, ��+ in a hyper-coordinate system as the universe of discourse as well as the � + -ball: �+ ⊏ ℝ�+ , for an arbitrary � ∈ ℕ⁄{ } and ∈ ℝ+ , so that: � ⊏ ℝ �̅ �+ � ≡ ⊚ ∍ { ⊏ ℝ ��+ The assumption made above are shown in Figure 2.1, below: ��+ �+ ���+ �� � − �̅ rcos� − ������ . � : [| � |] �= =| |= =� = [�� �] . �= We continue by writing down the proposition � � as the corresponding hypothesis: 9 Copyright©2012Fareed Saiepour. EVALUATION OF THE CONTENT OF BALL-BASED OBJECTS IN ℝ� � � : ∀� ∈ ℕ { }, ∀ ∈ ℝ+ , | � | = �� � . Moreover, the differential object �� ⊏ ℝ�+ , has been displayed such that dim� = � + , satisfying the following expression which involves a Cartesian product for its content: Considering � | �+ ���+ cos� × ( )| = | � |��| = | � ���+ cos� | | ( )| . and � � , we can simplify |��|, by writing: |��| = �� cos� � ���+ . It follows that: |= ∫ |��| ℝ ��+1 = ∫ �� cos� � ���+ = − �⁄ = ∫ �� −�⁄ �+ ∫ ��+1 =− cos�+ ��� = �� �� cos� � � sin� �+ �⁄ ∫ cos �+ ��� = −�⁄ Referring to the recursion of Lemma 2.7, we deduce that the proposition � � + { }. That is: | = ��+ � � + : ∀� ∈ ℕ { }, ∀ ∈ ℝ+ , | �+ �+ �+ �� �+ . is also true for all � ∈ ℕ . Considering the base-case as well as the hypothesis, by the principle of mathematical induction, we conclude that the proposition � � is also true for all � ∈ ℕ and the proof completes ■ The next theorem describes a relationship between the content of n-balls and their boundary for any � ∈ ℕ and any radius: ∈ ℝ+ and suppose that: Theorem 2.9 Let � ∈ ℕ, |� � � | = ��� ⊏ ℝ� . The following statement is true: �− . Proof For the case: � = , by referring to the fundamental theorem of calculus, the result becomes obvious, as expressed below: [|� � |] �= = |� | = |{− , }| = = [��� �− ] . �= Suppose that � ∈ ℕ { }, and that: � ≡ �− ⊚ . The latter expression has been displayed by using a hyper-coordinate system shown in Figure 2.2. Moreover, the figure shows that for any � ∈ ℝ+ , we have: Δ � ⊏ ℝ� , such that Δ � is the set-difference of �+� and � . That is: ∀� ∈ ℝ+ , Δ � ⊏ ℝ� ∍ Δ 10 � = �+� � . Copyright©2012Fareed Saiepour. EVALUATION OF THE CONTENT OF BALL-BASED OBJECTS IN ℝ� +� � �� � r − − −� � Δ �̅ +� r − − −� ������ . Recalling from the theory of Lebesgue measures, since: Δ |Δ � |=ห �+� � � ห=ห � �+� = , it follows that: � ห−| |. Using the result above, by considering the definition of derivatives, we obtain a result by taking limits as below: |Δ � | ห �+� ห − | � | � � | | lim = lim = � � � �→ �→ On the other hand, as � → , the content of Δ � , tends to the area of a rectangle with length and width: |� � | and �, respectively. In terms of limits, Since: lim |Δ �→ it follows that: lim �→ Comparing the results and |Δ � � | � = lim �→ | = lim |� �→ |� � � |∙� � |∙� , = |� together, we deduce that: |� � � Theorem 8, the required result is achieved and the proof completes ■ 11 |= | � � | � | . Finally, by applying Copyright©2012Fareed Saiepour. EVALUATION OF THE CONTENT OF BALL-BASED OBJECTS IN ℝ� Examples To evaluate the contents of any unit 7-ball and its boundary, since an arbitrary n-dimensional Euclidean space is, also an Affine space [9] , it suffices