IS 2401 LINEAR ALGEBRA

AND

DIFFERENTIAL

EQUATIONS

ASSIGNMENT - 02

Contents GROUP MEMBERS .................................................................................................................................... 3

INTRODUCTION ........................................................................................................................................ 4

FUNDAMENTAL THEORIES OF VECTOR INTEGRATION ................................................................. 5

del (∇) operator ......................................................................................................................................... 5

The gradient .............................................................................................................................................. 5

Curl ........................................................................................................................................................... 6

Divergence ................................................................................................................................................ 6

Basic Vector integration theories. ............................................................................................................. 6

Theorem 1: ............................................................................................................................................ 6

Theorem 2 : ........................................................................................................................................... 6

Theorem 1.3: ......................................................................................................................................... 7

Theorem 1.4: ......................................................................................................................................... 7

Theorem 1.5: ......................................................................................................................................... 7

Application of vector integration in fluid dynamics ................................................................................... 10

To find the rate of change of the mass of a fluid flows. .......................................................................... 10

Stock theorem ......................................................................................................................................... 11

Calculate the circulation of the fluid about a closed curve. ................................................................ 11

To analyze the vorticity of the fluid body ........................................................................................... 11

Bjerknes Circulation Theorem ................................................................................................................ 13

To analysis sea breeze ......................................................................................................................... 13

Application of vector calculus in Electricity and Magnetism ..................................................................... 14

Theorem: ............................................................................................................................................. 17

GROUP MEMBERS

Name Registration number 1) Fernando W.T.V.S EG/2013/2191

2) Perera A.L.V.T.A EG/2013/2278

3) Ismail T.A EG/2013/2209

4) Kapuge A.K.V.S EG/2013/2224

5) SurendraC.K.B.B EG/2013/2318

INTRODUCTION

The objective of this report is to create a simple explanation on application of Vector

Integration. To do this we have analyzed concepts of vector calculus, fluid dynamics, and the

Navier-Stokes equation. Upon finding such useful and insightful information, this report

evolved into a study of how the Navier-Stokes equation was derived.

The Navier-Stokes equation is named after Claude-Louis Navier and George Gabriel Stokes.

This equation provides a mathematical model of the motion of a fluid. It is an important

equation in the study of fluid dynamics, and it uses many core aspects to vector calculus.

Before explaining the Navier-Stokes equation it is important to cover several aspects of

computational fluid dynamics. At the core of this is the notion of a vector field. A vector field

is defined as a mapping from each point in 2- or 3-dimensional real space to a vector. Each

such vector can be thought of as being composed of a directional unit vector and a scalar

multiplier. In the context of fluid dynamics, the value of a vector field at a point can be used

to indicate the velocity at that point. Vector fields are useful in the study of fluid dynamics,

since they make it possible to discern the approximated path of a fluid at any given point.

FUNDAMENTAL THEORIES OF VECTOR INTEGRATION

del (∇) operator

Del is defined as the partial derivatives of a vector. Letting i, k, and j denote the unit vectors

for the coordinate axes in real 3-space, the operator is defined.

∇= 𝑖𝜕

𝜕𝑥+ 𝑗

𝛿

𝛿𝑦+ 𝑘

𝜕

𝜕𝑧

Note that here it has indicated uppercase letters to denote vector fields, and lower case letters

to denote scalar fields.

The gradient

The gradient is defined as the measurement of the rate and direction of change in a scalar

field. The gradient maps a scalar field to a vector field. So, for a scalar field f,

𝑔𝑟𝑎𝑑(𝑓)=∇(𝑓)

As an example of gradient, consider the scalar field 𝑓=𝑥𝑦2+𝑧.We take the partial derivatives

with respect to x, y, and z.

𝑑/𝑑𝑥=𝑦2, 𝑑/𝑑𝑦=2𝑥 𝑑/𝑑𝑥=1

So, the gradient is:

(𝑓)=𝑦2𝑖+2𝑥𝑗+𝑘

Curl

Curl is defined as the measurement of the tendency to rotate about a point in a vector field.

The curl maps a vector field to another vector field. For vector F, we define

𝑐𝑢𝑟𝑙(𝐹)=∇ ×𝐹 .

Divergence

Divergence is models the magnitude of a source or sinks at a given point in a vector field.

Divergence maps a vector field to a scalar field. For a vector filed F,

𝑑𝑖𝑣(𝐹)=∇∙𝐹

Basic Vector integration theories.

