Types of Derivatives : Concepts and Applications ( II )

The notion of differential geometry is known to have played a fundamental role in unifying aspects of the physics of particles and fields, and have completely transformed the study of classical mechanics. In this paper we applied the definitions and concepts which we defined and derived in part (I) of our paper: Types of Derivatives: Concepts and Applications to problems arising in Geometry and Fluid Mechanics using exterior calculus. We analyzed this problem, using the geometrical formulation which is global and free of coordinates.


Introduction
In our previous paper, we defined three types of major derivatives such as the Exterior Derivative, Lie Derivative and Covariant Derivative (Kolá r, et al., 1999;Warner, 2013).We divided our work into two parts, where in the first section we started by defining a Differentiable Manifold structure, (Arkani-Hamed, et al., 2010) then The Tangent Bundle, where we built the Bundle from the tangent space defined on a Differentiable Manifold; we defined the Cotangent Bundle in a similar fashion to the Tangent Bundle, then considered Smooth Vector Fields and finally concluded our structure by defining Tensor fields and Riemannian Manifolds (Beig, R.).
In the second part, we defined The Covariant Derivative in which we defined Covariant derivatives of covectors and Tensors, Lie Derivatives:Lie Derivatives of tensor fields and Differential Forms.Finally, in the third part we defined The Exterior Derivative and shed light on the properties of the respective Derivative.
Exterior calculus is a concise formalism to express differential and integral equation on smooth and curved spaces in a consistent manner, while revealing the geometrical invariants at play.One of the main goals of developing a geometric theory of fluid is to put all the existing computational techniques in one abstract setting.This rationalization of computational mechanics will be theoretically interesting for its own sake.
Rewriting equations of fluid mechanics in terms of differential forms enables one to clearly see the geometric features of the fluid field theory.(Kobayashi, S., & Nomizu, K., 1963) Let M be a smooth differentiable manifold of dimension m.The Tangent Bundle TM is defined to be TM =∪ p∈M T p M = {(p, υ)|p ∈ M, υ ∈ T p M}.

Definition
(1) (Lang, S., 1999) AVector field υ on M is a section of the tangent bundle TM, ie υ: M → TM such that π ∘ υ(p)for every p ∈ M. In other words Let x: U → R m be a local chart of M, and p ∈ U, then υ(p) = ∑ υ p m i=1

Definition
A covariant derivative is an operator V on tensor fields which satisfies the following conditions: 1) If T is of rank (r, s), then ∇T is of rank (r, s + 1); the covariant rank increases by 1.
2) For any function 3) For any function f and tensor T, ∇(fT 4) More generally, for any tensors Sand T, ∇(S ⊗ T) = ∇S ⊗ T + S ⊗ ∇T.Derivative (O'neill, B., 1983) If φ is a local diffeomorphism M→ N, we may define a pull-back map φ * ∶ ΓT r,s N → ΓT r,s Mon mixed field as follows.For T ∈ ΓT r,s N we define

Definition
Let X ∈ (M) be smooth vector field.Then for every m ∈ Mwe denote by t ↦ φ X t (m) the (maximal) integral curve for X with initial point m. the domain of this integral curve is an open interval I X,m containing 0. Let T ∈ ΓT r,s M and we define the Lie derivative of T with respect to X by note that φ X t is a diffeomorphism from a neighborhood of m onto a neighborhood of φ X t (m).Accordingly, the expression is a well-defined element of (T m M) r,s which depends smoothly on t (in a neighborhood of 0).Accordingly, (ℒ × T) m defines a tensor in (T m M) r,s .Moreover, by the smoothness of the flow of the vector field X it follows that the section ℒ × T of the tensor bundle T r,s M thus defined is smooth.In other words, we have defined a linear map.
ℒ X ∶ ΓT r,s M → ΓT r,s M, called The Lie derivative.In a similar way it is seen that the Lie derivative defines a linear map ℒ X ∶ E k (M) → E k (M) (Cartan's formula) let X be a smooth vector field on M. then on E(M), ℒ X = i(X) ο d + d ο i(X).
The proof was given in paper (I)Applications.