to consider the unit 7-ball centred at the origin as an object in the space ℝ . That is: ⊏ ℝ . Consequently: � ⊏ ℝ . Firstly, applying the result of | by proceeding as below: Theorem 2.8, we evaluate | | |=[� ] = = = � =∏ =∏ � �= �= � � ‼ ‼ ‼ ‼ ‼ ‼ ‼ �− ‼ ‼ ∙ ∙ ∙ ∙ ∙ ∙ ∙∏ ∙ = ‼ ‼ ‼ ‼ ‼ ‼ ‼ �‼ ‼ �= |= Next, applying Theorem 2.9 as well as the result: | ห=[ � ห� �− ‼ ∙ �‼ ] = = �3 , we evaluate ห� ห= � = ห � ∙ � ∙� = � = . ห as below: □ Although the results of Theorem 2.8 and Theorem 2.9 seem quite simple, when evaluating �� for higher dimensions, the number of double-factorials involved may reduce the simplicity. In order to remove this complexity, depending on the parity (oddness & evenness) of � ∈ ℕ, the values of �� can be investigated, individually, followed by taking Notation 2.4 into account. To serve the purpose, we draw our attention to the next lemma: Lemma 2.10 Let � ∈ ℕ. The following statement is true: �+ ⁄ �� = �⁄ � �− ⁄ ∙� �‼ �⁄ � . Proof Using Notation 2.6 and applying Lemma 2.5, we proceed as below: � �� = ∏ �= If n is odd, then: ∏��= Combining the results � �− ‼ �− = ⋅ �‼ �− � � and = �+ ⁄ and ⋅� �− ⁄ �‼ ⋅ together, we obtain: �� = �‼ ⋅ Considering the parity of � ∈ ℕ in and � � = � �= together, we obtain: �� = On the other hand, if n is even, then: ∏��= Combining the results � ‼ ‼ ‼ ⋅ ∙∙∙ ⋅ ⋅ ⋅∏ ‼ ‼ ‼ �⁄ ⋅ � �⁄ �+ ⁄ �⁄ � � ‼ = ⋅∏ �‼ �= ⋅� �− � � ⁄ ⋅ � �⁄ , when applying Notation 2.4, for each � ∈ ℕ it follows that: 12 Copyright©2012Fareed Saiepour. EVALUATION OF THE CONTENT OF BALL-BASED OBJECTS IN ℝ� �� = �‼ �+ ⁄ ⋅( �⁄ )� ⋅ (� �− ⁄ �+ ⁄ � � ⁄ )� = �⁄ � ⋅� �‼ �− ⁄ �⁄ � ■ This chapter ends by pointing out an interesting feature about the bounds and the descending order of the values of �� , as � ∈ ℕ varies. The diagram in Figure 2.3 displays the values of �� for � ∈ { , … , }. 6 5 4 3 2 1 0 0 1 2 3 4 5 6 7 8 9 10 11 12 ������ . The figure shows that �� attains a maximum value at � = . Precisely speaking, by applying Lemma 2.10, we can write the following expression: ⁄ 5 max{�� ∍ � ∈ ℕ} = � = ⁄ 5 ⋅� ‼ ⋅� = = � = . sf . On the other hand, it is also possible to express the infimum for �� as n tends to infinity by: lim �� = �→∞ other words: inf{�� ∍ � ∈ ℕ} = . In Table 2.4, preserving the descending order, the values of �� for � ∈ { , … , an accuracy up to 5 significant figures: � � 5 6 4 7 3 8 9 2 10 5.2638 5.1677 4.9348 4.7248 4.1888 4.0587 3.2985 3.1416 2.5502 Since for each � ∈ ℕ, we have: | � � �� . 1 2 11 . In } have been displayed with 12 13 14 15 1.8841 1.3353 0.9106 0.5993 0.3814 | = �� , it follows that: ∀� ∈ ℕ { }, | |>| � |. In a similar manner it is possible to derive expressions for the corresponding boundary sets. By Theorem 2.9, since for each � ∈ ℕ, we have: |� � | = ��� , preserving the descending order, Table 2.5 displays values of ��� with an accuracy up to 5 significant figures for � ∈ { , … , }. 