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1 Volume 5, Issue 3, March 2015 ISSN: X International Journal of Advanced Research in Computer Science and Software Engineering Research Paper Available online at: Special Issue on 2 nd International Conference on Electronics & Computing Technologies-2015 Conference Held at K.C. College of Engineering & Management Studies & Research, Maharashtra, India An idiosyncratic use of Green s Theorem and Gauss s Divergence Theorem for Applications in Applied Physics Yogesh Karunakar A.P. Dept. of ETRX K.C.COE India Abstract: There are many inter-connected goals in presenting this paper. Green s theorem is a very interesting theorem which shows a relationship between a closed area and the path surrounding this closed area. It also help in the theory whenever one wants to switch from one kind of integral to another (double integrals to line integrals and vice-versa). This transformation can be done using the Green s theorem. Green's theorem has been and is still being used in various area estimation algorithms. There is even an old area estimating instrument called the planimeter used in irregularly shaped areas, which is justified by Green's theorem Gauss Divergence Theorem transforms surface integrals into volume (triple) integrals. Its main objective is to find out flux (Flux is surface bombardment rate), which is used in variety of applications to analyze properties of vector field and flow. This theorem provides deep insights to the nature of flow and the basis of many principles in physics and engineering and plays important role in electromagnetic and electrostatic theory. This theorem is applied to Gauss s Law. Gauss's law also has a close mathematical similarity with a number of laws in other areas of physics, such as Gauss's law for magnetism and Gauss's law for gravity. Keywords: I. INTRODUCTION TO CONCEPTS There are many inter-connected goals in implementing this project. Here we evaluate two major theorems of maths: 1. Green s Theorem Green s theorem is a very interesting theorem, which shows a relationship between a closed area and the path surrounding this closed area. Double integrals over a plane region may be transformed into line integrals along the boundary of region and vice-versa. This is of practical interest because it may help to make the evolution of an integral easier. It also helps in the theory, whenever one wants to switch from one kind of integral to another. The transformation can be done by the Green s theorem. 2. Gauss s Theorem Gauss Divergence Theorem is first biggest integral theorem; it is called as Gauss Divergence theorem because it involves the divergence of a vector function. Invented by Carl Friedrich Gauss, the great mathematician.this theorem is also called as conservation law which states that the volume integral of the divergence is equal to the net flow across the volume s boundary. Divergence theorem states that the outward flux of a vector field through a closed surface is equal to the volume integral of the divergence of region inside the surface. Intuitively, it states that the sum of all sources minus the sum of all sinks gives the net flow out of a region. If a fluid is flowing in some area and we wish to know how much fluid flows out of a certain region within that area; then we need to add up the sources inside the region and subtract the sinks. The fluid flow is represented by a vector field and the vector field's divergence at a given point describes the strength of the source or sinks there. So, integrating the field's divergence over the interior of the region should equal the integral of the vector field over the region's boundary. The divergence theorem says that this is true. The idea behind Green s theorem When C is an oriented closed path (i.e., a path where the endpoint is the same as the beginning point), the integral Represents the circulation of F around C. If F were the velocity field of water flow, for example, this integral would indicate how much the water tends to circulate around the path in the direction of its orientation. 2015, IJARCSSE All Rights Reserved Page 315

2 One way to compute this circulation is, of course, to compute the line integral directly. But, if our line integral happens to be in two dimensions (i.e., F is a two-dimensional vector field and C is a closed path that lives in the plane), then Green s theorem applies and we can use Green s theorem as an alternative way to calculate the line integral. Green s theorem transforms the line integral around C into a double integral over the region inside C. However, it s not obvious what function we should integrate over the region inside C so that we still get the same answer as the line integral. To figure out what we should integrate, the notion of circulation is quite helpful. Think of the integral CF ds as the macroscopic circulation of the vector field F around the path C. Now, imagine you came up with a microscopic version of circulation around a curve. This microscopic circulation at a point (x, y) has to tell you how much F would circulate around a tiny closed curve centered around (x, y). We could picture the microscopic circulation as a bunch of small closed curves (shown below in green), where each curve represents the tendency for the vector field to circulate at that location (imagine that the small curves were really, really small, much smaller than pictured). Green s theorem is simply a relationship between the macroscopic circulation around the curve C and the sum of all the microscopic circulation that is inside C. If C is a closed curve in the plane (remember, we are talking about two dimensions), then it surrounds some region D (shown in red) in the plane. D is the interior of the curve C. Green s theorem says that if you add up all the microscopic circulation inside C (i.e., the microscopic circulation in D), then that total is exactly the same as the macroscopic circulation around C. Adding up the microscopic circulation in D means taking the double integral of the microscopic circulation over D. Therefore, we can write Green s theorem as The idea behind Gauss s Divergence theorem: In vector calculus, the divergence, also known as Gauss theorem, In vector calculus the divergence theorem, also known as Gauss' theorem (Carl Friedrich Gauss), Ostrogradsky's theorem (Mikhail Vasilievich Ostrogradsky), or Gauss- Ostrogradsky theorem is a result that relates the flow (that is, flux) of a vector field through a surface to the behavior of the vector field inside the surface. More precisely, the divergence theorem states that the outward flux of a vector field through a closed surface is equal to the volume integral of the divergence of the region inside the surface. Intuitively, it states that the sum of all sources minus the sum of all sinks gives the net flow out of a region. The divergence theorem is an important result for the mathematics of engineering, in particular in electrostatics and fluid dynamics. If a fluid is flowing in some area, and we wish to know how much fluid flows out of a certain region within thatarea, then we need to add up the sources inside the region and subtract the sinks. The fluid flow is represented by a vector field, and the vector field's divergence at a given point describes the strength of the source or sinks there. So, integrating the field's divergence over the interior of the region should equal the integral of the vector field over the region's boundary. 2015, IJARCSSE All Rights Reserved Page 316