The Geometric Setup
We give a geometric model of the basic kinematics used in modeling the fluid flow.
3.1 The Fluid Space (Haller, G., 2001) Assume our fluid flows in a smooth manifold M(M denotes a differentiable n-manifold).
A fluid particle is a point in the manifold.Points in a domain D ⊂ M represent the geometric positions of material particles;these points are denoted by x ∈ M and called particle labels.

Fluid Motion (Geometric Notion of the Fluid Motion)
The Fluid moves in a manifold whose points represent the fluid particles.Let x ∈ M be a point in M (M is the space in which the fluid moves) and consider the particle of fluid moving through x at time t = 0.As t increases, we denote byϕ t (x) the curve followed by the fluid particle, which is initially at x ∈ M. For fixed t, each ϕ t will be a diffeomorphism of M. Thus the fluid motion is a smooth one parameter family of diffeomorphisms ϕ t : M→M; with ϕ 0 =Id.t→ ϕ(t) is a one parameter family of diffeomorophisms of M.For each value oft,we define a vector field X t ∈ (M) as follows: For x ∈ M, X t (x) is the tangent vector to the curve u → ϕ(u)ϕ(t) −1 x at u = t, X, is the velocity field corresponding to the fluid motion defined by ϕ(t): that is, X t (x) is the velocity vector of the particle that, at time t, is at the point x.
Note: the inverse maps be computed by reversing time, A flow is called steady (or stationary)if its vector field satisfies: i.e. the "shape" of the fluid flow is not changing.Even if each particle is moving under the flow, the global configuration of the fluid does not change.
Let the one-form α ∈ Ω 1 (M) describe the velocity of a fluid.We define vorticity as The vorticity is a 2-form UJ is dual to the vorticity vector field (.The trajectories of vorticity field are called vortex lines.A flow is called irrotational if From the Poincare lemmaon some open subset U ⊂ M, there exist φ ∈ ℱ U such that α = dφ (10) φ ∈ ℱ U is called velocity potential.

Continuity Equation on Manifolds
Reformulating the continuity equation from the point of view of vector fields and differential forms on manifolds.
Let D ⊂ M be a sub region of MConsider a fluid moving in a domain D ⊂ M and suppose ω is a fixed¬volume element differential form on M, which is point-wise nonvanishing.Then, the total mass of the fluid in the region D at time t is m(D, t) = ∫ D ρ t ω .
(11) Where ρ t describes the mass-density of the fluid at time t.from the principle of mass conservation in fluid dynamics, the total mass of the fluid, which at time t = 0 occupied a region D,remains unchanged after time t.Thus, the total mass of the fluid at time t = 0 occupinga region D is maintained with time, i.e. ∫ ϕ t (D) ρ t ω = ∫ D ρ 0 ω .
(12) Where ϕ t the one-parameter family of diffeomorophisms.Equation ( 12) isis the integral invariant for conservation of mass.
By the change of variable formula, the left hand side of this relation is equal to ) differentiating,with respect to t, we get Then ( 14) takes the form by change of variable formula Since D is an arbitrary open set, this can be true only if the integrand is zero; that is This is the equation of continuity in invariant form.Equation ( 17) can be written as whereω t ∶= ρ t ω is a 1-parameter family of n-forms on M.
(17) takes the following form L X (ρω) = 0 (19) From properties of the Lie derivative this reduces to (L X ρ)ω + ρL X ω = 0 (20) Since ρis a constant function,the first term on the left-hand side of equation ( 20) is vanish, then equation ( 20) take the following form (21) This is the geometric form of equation of continuity for incompressible fluid.According to (Cartan's formula) the Lie-derivative applied to volume form can may be written as Now, whenever M = ℝ 3 .In fact substituting (24)into the continuity equation ( 17), we get the following Sinceω = dx 1 dx 2 dx 3 (volume form), therefore Which is the usual equation of continuity.(Verhulst, F., 2006) In this sub section, we use the Cartan's theory to show that it is possible to rewrite the continuity equation as an exterior differential system.