13 Copyright©2012Fareed Saiepour. EVALUATION OF THE CONTENT OF BALL-BASED OBJECTS IN ℝ� � � 7 � 8 33.1 32.5 6 31 9 5 10 11 4 12 3 13 14 2 15 16 17 1 18 19 20 29.7 26.3 25.5 20.7 19.7 16 � 12.6 11.8 8.39 6.28 5.72 3.77 2.4 2 1.48 0.89 0.52 �� . Using the table above as well as Lemma 2.10, we can also write corresponding expressions for the bounds of ��� , as below: ⁄ 7 ⁄ 7 ⋅� � ⋅� = × = = . sf . max{��� ∍ � ∈ ℕ} = � = × ‼ Thus: | > |� � | . ∀� ∈ ℕ { }, |� Moreover, since: lim ��� = , it follows that: inf{��� ∍ � ∈ ℕ} = �→∞ . Finally, the diagram in Figure 2.6, describes a comparison between the values of | � | and |� for � ∈ { , … , } (the dark and light dots represent the n-balls and their boundaries, respectively): � | 35 30 25 20 15 10 5 0 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 ������ . 14 Copyright©2012Fareed Saiepour. EVALUATION OF THE CONTENT OF BALL-BASED OBJECTS IN ℝ� 3.The content of ball-based objects (elliptic balls, cylinders & cones) Definition 3.1 Let � ∈ ℕ and , ∈ ℝ+ . We write: � ; ⊏ ℝ� , to denote the elliptic n-ball of majorradius r and minor-radius s as an object in ℝ� , which is defined as below: � ; �− � �� = {� ∈ ℝ ∍ ∑ �= + �� ≤ } . Notation 3.2 Let � ; ⊏ ℝ� for some � ∈ ℕ and , ∈ ℝ+ . To represent � ; in a hyper� �− coordinate system, regardless of the special case � = , we use the expression: , ≡ ⊚ , such that: �− ⊏ ℝ �̅ and ⊏ ℝ �� , as shown in Figure 3.1: �� � ; �̅ − �− − ������ . Remarks As a special case for Definition 3.1, if � = , it is important to note that the major-radius vanishes as �− becomes dimensionless. For this purpose, we have: ; ≡[ By: ℝ ≡ [ℝ �� ] � ; ] �= ≡[ �− ⊚ ] �= ≡ ⊚ ≡ ⊏ ℝ ≡ [ℝ �� ] �= . , we mean that the real line is a vertical 1-dimensional Euclidean space. �= The next theorem introduces a similar method to the one given in Theorem 2.8 for evaluating the content of elliptic n-balls: 15 Copyright©2012Fareed Saiepour. EVALUATION OF THE CONTENT OF BALL-BASED OBJECTS IN ℝ� Theorem 3.3 Let � ⊏ ℝ� for some � ∈ ℕ and , ∈ ℝ+ . The following statement is true: ; | � | = �� ; �− . Proof To verify the case: � = , by referring to the remarks after Notation 3.2, we proceed as below: [| � ; |] �= =| |=| ; |=| ⊚ |= =� = [�� �− ] �= . For the case: � = , we recall the area of an ellipse with radii r and s. Precisely speaking, the content of an elliptic-disc or an elliptic 2-ball which we denote by E. That is: |�| = |{ � , � � ∈ℝ ∍ The procedure is like the previous case, as follows: [| � ; |] �= =| ; |=� + � =� ≤ }| = � . �− ] = [�� Now suppose that: � ∈ ℕ { , } and consider Figure 3.2. �� � ��� � − �= . rcos� ; �� �̅ − ������ . The differential object �� ⊏ ℝ� has been displayed in the figure such that dim� = �. For |��|, we can write: 16 Copyright©2012Fareed Saiepour. EVALUATION OF THE CONTENT OF BALL-BASED OBJECTS IN ℝ� |��| = | �− cos� × ( ��� )| = | �− cos� |��� . When proving the proposition � � + in Theorem 2.8, we used the fact that: ���+ = � sin� . Without the requirement for mathematical induction, we use a similar argument by stating that: ��� = � sin� . For this purpose, we proceed as below to complete the proof: | � ;s | = ∫ |��| = ∫ ��− ℝ �� − �⁄ = ∫ ��− −�⁄ �− cos� �− ��� = cos � ��� = ��− ∫ ��− �� =− �− �⁄ cos� �− ∫ cos� � �� = −�⁄ � sin� �− � ��− = �� �− ■ � ⊏ ℝ� and � ; ⊏ ℝ� for some � ∈ ℕ { } and , ∈ ℝ+ . Also suppose that: � ≡ �− � �− �− ⊚ and ; ≡ ⊚ . In both