Calculus Group Language

Abstract

Differentiation and Integration in Calculus Language are Identified to Cyclic Group of order two (isomorphic to Integers mod two). This is done by identifying the symmetry of Calculus in the two operations of Differentiation and Integration. That is, Integration and Differentiation are symmetries of each other which may be unified by Identification. This simple case of Identification is extended to further Identify compositions of various degrees. These compositions identify various forms of constructions such as those seen in Numbers (Real Number Line Construction), Groups (Composition of finite Simple Groups), Calculus (Differentiable Structures) and other Languages.

Fundamental Identity

Let the Fundamental Identity be denoted by \(E\). In group (symmetry) Language, this is the trivial group: it comprises of only one element \(E\) itself with the trivial operation \(\cdot\) giving \(E \cdot E = E\). This is named \(C_1\) and written as \(C_1\,\, =\,\, \{\,\, E\,\,\}\). To identify with other structures, at least one other element must be produced as a form of Identity. However, having another identity element in the same group is redundant: \(E_1 \cdot E_2 = E_1 =E_2\). This makes it necessary to “Differentiate” the Identity into two or more elements. Making any “partition” (we have seen in previous posts that creating a boundary is Differentiation) of Identity automatically creates two sides, one on either side of the boundary of partition. This defines an element and its inverse. In the simplest case, the Partition divides equally: that is, the element is its own inverse. This forms the Cyclic group of order 2, which is the simplest non-trivial group.

• In Language of Algebra (Equations), the Identity is \(1\) and the trivial Equation is \(x=1\) where \(x\) acts as the identifier of elements satisfying the Equation.

• In Calculus Language, Integration is the Identity \(E\) and Differentiation is \(D\).

• In Usual Calculus Language, \(1\) as an operation on Functions is the Identity giving \(1 \cdot f = f\) for any function.

  • In Calculus of Functions, \(d\) as an operation on functions gives \(d \cdot (f)= 1\cdot (f)\) as the Eigenvalue equation. Name the solution of this Equation to be the Identity (\(E\)) function of calculus (\(\exp\)) which will later be seen as the Exponential function.

• In Orientations Sign Language, \(+\) is the Identity. Thus, acting on any orientation with \(+\) maintains the orientation.

• In Parity Language, Even is the Identity.

• Observe that the Identity preserves itself under the Differentiation operation.

  • In Algebra, \(1\) stays as one of the roots on Differentiting the roots.

  • In Calculus, Integration stays preserved as one of the operations on Differentiation.

  • In Orientation, \(+\) stays as one of the signs on Differentiation (to (\(-\), \(+\))).

Identifying Fundamental Calculus Language as a Fundamental Cyclic Group of order 2

• In the usual Calculus Language we have two operations, one being Differentiation denoted by \(D\) and the other being Integration \(I\). Here we start by identifying these two as the same operation and then further proceed with the usual notions in the next secions.

• Let \(E\) be the Identity. Then the group \(\{D, I, E\}\) with relations \(I \,\,\cdot \,\, D \,\,= \,\, E\) and \(I=D\) identify the Calculus Language at any arbitrary but fixed degree as a Symmetry of differentiation \(D\) and integration \(I\).

• The above presentation of Calculus as a group is identified as \(\{E, I\}\), with \(E\) as identity and \(I^2=E\). This is a Cyclic Group of order 2, that is \(C_2\). The group operation table is: \(\begin{array} {|c|c|c|} \hline \cdot & E & I \\ \hline E & E & I \\ \hline I & I & E \\ \hline \end{array}\)

or equivalently,

• We shall identify how \(E \cdot E = E\) and \(D \cdot D = E\) appears in various forms in Calculus.

• In the above, Differentiation (\(D\)) and Integration (\(I\)) at the same Degree (of the same measure) are identified as duals to each other to produce Identity (\(E\)) of Lower Degree (less measure).

• Thus \(C_2\) is identified as the fundamental Group of Calculus Language.

We see through the above representation of Differentiation as an element of a group, \(E \cdot E = E\) and \(D\,\, \cdot D\,\, =E\) or \(I \,\, \cdot \,\, I =E\).

• In Language of Equations (Roots), the Identity \(E\) has been Differentiated to obtain two roots: one itself and the other a pair of indistinguishable parts \(D\) and \(D\) which are inverse to each other. Thus the single equation \(E \cdot E = E\) gets differentiated into a pair of two equations:

In Algebra (the Language of Eqautions), these are Identified as a Single equation \(x^2 = 1\) and the roots are given by \(1\) and \(-1\), where \(1\) is the identity and acts as its own root and \(-1\) corresponds to \(D\). This shows how we start from \(\{E\}\), the trivial group which can be partitioned (allowing redundancy) to \(\{ E, E\}\) which is two copies of the same Identity element, both being inverses to each other to give \(E\,\, \cdot\,\, E\,\,= \,\,E\). One of the \(E\)’s is then Differentiated (partitioned) further into two halves \(\{E_{*}, E_{*}\}\) which are renamed as \(\{D, D\}\) to recognize that they are at a lower degree (D stands for down or differentiated). Correspondingly, in Equations we start with \(1\) as the Identity. \(x\) is the identifier for all elements in the group, that is, all elements in a group may be identified to \(x\). Thus, \(x = 1\). To go further, differentiating \(x\) into two halves requires producing two \(x\)’s, this is done by identifying the elements with themselves or the equation with itself (symmetry) : \(x \cdot x = 1 \cdot 1\) which is \(x^2 =1\) and gives \(\{-1,1+\}\) as roots. Thus the cyclic group of order \(2\) called (\(C_2\)) in Calculus is the pair of roots \(\{-1,1+\}\) in Algebra. That is, \(-1\) corresponds to \(D\) and \(1+\) corresponds to \(E\). The \(-1\) is analogous to reflection in Symmetry Language. Allowing redundancy, or Identifying two copies of the same is required for differentiating. In other words, increasing Degrees of freedom (up) gives corresponding roots (down). Or, Integration gives Differentitation. This flow of thought can be reversed to see Differentiation as Identification (Integration).

• In Symmetry Language, an single object taken to be Identity (\(E\)) can be reflected to obtain a copy which is renamed to distinguish from the original object. Visually, the change in orientation occurs naturally. Reflection as an operation itself forms the cyclic group of order two \(C_2\). As an application, right handed orientation can be reflected to form left handed orientation. Right can be reflected to left.

• In Calculus of Functions Language, the Operator Equation \(D \cdot D = E\) when Identified with it’s Spectrum gives the Eigenvalue Equation by acting on function \(f\) to give

Here for the Spectrum taking Identity as 1 for simplicity, it can be replaced by \(\lambda\) later. “Two Roots of 1” in the above Eigenvalue equation correspond to \(\exp\{\,\,1\,+\,\,\}\) and its reflection, call it the \(\exp\{\,\,-\,1\,\,\}\) function. Corresponding to \(\lambda\) as Identity of Spectrum, we get \(\exp\{\lambda+\,\,\}\) and its reflection \(exp\{-\lambda\,\,\}\) as the roots satisfying the Eigenvalue equation.

• In the Orientations Sign Language, this corresponds to \(C_2 = \{ \,\, -\,\,,\,\, + \,\, \}\) where \(+\) is the Identity and \(-\) corresponds to \(D\).

• In Geometry \(S^0\) is \(C_2\), where the left hand point corresponds to \(D\) and the right hand point corresponds to the Identity \(E\).

• In Parity Language, Odd is \(D\) and Even is Identity \(E\).

• In Particle Physics, Fermion is \(D\) and Boson is Identity \(E\).

Identifying the Differential Forms of Fundamental Calculus (Cyclic Groups of finite order)

To put in order the story so far: We start with the Trivial Group consisting of only one element called the Identity \(E\). Next we “Differentiate the Identity” into two parts: an element and it’s inverse. In the simplest case, this Differentiation is Symmetric so as to give the element as its own inverse resulting in creation of a Cyclic Group of order 2, that is \(C_2 = \{\,\,E\,\,,\,\, D\,\,\}\) where \(D \,\,\cdot\,\, D \,\,=\,\,E\) . This Fundamental Cyclic Group appears in various forms in different Languages- \(S^0\) in Geometry, in Algebra \(\{\,\,-1\,\,,\,\,1+\,\,\}\), Orientations \(\{\,\,-\,\,,\,\,+\,\,\}\), Binary Language \(\{\,\,1\,\,,0\,\,\}\), Morse Code \(\{\,\,-\,\,,\,\,\cdot\,\,\}\), Parity \(\{Odd, Even\}\), Physics- Spin down and Spin up, Bosons and Fermions and so on.

Now we Further Differentiate the element \(D\) in Cyclic Group of Order 2 to give two distinguishable sub-elements. Together with \(E\) and \(D\), these will be called “fourth roots of Identity” (think of roots of trees coming from Identity). Since we have already established the correspondence with Equations (Algebra), we may see this process first comfortably through equations. Through equations, we see this as follows:

• Start with where we left: the Cyclic Group of order two having the following relation: \(x \,\,\cdot\,\, x = 1\)

  • We named one root to be \(-1\) and the other root to be \(1+\). In symmetry language \(-1\) is obtained by reflecting \(1+\).

  • We repeat the procedure of Identification to obtain roots. Here we are left to Identify the equation with \(-1\) so we do that as:

  \[x \,\,\cdot\,\, x = -1\]

  • Name the two roots to the above equation as \(-i\) and \(i+\). Thus \(\,\,-i\,\, \cdot\,\, -i \,\,= \,\,i\,\, \cdot \,\,i = \,\,-\,1\)

  • Identifying all together, we get the Cyclic group of order \(2 \cdot 2\) or \(2^2\) or \(4\). This is called \(C_4\) and written as \(\{ \,\,-\,i\,\,, \,\,-\,1\,\,,\,\, \,\,i\,+\,\,,\,\, 1\,+\,\, \}\)

  • This corresponds to the pair of two equations:

• The above procedure is equivalent to Differentiating “D” in Calculus Language.