A. Compressible (general) case:
To reformulate the continuity equation as an exterior differential system(alternate geometric approach), we set.
We define the 3-form: 28) By applying the exterior derivative of both sides in ( 28) Using the properties of wedge product, this becomes Therefore div= 0 corresponds to Thus the continuity equation correspond to the closed form  = 0 (32) Suppose we express the flow in terms of initial conditions (or other parameters) by  = (,  1 , … ,  3 ), (33) so that the a' are the parameters and (38) This means that the 3-form is an integral-invariant for the flow   , which represents property of conservation of mass Equation ( 32).We consequently have the following result.Given two 3-chains  3 ,  3 ′ ∈  3 () which are in 1 -1 correspondence such that corresponding points lie on the same trajectory of the flow {  }, then: If now  3 =  3 (0)   0 = 0, then, by (2.37), And following up  3 (0) =  3 (1) at time  1 , we have: Which expresses that mass is preserved in the flow {  }, another form of the conservation of mass.

B. Inviscid, incompressible, and irrotational (potential flow) case:
A potential flow describes what the flow would be like if it were inviscid, incompressible, and irrotational.which in vector calculus is described by Laplace equation  2  = .Reformulating the problem in a differential geometry terms we consider the scalar function  ℎℎ   0 −    2 −  , following contact variables Let M be a manifold with variables (, , , , , , ), on M we define the contact form  =  −  −  − , (43) And the 2-form  =  +  + , (44) By taking the exterior derivative of (44) and using the anti-symmetric property of wedge product we obtain By using the anti-symmetric property of wedge product, this becomes Note that  = 0. Then Laplace equation is equivalent to the relation  = 0 (47) This is the coordinate-free version.(Datta, A., & Majumdar, A., 1980) Navier-Stokes equation is the most general equation for description of fluid phenomena , which as special case comprises Euler's equation of motion.Let M be a differentiable n-manifold,  be a differential 1-form on M, co be a volume-element differential form on Mand D be a domain in M.

Momentum Equation on Manifolds
Where∈  is the 3-index, skew-symmetric tensor (with ∈ 123 = 1).Equations ( 63) and ( 64) correspond to the conventional continuous approach where: Now Which is the usual momentum equation of motion.(Gilkey, P. B., 1975;Yamabe, H., 1960) In this section we express the Euler's equation in the language of differential one-forms on Riemannian manifold.The formulation is obtained in the absence of body forces.

Euler Equation on Riemannian Manifold
Consider an n-dimensional manifold M with a Riemannian metric g.By identifying the differential forms with their dual vector fields, we rewrite Euler equation in terms of differential 1-forms.Let  ∈  1 () be the one form associated to X ( is the one-form dual to X), describe the velocity of an ideal fluid.We seek an invariant meaning for the sum of the last three terms on the left-hand side of equation ( 26).For fixed i this expression equals (77) meaning that d is preserved by X.
Since   − , we get   = −   = 0 (78) Thus f is invariant under the flow of X.We will call Euler vector fields to the solutions to the equations of an ideal steady incompressible fluid on a manifold.
( 1 , 2 , 3 ) ( 1 , 2 , 3 )  1  2  3 = (, ) 1  2  3(36)Since  = 0 we deduce that Consider a fluid moving in a domain  ⊂ .For any continuum there are two types of forces acting on a piece of material. (  ) + (  ) =  ((  )  ) +  ( geometric version of momentum equation of fluid motion, which is coordinate-free.If M= ℝ 3 , with coordinates   ,  = 1,2,3. 67) and (68) into the Euler equations (65), we get the following exterior differential system Euler equation in terms of differential 1-forms.According to (Cartan's magic formula) the Lie-derivative applied to one-form a may be written as  =   +   =   + (())=  + (()) (70) Since  = .So substituting (70) into the Euler equation (69), we get the following alternative form of the Euler equation (69)and exterior calculus, we derive in the following a set of invariant fluid equations.A. Bernoulli equation.For a steady (time-independent) flow of a perfect (an inviscid and incompressible) fluid, Euler equation (69) becomes is constant along each streamline (integral curves of X) of the fluid flow.This is the Bernoulli's principle.The function f is known as the Bernoulli function of X.Also (  ) =   = (−) = 0