objects, we have: ⊏ ℝ �̅ . In other �− � � words, is the cross-section of and ; , with respect to the subspace ℝ �̅ . For this purpose, we call the objects � and � ; , ball-based objects as their cross-section with respect to the given subspace ℝ �̅ is an � − -ball. We continue this chapter by introducing and evaluating contents of two other types of objects, as follows: Let Definition 3.4 � ∈ ℕ and , ℎ ∈ ℝ+ . We write: � height h as an object in ℝ� , which is defined as below: � Notation 3.5 Let � � �− ; ℎ = {� ∈ ℝ ∍ (∑ �= ; ℎ ⊏ ℝ� , to denote the n-cylinder of radius r and �� ≤ ℎ |�� | ≤ )} . ; ℎ ⊏ ℝ� for some � ∈ ℕ and , ℎ ∈ ℝ+ . To represent coordinate system, regardless of the special case � = , we use the expression: such that: �− ℎ ⊏ ℝ �̅ and ⊏ ℝ �� , as shown in Figure 3.3. Remarks As a special case for Definition 3.4, if � = , it is important to note that dimensionless. For this purpose, we have: ; ≡[ � ;ℎ ] �= ≡[ �− ℎ × ( )] �= ≡ ℎ × ( )≡ � ,ℎ ≡ �− ; ℎ in a hyper�− ℎ × , becomes ℎ ( ) ⊏ ℝ ≡ [ℝ �� ] Regarding Notation 3.5 and considering the hyper-coordinate system, we can define notation, as below: 17 � � �= . ; ℎ using a norm- Copyright©2012Fareed Saiepour. EVALUATION OF THE CONTENT OF BALL-BASED OBJECTS IN ℝ� � ℎ |�� | ≤ )} . ; ℎ = {� ∈ ℝ� ∍ (‖�̅ ‖ ≤ ℎ/ �� � ℎ/ �− ;ℎ �̅ − −ℎ/ Theorem 3.6 Let � ������ . ; ℎ ⊏ ℝ� for some � ∈ ℕ and , ℎ ∈ ℝ+ . The following statement is true: | � ; ℎ | = ��− �− ℎ. Proof If � = , to verify the case: � = , by referring to the remarks after Notation 3.5, we proceed as below: [| � ; ℎ |] �= =| ; |=| ℎ ℎ × ( )| = | ( )| = ℎ = � ℎ = [��− �− ℎ] �= . For the case: � = , we recall the area of a rectangle with length and width: r and h, respectively. Precisely speaking, the content of a 2-cylinder is: rh. �� � ;ℎ ℎ/ �� ��� �̅ − −ℎ/ ������ . 18 Copyright©2012Fareed Saiepour. EVALUATION OF THE CONTENT OF BALL-BASED OBJECTS IN ℝ� Now suppose that: � ∈ ℕ { , } and consider Figure 3.3. As in the proof of Theorem 2.8 and Theorem 3.3, this time we have: �� = ℝ� ∍ dim� = � and: �− |��| = | Applying integration, we proceed as below: | � ;s | = × ( ℎ⁄ �− ∫ |��| = ∫ ��− −ℎ⁄ ℝ �� ��� )| = | ��� = ��− �− �− |��� . ℎ⁄ �− ∫ ��� = ��− −ℎ⁄ ℎ ■ We finalise this chapter by introducing and evaluating the content of ball-based n-cones for each � ∈ ℕ. Definition 3.7 Let � ∈ ℕ and , ℎ ∈ ℝ+ . We write: height h as an object in ℝ� , which is defined as below: � Notation 3.8 Let � �− ; ℎ = {� ∈ ℝ ∍ (∑ � �= �� � ; ℎ ⊏ ℝ� , to denote the n-cone of radius r and ≤( − ℎ |�� | ≤ )} . �� ) ℎ ; ℎ ⊏ ℝ� for some � ∈ ℕ and , ℎ ∈ ℝ+ . To represent coordinate system, regardless of the special case � = , we use the expression: such that: �− ⊏ ℝ �̅ and ℎ ⊏ ℝ �� , as shown in Figure 3.3. �� h/2 ℎ/ � � � ,ℎ ≡ ; ℎ in a hyper- �− ⨻ ℎ , ;ℎ �− �̅ − -h/2 ������ . Regarding Notation 3.8 and considering the hyper-coordinate system, we can define notation, as below: 19 � ; ℎ using a norm- Copyright©2012Fareed Saiepour. EVALUATION OF THE CONTENT OF BALL-BASED OBJECTS IN ℝ� � Theorem 3.9 Let � ; ℎ = {� ∈ ℝ� ∍ (‖�̅ ‖ ≤ ( − ℎ |�� | ≤ )} . �� ) ℎ ; ℎ ⊏ ℝ� for some � ∈ ℕ and , ℎ ∈ ℝ+ . The following statement is