• In the Language of Calculus of functions, Differentiating the Identity \(E\), satisfying the relation \(E \,\,=\,\, D\,\, \cdot\,\, D\) into two elements is a repeat of the procedure we did to obtain Cyclic group of order two from the trivial group. That is, we may insert a Cyclic Group of order \(2\) with \(D\) as the identity. This produces the Cyclic group of order 4 corresponding to the one obtained through Algebraic equations above. Instead of repeating the writing of equations and group table, we see the construction in Calculus of functions through the Eigenvalue Equations:

  • We had the following Eigenvalue equation:
  \[(\,\,D \,\,\cdot\,\, D\,\,)\,\, f \,\,= \,\,(\,\, 1\,\,)\,\, f\]

  Whose solutions were named as \(\exp\{\,\,-\,1\,\,\}\) and \(\exp\{\,\,1\,+\,\,\}\).

  • We further Identify the left hand side of the above with \(-1\) as the new Identity of the Spectrum to see the new Eigenvalue Equation:
  \[(\,\,D \,\,\cdot\,\, D\,\,)\,\, f \,\,= \,\,(\,\, -\,1\,\,)\,\, f\]

  Borrowing the symbols from Algebra Language, we name the two solutions to this as \(\exp\{-i\}\) and \(\exp\{i+\}\).

  • We may identify these separate solutions by defining a new function which takes values in two dimensions. That is instead of working with a function from one variable to one variable, we define a new finction \(F \colon U \to V\), where \(U \subseteq \mathbb{R}\) and \(V \subseteq \mathbb{R}^2\). Let us name the function in the first component to be cosine (denoted by \(\cos\)) and the function in the second component to be sine (denoted by \(\sin\)). Thus \(F \,\, = \,\,(\,\,\cos\,\,,\,\, \sin\,\,)\). If we are to compare with Cyclic group of order \(2\), here \(\cos\) shall behave as (Even) and \(\sin\) shall behave as (Odd). We are to assign two reflections corresponding to two Differentiations so as to reproduce the behaviour seen in the Cyclic Group of Order 4. Let us use the signs \(-\) and \(+\) to denote a pair of reflections. Composing signs with the individual element functions, we obtain the four elements which are \(\{\,\,-\,\,\sin\,\,,\,\,-\,\,\cos\,\,,\,\,\sin\,\,+\,\,,\,\,\cos\,\,+\,\, \}\). Let the Differentiations (reflections) be as follows:

On further composing with Differentiation (reflection), This implies that :

Combining these as a two component function gives:

The last equation is \((d \cdot\,\, d \,\,) F = - F\) which is the Eigenvalue Equation we were trying to represent.

This suggests that \((\,\, \cos\, +\,\,,\,\,\sin\,+\,\,)\) corresponds to \(\exp\{\,\,i\,\,\}\) and its second component sign reflection \((\,\, \cos\, +\,\,,\,\,-\,\,\sin\,\,)\) corresponds to the reflection \(\exp\{\,\,-\,i\,\,\}\).

• Another way to see the above is by the Algebra homomorphism \((\,\, \cos\, +\,\,,\,\,\sin\,+\,\,)= \,\, \cos \,\,+ \,\,i\, \sin\) since reflection of \(i\) to \(-i\) gives the corresponding result as \(\,\, \cos \,\,+ \,\,-\,i\, \sin\). Hence \(\exp(i)\,\, = \,\, \cos \,\,+ \,\,i\, \sin\) and \(\exp(-i) \,\, =\,\, \cos \,\,+ \,\,-\,i\, \sin\).

Thus we obtain the Fundamental Identity:

Thus we see the Fundamental Law:

Using the naming of \(i\) to satisfy the equation \(i \cdot i =\,\,-\,1\), we see

This is also known as Pythagoras Theorem.

• The procedure seen above can be repeated to generate the corresponding Cyclic groups of order \(n\) by simply mapping it to the algebraic equation \(x^n= 1\) which produces the \(n^{th}\) roots of Identity as the elements of the cyclic group.

Identifying the Usual Calculus of functions as a Cyclic Group of Integers

The procedure above in the limiting case reproduces the cyclic group of all Integers.

• The usual calculus of functions one uses a differentiated form of Fundamental Calculus. Differentiating the above cyclic Group of order \(2\) into a cylic group of infinite order gives the usual calculus with \(d\) as the element corresponding to one differentiation and \(i\) as the inverse of differentiation, that is \(i = d^{-1}\). That is, Calculus of functions is identified as the group isomorphic to the Integers (\(\mathbb{Z}\)).

Reference to usual Number Groups