3 II. MATHEMATICAL STATEMENT A region V bounded by the surface S= V with the surface normal n Suppose V is a subset of Rn (in the case of n = 3, V represents a volume in 3D space) which is compact and has a piecewise smooth boundary. If F is a continuously differentiable vector field defined on a neighborhood of V, then we have The left side is a volume integral over the volume V, the right side is the surface integral over the boundary of the volume V. The closed manifold V is quite generally the boundary of V oriented by outward-pointing normal and n is the outward pointing unit normal field of the boundary V. (ds may be used as shorthand for nds.) By the symbol within the two integrals it is stressed once more that V is a closed surface. In terms of the intuitive description above, the left-hand side of the equation represents the total of the sources in the volume V, and the right-hand side represents the total flow across the boundary V. By applying the divergence theorem in various contexts, other useful identities can be derived (cf. vector identities). Applying the divergence theorem to the product of a scalar function f and a vector field F, the result is a special case of this is,, in which case the theorem is the basis for Green's identities. 2. Applying the divergence theorem to the product of a scalar function, f, and a non-zero constant vector, the following theorem can be proven: Applying the divergence theorem to the cross-product of a vector field F and a non-zero constant vector, the following theorem can be proven: 1. Divergence In vector calculus, divergence is an operator that measures the magnitude of a vector field's source or sink at a given point, in terms of a signed scalar. For example, consider air as it is heated or cooled. The relevant vector field for this example is the velocity of the moving air at a point. If air is heated in a region it will expand in all directions such that the velocity field points outward from that region. Therefore the divergence of the velocity field in that region would have a positive value, as the region is a source. If the air cools and contracts, the divergence is negative and the region is called a sink. More technically, the divergence represents the volume density of the outward flux of a vector field from an infinitesimal volume around a given point. Definition of divergence: In physical terms, the divergence of a three dimensional vector field is the extent to which the vector field flow behaves like a source or a sink at a given point.if the divergence is nonzero at some point then there must be a source or sink at that position. More rigorously, the divergence is defined as derivative of the net flow of the vector field across the surface of a small region relative to the volume of that region. Formally, of an arbitrary shaped region in R3 is called divergence of v of the divergence of the vector field defined by V. Another common notation for the divergence of v is, From this definition it also becomes explicitly visible that div F can be seen as the source density of the flux F. The intuition that the sum of all sources minus the sum of all sinks should give the net flow outwards of a region is made precise by the divergence theorem. Application in Cartesian co-ordinates: Cartesian coordinates on a 3-dimensional Euclidean space, and let i, j, k be the corresponding basis of unit vectors Let x, y, z be a system of. The divergence of a continuously differentiable vector field F = U i + V j + W k is defined to be the 2015, IJARCSSE All Rights Reserved Page 317