true: � | ;ℎ | = � ⋅ ��− �− ℎ. Proof The cases for � ∈ { , , }, can be verified by direct proof. Considering � = , we verify the case � = , by referring to Notation 3.8 and proceeding as below: [| � ; ℎ |]�= = | ;ℎ | = | �− ℎ ℎ ( ) | = | ( ) | = ℎ = � ℎ = [ ⋅ ��− � ⨻ �− ℎ] �= . For the case � = , we recall the isosceles triangle with base 2r and height h and use the fact that its area is equal to: ⋅ ⋅ ℎ, as below: � [| ; ℎ |]�= = | ;ℎ | = ⋅ ⋅ℎ = ⋅ � ℎ = [ ⋅ ��− � ⋅� ⋅ℎ = ⋅� �− ℎ] �= . For the case � = , we recall the cone with radius r and height h and use the fact that its volume is equal to: ⋅� ⋅ ℎ, as below: [| � ; ℎ |]�= = | ;ℎ | = We complete the rest of the proof by considering � ∈ ℕ �� h/2 − ��� ℎ = [ ⋅ ��− � { , , } as in Figure 3.6. � �− ℎ] �= . ;ℎ �� �̅ -h/2 ������ . Since the method is like the proof for | � ; ℎ | in Theorem 3.3. using the differential object ��, we give no further explanation and proceed as below, followed by applying integration and involving more techniques from calculus: 20 Copyright©2012Fareed Saiepour. EVALUATION OF THE CONTENT OF BALL-BASED OBJECTS IN ℝ� |��| = | ∴| � �− ⨻ ;ℎ | = ℎ⁄ ∫ |��| = ∫ | ℝ �� = ��− = ��− = ��� )| = | ( � �− �− ⋅ ��− −ℎ⁄ ℎ⁄ �− ( − �� ��� ) × ( )| = | ℎ ℎ⁄ �� ( − ) | ��� = ∫ ��− ℎ �− �� �− ��� = ��− ∫ ( − ) ℎ −ℎ⁄ −ℎ �− �� ℎ � � − [ �− � ℎ⁄ ]� � =−ℎ ⁄ = � ℎ ( − ) +( + ) � �− −ℎ⁄ ⋅ ��− = � ℎ⁄ ∫ �� =−ℎ⁄ �− ⋅ ��− �− �� ) | ��� . ℎ �� �− ��� ( − ) ℎ �� �� �− � − ℎ ( − ) ℎ − ℎ ℎ [( �− ( − � �� − ) ] ℎ ℎ ■ ℎ⁄ �� =−ℎ⁄ Comparing the result of Theorem 3.9 with the result Theorem 3.3 leads us to the following interesting consequence: Corollary 3.10 Let is true: � ;ℎ ⊂ � | ; ℎ ⊏ ℝ� , for some � ∈ ℕ and , ℎ ∈ ℝ+ . The following statement � ;ℎ | = � ⋅| � ;ℎ | . Proof It suffices to compare the results of Theorem 3.3 and Theorem 3.9 ■ 21 Copyright©2012Fareed Saiepour. EVALUATION OF THE CONTENT OF BALL-BASED OBJECTS IN ℝ� Bibliography [1] Gray, A. "Euclidean Spaces." §1.1 in Modern Differential Geometry of Curves and Surfaces with Mathematica, 2nd ed. Boca Raton, FL: CRC Press, pp. 2-5, 1997. [2] Lay, David C. (2006). Linear Algebra and Its Applications (3rd ed.). Addison–Wesley. ISBN 0-321-28713-4. [3] Bourbaki, Nicolas (1987). "Chapters 1–5". Topological vector spaces. Springer. ISBN 3-540-13627-4. [4] P. Dugac (1989). "Sur la correspondance de Borel et le théorème de Dirichlet–Heine–Weierstrass–Borel–Schoenflies– Lebesgue". Arch. Int. Hist. Sci. 39: 69–110. [5] Gauld, D. B. (1974). "Topological Properties of Manifolds". The American Mathematical Monthly. Mathematical Association of America. 81 (6): 633–636. [6] Solovay, Robert M. (1970). "A model of set-theory in which every set of reals is Lebesgue measurable". Annals of Mathematics. Second Series. 92 (1): 1–56. [7] Devlin, Keith J. (1979). Fundamentals of contemporary set theory. Universitext. Springer-Verlag. ISBN 0-387-90441-7 [8] Callan, David (2009), A combinatorial survey of identities for the double factorial, arXiv:0906.1317 [9] Tarrida, Agusti R. (2011), "Affine spaces", Affine Maps, Euclidean Motions and Quadrics, Springer, ISBN 978-085729-709-9 22 Copyright©2012Fareed Saiepour.