Theorem 1:

Let γ be an oriented curve in R3 (R- Real) with initial and final points P0and p1,

respectively. Let h(x, y, z) be a scalar function. Then,

∫∇ℎ. 𝑑𝑟 = ℎ(𝑃1) − ℎ(𝑃0)𝛾

Theorem 2:

Let M be an oriented surface in R3 (R - Real) with boundary given by the closedcurve γ,

withorientation induced from that of M. Let F(x, y, z) be a vector field.Then,

∬ (∇ × 𝐹). 𝑛𝑀

𝑑𝑆 = ∮𝐹. 𝑑𝑟𝛾

Theorem 1.3:

Let E be a bounded solid region in R3(R- Real) with boundary given by theclosed surface M,

with theoutward pointing orientation. Let F(x, y, z) be a vectorfield. Then,

∭(∇. 𝐹) 𝑑𝑉 = ∯ 𝐹. 𝑛 𝑑𝑆𝑀𝐸

Theorem 1.4:

A vector field F in R3 is said to be conservative or irrational ifany of the following

equivalent conditions hold:

∇ × F = 0 At every point.

∫ 𝐹. 𝑑𝑟𝛾

Is independent of the path joining the same two endpoints.

∮ 𝐹. 𝑑𝑟 = 0𝛾

For any closed path γ.

F = ∇h For some scalar potential h.

In fact this theorem is true for vector fields defined in any region where all closedpaths can

be shrunk to a point without leaving the region.

Theorem 1.5:

A vector field F in R3 is said to be solenoidal or incompressible ifany of the following

equivalent conditions hold:

∇.F = 0 At every point.

∬ 𝐹. 𝑛 𝑑𝑆𝑀

Is independent of the surface M having the same boundary

curve.

∯ 𝐹. 𝑛 𝑑𝑆 = 0𝑀

For any closed surface M.

F = ∇ × A For some vector potential A.

Similarly, this theorem is actually true for vector fields defined in any regionwhere all closed

surfaces can be shrunk to a point without leaving the region. The above two theorems should

look very similar. Everything is shifted up byone dimension and the curl is replaced by the

divergence, but the theorems areidentical in form.

APPLICATION OF VECTOR INTEGRATION IN FLUID DYNAMICS

To find the rate of change of the mass of a fluid flows.

Since the fluids are not rigid like solid parts in the fluid body can move in different velocities

and fluid does not have the same density all over the body. We can fiend the total mass in the

fluid region by integrating the density over R.

∭ 𝜌(𝑥, 𝑦, 𝑧)𝑑𝑥𝑑𝑦𝑑𝑧𝑅

If the region R is not changing with the time (assume that R is a control volume or fixed

volume), only way that mass going to change is by the fluid entering and leaving the R

through its boundary surface M. If we let v(𝑥, 𝑦, 𝑧, 𝑡) be a time dependent vector field which

the v will give the velocity at any point we can say that the flux integral of 𝜌v over M will

give the rate of change of mass flow.

𝑑𝑚

𝑑𝑡= ∰ 𝜌𝒗 𝒏𝑑𝑆

𝑀

So we can see the vector integration is used to fiend the rate of change of the mass of a fluid

flow.

Stock theorem

Calculate the circulation of the fluid about a closed curve.

Stock theorem is used in this. The application is circulation of the fluid about a closed curve

γ. This is just the line integral of v over γ, which we can rewrite for any surface m which has

γboundary.

∬ ∇ × 𝒗𝑀

𝒏𝑑𝑆

To analyze the vorticity of the fluid body

As the Wikipedia says vorticity is a pseudo vector field that describes the local spinning

motion of a fluid near some point (the tendency of something to rotate), as would be seen by

an observer located at that point and traveling along with the fluid in fluid dynamics.

In hear the Stoke’s theorem is used in calculation. It states that the circulation about any

closed loop is equal to the integral of the normal component of velocity over the area

enclosed by the contour.

∮ 𝒗. 𝑑𝑙 = ∬(∇ × 𝒗)𝐴

. 𝒏𝑑𝐴

Bjerknes Circulation Theorem

To analysis sea breeze

In fluid dynamics, circulation is the line integral around a closed curve of the velocity field. It

is obtain by taking the line integral of Newton’s second law for a closed chain of fluid partial.

It is known as the Bjerknes Circulation Theorem.