4 scalar-valued function: Although expressed in terms of coordinates, the result is invariant under orthogonal transformations, as the physical interpretation suggests. 2. Flux: In the study of transport phenomena (heat transfer, mass transfer and fluid dynamics), flux is defined as the amount that flows through a unit area per unit time. Flux in this definition is a vector. And In the field of electromagnetism and mathematics, flux is usually the integral of a vector quantity over a finite surface. It is an integral operator and acts on a vector field as do the gradient, divergence and curl found in vector analysis. The magnetic flux is thus the integral of the magnetic vector field B over a surface, and the electric flux is defined similarly. Mathematics: Flux has a primary mathematical definition in terms of a surface integral which uses the vectors that represent Where is the vector field, is the normal unit vector which is perpendicular to the surface S, and ds is the differential surface element. There are many fluxes used in the study of transport phenomena. Seven of the most common forms of flux from the transport literature are defined as: Momentum flux, the rate of transfer of momentum across a unit area (N s m 2 s 1). (Newton's law of viscosity,) Heat flux, the rate of heat flow across a unit area (J m 2 s 1). (Fourier's law of conduction)[5] (This definition of heat flux fits Maxwell's original definition.[6]) Chemical flux, the rate of movement of molecules across a unit area (mol m 2 s 1). (Fick's law of diffusion) Volumetric flux, the rate of volume flow across a unit area (m3 m 2 s 1). (Darcy's law of groundwater flow) Mass flux, the rate of mass flow across a unit area (kg m 2 s 1). (Either an alternate form of Fick's law that includes the molecular mass, or an alternate form of Darcy's law that includes the density) Radiative flux, the amount of energy moving in the form of photons at a certain distance from the source per steradian per second (J m 2 s 1). Used in astronomy to determine the magnitude and spectral class of a star. Also acts as a generalization of heat flux, which is equal to the radiative flux when restricted to the infrared spectrum. Energy flux, the rate of transfer of energy through a unit area (J m 2 s 1). The radiative flux and heat flux are specific cases of energy flux. Electromagnetism: Flux is the magnitude of a river's current, that is, the amount of water that flows through a cross-section of the river each second. The amount of sunlight that lands on a patch of ground each second is also a kind of flux. The flux visualized. The rings show the surface boundaries. The red arrows stand for the flow of charges, fluid particles, subatomic particles, photons, etc. The number of arrows that pass through each ring is the flux. As a mathematical concept, flux is represented by the surface integral of a vector field, Where: E is a vector field of Electric Force, da is the vector area of the surface S, directed as the surface normal, Φf is the resulting flux. If the surface encloses a 3D region, usually the surface is oriented such that the influx is counted positive; the opposite is the out flux. The divergence theorem states that the net out flux through a closed surface, in other words the net out flux from a 3D region, is found by adding the local net outflow from each point in the region (which is expressed by the divergence). disciplines in which we see currents, forces, etc., applied through areas. ANALYSIS & DISCUSSION Using Green s Theorem 1 Use Green s Theorem to calculate the area of the disk D defined by x2 + y2 4. Solution: Since we know the area of the disk of radius 2 is 4π, we better get 4π for our answer. The boundary of D is the circle of radius 4. We can parameterized it in a counterclockwise orientation. Using Green s theorem, 2015, IJARCSSE All Rights Reserved Page 318

5 Area of region A=1/2 c (xdy-ydx) In polar co-ordinate, x= r cosθ & y= r sinθ, r is radii, dx= -r sinθ dθ & dy= r cosθ dθ, A= 1/2 c (xdy-ydx) = 1/2 c[(r cosθ) (r cosθ)-(r sinθ) (-r sinθ)] dθ = 1/2 c[(2 cosθ) (2 cosθ) + (2 sinθ) (2 sinθ)] dθ = 1/2 02π [4(sin 2θ+cos 2θ)] dθ = 2 02πdθ = 4π 2. A particle moves once counterclockwise about the circle x2 +y2 =4, R about the origin, Under the influence of the force field f(x, y) = (y+3x)i + (2y-x).calculate the work done. Solution: 3. Use Green's theorem to find the counterclockwise circulation and outward flux for the field F = (y 2 - x2) i + (x 2 + y 2) j and the curve C: the triangle bounded by y = 0, x = 3, and y = x. Solution: Let M = y 2 - x 2 and N = x 2 + y 2. The bounds of integration for this region will be y<=x<=3, 0<= y<= , IJARCSSE All Rights Reserved Page 319

6 4. Find area of plane bounded by curve y=x^2 and line x=y where 0<=x<=3 Solution: for finding area there must be f1= - y and f2=x. Area=9 2015, IJARCSSE All Rights Reserved Page 320

7 Evaluate Evaluate (x+y^2 i +y-z j +x^2+y k).da, where S is surface of Sphere bounded by the plane { (x,y,z) : x^2+y^2=4}.use outward normal nsolution: Given the ugly nature of the vector field, it would be hard to compute this integral directly. Let F=(x+y^2 i +y-z j +x^2+y k). However, the divergence of F is : div F = 2 We use the divergence theorem to convert the surface integral into a triple integral We transform into spherical coordinate, x=ρ*cos (Φ)*sin (Θ),y= ρ*sin(φ)*sin(θ), z= ρ*cos(φ) ; Hence, limits are 0 ρ 2, 0 Φ pi, 0 Θ 2*pi. Test Results III. CONCLUSION The work can be extended to applications like computer graphics, some animation sequences and mechanics. The results from the above research have shown that calculation of surface charge from point sources can be easily found out and visualized. The outcomes are quite promising for more research into these areas and much work has to be still carried out exploring and exploiting the above theorems for much better results 2015, IJARCSSE All Rights Reserved Page 321

8 REFERENCES [1] Anton, Divens, Davis. CALCULUS.seventh edition. Wiley Publications. [2] Mc.Quirrie. MATHEMATICAL METHOD FOR ENGINEERS AND SCIENTIST. [3] J.N.Sharma and A.R.Vasishtha MATHEMATICAL ANALYSIS. Sheth Publications. [4] Brewer, Jess H. ( ). "DIVERGENCE of a Vector Field". Vector Calculus. 2015, IJARCSSE All Rights Reserved Page 322