∫(𝑑𝑣

𝑑𝑡= −2Ω × 𝒗 −

1

𝜌∇𝑝 × 𝒈 × 𝑭)𝑑𝑙

This theorem use vector integration. This theorem is used in analyzing the bartropic fluids.

The definition of the baratropic fluids is that they are useful model for fluid behavior in a

wide variety of scientific fields, from meteorology to astrophysics. Most liquids have a

density which varies weakly with pressure or temperature, which is the density of a liquid, is

nearly constant, so to first approximation liquids are barotropic.

The sea breeze analysis can be explain using the barotropic flow

Figure: Sea breeze illustration

The sea breeze will develop in which lighter fluid the warm land air is made to rise and

heavier fluid sea air is made to sink. So the air from see will come to land to fill the free place

this occurs sea breeze.

APPLICATION OF VECTOR CALCULUS IN ELECTRICITY AND

MAGNETISM

In this discussion we will discuss the mathematical consequences of theorems.Let us take

Electric and Magnetic field in space as E(x,y,z,t) and B(x,y,z,t) where (x,y,z) represents the

position in space and t represents the time. Further let ρ(x,y,z,t) be charge density and

J(x,y,z,t) the current density in space. Current density is a vector field since current is given

by both magnitude and direction.

The equations governing Electricity and Magnetism are;

∇ · E =𝜌

∈0 Gauss, law

∇ × E = −𝜕𝐁

𝜕𝑡 Faraday’s law

∇・B = 0

∇ × B = μ0J + μ0ϵ0𝜕𝐸

∂t Ampere-Maxwell Law

Where; ϵ0 = 8.85×10−12𝑐2

𝑁𝑚2is the permittivity of free space andμ0 = 4π×10−7

𝑁𝑠2

𝐶2 is the

permeability of free space.

Magnetic field B is always solenoidal, and can be written as the curl of a vector potential B =

∇×A. Thus we can show that magnetic flux through any closed surface is always zero by use

of following theorem.

Figure: Electricity and Magnetic field

Theorem:

A vector field F in 3 dimensional spaceis said to be solenoidal or incompressible if any of the

following equivalent conditions are true:

∇・F = 0 at every point

∬ 𝐅・𝐧𝑑𝑆𝑀

is independent of the surface M having the same boundary curve

∯ 𝐅・𝐧𝑑𝑆𝑀

= 0 for any closed surface M

F = ∇ × A for some vector potential A

Since the divergence of any curl is zero, we can write using Maxwell’s equation;

∇・ (∇ × E) = ∇・ (−∂𝐁

∂t) = −

∂

∂t(∇ ・𝐁) = 0

For the magnetic field we get;

∇・ (∇ × B) = ∇・ (μ0J + μ0ϵ0𝜕𝑬

∂t)

∇・ (∇ × B) = μ0∇・J + μ0ϵ0𝜕

∂t (∇・E)

∇・ (∇ × B) = μ0 (∇・J+∂ρ

∂t)

For the consistency of divergence of curl to be zero it is required that∇・J+∂ρ

∂tto be zero.

This is ideally the conservation of charge.

Now let us consider constant electric E and magnetic B fields. Then the two time derivatives

get drop out of Maxwell’s equation. In this case the curl of electric field is zero. Thus we can

write E = −∇ϕ. Where ϕ is some scalar potential function ϕ(x,y,z). The minus sign is used for

the easiness thus; we can represent the flow of positive charge from higher potential point to

lower potential. In this constant field, over a closed path the cyclic integral evaluates to zero.

Now we have;

∇・E = −∇・∇ ϕ = −∇2 ϕ =ρ

ϵ0

When the object is highly symmetric we can use Gauss’s Law and Ampere’s Law to calculate

electric and magnetic fields. Consider a uniformly charged solid sphere of radius R. since

there is no any preferred direction from symmetry we can say that electric charge outside the

sphere is radially directed which only depend on the radius r from the origin. So E· n = E(r)

because the electric field is parallel to the normal vector. Now we can integrate both side of

Gauss’s Law over a solid sphere Br of some constant radius

r > R and use the divergence theorem:

∭ (∇ ∙ 𝐄)𝑑𝑉𝐵𝑟

= ∭𝜌

ϵ0𝐵𝑟

𝑑𝑉 = 𝑄

ϵ0

∯ 𝐄 ∙ 𝐧𝑑𝑆 = 𝑀

∯ 𝐸(𝑟)𝑑𝑆 = 4𝜋𝑟2𝐸(𝑟)𝑀

Where, Q is the total charge of the sphere.

E(r) is a constant on the sphere of radius r since ρ is constant in the charged sphere and zero

outside it

We can do an analogous calculation for magnetic fields. Suppose we have an infinitely long

thick wire (an infinitely long cylinder) of some radius R. Current is flowing through this

cylinder with some uniform current density J. Now because the force on a moving charge due

to a magnetic field is perpendicular to both the direction of motion of the charge and the

direction of the field, symmetry tells us that the magnetic field due to this infinite wire must

be tangential to circles perpendicular to and centered on the wire. That is, if we point the

thumb of our right hand in the direction of the current, the field lines go around the wire in

the direction of our fingers. By symmetry, the magnitude of the magnetic field depends only

on the perpendicular distance r from the wire. Now we integrate both side of Ampere’s Law

over a solid disc Dr of some constant radius r > R and use Stokes’

Theorem:

∬ (∇ × 𝑩)𝑑𝑠 =𝐷𝑟

∬ μ0𝐉ds𝐷𝑟

=μ0𝑰

∮ 𝑩𝑑𝑥𝑟

= ∮ 𝐵(𝑟)𝑑𝑟 = 2𝜋𝑟

rB(r)

WhereIis the total current through the wire, since J is constant in the wire and zerooutside it,

and B(r) is a constant on the circle of radius r. Thus we see

B(r) = μ0I/2𝜋𝑟

Which is the same at the magnetic field due to an infinitely thin wire with current I Inside the

wire the field is slightly more complicated. As a final illustration of the use of vector calculus

to study electromagnetic theory, let us consider the case where the fields are time varying, but

we are in free space where the charge and current densities are both zero. We will need to

make use of the following identity for a vector field F, which can be easily proved by writing

down the definitions and checking each component:

∇ × (∇ × 𝐅) = ∇(∇ × 𝐅) − ∇2𝑭

We apply this identity to both the electric and magnetic fields, and use all of Maxwell’s

equations to simplify the results, remembering that both ρ and J are assumed to be zero:

∇ × (∇ × 𝐄) = ∇(∇. 𝐄) − ∇2𝑬 = −∇2𝑬

= ∇ (−𝜕𝑩

𝜕𝑡) = −

𝜕

𝜕𝑡(∇ × 𝑩) = −𝜇° ∈°

𝜕2𝑬

𝜕𝑡2

and similarly:

∇ × (∇ × 𝐁) = ∇(∇. 𝐁) − ∇2𝑩 = −∇2𝑩

= ∇ (𝜇° ∈°𝜕𝑬

𝜕𝑡) = 𝜇° ∈°

𝜕

𝜕𝑡(∇ × 𝑬) = −𝜇° ∈°

𝜕2𝑩

𝜕𝑡2

Thus we see that each of the three components of both the electric and magneticfields satisfy

the differential equation

𝜕2𝑓

𝜕𝑡2= 𝐶2∇2𝑓

Figure: Electric field and Magnetic field

Forc = 1

√(𝜇°∈°)

This equation represents the motion of a wave with speed c. Hence we see that in free space

the electric and magnetic fields propagate as waves with speed

1

√(𝜇°∈°)=

1

√(4𝜋×10−7𝑁82)(8.85×10−12 𝑐2

𝑁𝑚2)

= 2.99863 × 108 𝑚

𝑠

Figure: Gauss Figure: Faraday Figure: Maxwell Figure: Stokes

This is exactly the speed of light. Maxwell studied on electromagnetic waves and was able to

deduce that light is an electromagnetic wave upon the experimental information of speed of

light back in 1880.Allelectromagnetic waves: gamma rays, X-rays, ultraviolet rays, light,

infrared rays, microwaves, radio waves; are propagating electric and magnetic fields. The

only difference is the frequency from wave to wave is different. They All travel at the same

velocity. The energy of the wave is proportional to the frequency, which is why X-rays are

far more harmful to us than radio waves.

REFERENCE

http://en.wikipedia.org/wiki/Stokes'_theorem

http://wxmaps.org/jianlu/Lecture_6.pdf

http://www.math.ubc.ca/~cass/courses/m266-99a/ch8.pdf

http://www.cs.umd.edu/~mount/Indep/Steven_Dobek/dobek-stable-fluid-final-2012.pdf