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I. Cheat sheet - What you need to know about vector analysis from conservative field to the divergence theorem

SUMMARY : what you need to remember about VECTOR FIELDS (click on image)
to deal with Maxwell equations. more on dot and cross product.
applications of double integrals/triple integrals

picture to show the direction u along which the rate of change is computed (see summary)/This is a paraboloid z = f(x,y)
the yellow vector u gives the direction u along which the rate of change of f is computed.
Note that the vector u is drawn in the plan x,y.


cylindrical and spherical coordonates:
source: calculus website

Vector fields like velocity field (flow of water) , electric field, magnetic field are vector fields  with a magnitude and direction at every point.
 To describe a vector field
1) You must specify how the field spreads our or DIVERGES
(computing divergence of a field = computing  the flux of a field through a close surface (like a bag)  = number of field line leaving  OUT
the surface (from inside to outside))
If the divergence is not zero, it means there is a " source" or a " sink" inside the close surface.  
2) you must specify how the field circles around or CURLS.
(Water can curl around a vortex. a magnetic field can curl a current = compute the curl of the field).
(computing  the curl  = computing the line integral of the field/velocity  along a close line).
(In 1873 Maxwell wrote down 4 equations which specify the CURL and DIVERGENCE of electric and magnetic fields)

EXAMPLES:

I- example of divergence of a field / to see divergence of a magnetic field or electric field, see below Maxwell equations


Flux of water through a closed surface = 0
because no water is being created in the surface and
no water is being removed and the density stays
constant.

Like wise, The total of flux of light (from star or bulb) through a close surface is proportional to the intensity of the light. The light radiates outward.


Water  can  flow radially from a source. The flux measures how much water is being created.


II. application of grad f :  E = - gradV
V is the energy per unit charge or electric potential . It is a scalar field . E is a vector field. It is the electric force per unit charge.
We define the difference in potential (energy change  per unit charge) between 1 and 2 by the work done to bring
a test charge (1C) from 1 to 2 against an electric field E. (If the electric field is produced by a positive charge, you have to push hard
to move a positive test charge from 1 to 2. your energy is transformed into electric potential energy and if you let go, the charge
will be repelled and will gain kinetic energy) .

Our work is - work done by E or:  V2-V1 = - Sum (E.ds) or line integral along the oriented path that connects 1 to 2.
E is a conservative field because the line integral does not depend on the path betwen 1 and 2.
let's work in a (x,y) plane. dV is the work done along a line element ds by the field E.

dV = - E.ds    Since the path does not matter dV = - (Ex dx + Ey dy ) (( move along the x axis then along the y-axis to go from 1 to 2)
dV is a total differential (see vector analysis) so - dv/dx = Ex and -dV/dy = Ey   (should be curled derivatives)
or 
so E = - grad V where grad is the operator gradient. All conservative fields derive from a potential.

So Eu is the rate of change of the potential in the direction u. (Eu=- gradeV . u    dot product, see 1st picture above).
You can imagine the potential to be a mountain and the slope of the mountain in a given direction is the component of E along that direction.
The direction of the greatest rate of change is given by E=-gradV and shows which way a charge will move.
(along the steepest slope like water flowing down a mountain). A positive charge will move toward lower potentials and a negative
charge toward higher potentials (uphill). So given a potential, we can compute E. The equipotential lines V = constant (or surfaces)
are always perpendicular to the field lines. (grad V show the direction of the greatest change of V, has to be perpendicular).
Likewise, in a well a marble will move along the greatest slope.
see here these great applet in 3D that show potential V in 3D as well or mountain. the field lines show the path
that the charges follow. The also show the direction of greatest change of V. You can als see the equipotential lines.
Along these line, the energy of the charge does not change, no E along these line, no foce experinced by a charge.

See this great applet to visualize V in 3D as a well (Q is netative) or as a well (Q is positive). You can visualize the field lines
along which charges will move. They indicate the greatest rate of change. You can also see the equipotential lines. (no E along these lines).

the mountain is a positive pontential created by a positive charge Q. The white dots are positive test charges moving away from Q.
The well is a negative potential creates by a negative Q. The lines are the electric field lines. Show the greatest slopes. The charges
floow these path. This is the potential created by a dipole.
-------------------------------------------------------------------------------------------------------------------
more intuitive
Let's say you are in an electric field in space holding a test charge (+ 1C). If you take a step along the x-direction
if you don't measure a difference in potential that means you are walking along an equipotential line and the component of E along x is 0.
(the only way to get a 0 difference in potential).
Ex=0. If you do feel a difference in potential then the rate of change in this direction is E x.
Ex = - dV/dx     (derivative of V in the direction x, should be curled derivative )
Likewise: Ey = - dV/dy and Ez=-dV/dz
so E = - grad V where grad is the operator gradient. (see vector analysis ).
----------------------------------------------------------------------------------------------------


III)Understanding the operators CURL of a vector field (velocity and force field )

Curl of a velocity
(in italic vectors)
If the vector field is  a velocity field V(x,y,z),  the CURL applied to V find the rotation part of the motion.
It computes the angular velocity w (vector). So if there is no rotation in a velocity field, the CURL is 0.

Consider for example a uniform motion around the z-axis with a angular velocity w. d
Ɵ/ dt = w  / The  radius is 1.

The projection on the X,Y plan is the circle of radius one.
If you parametrize the motion you get: x = cos(
wt ) and y = sin (wt  )    so Vx = - wsin(wt)  and Vy=wcos(wt)  (components along X and Y axis)
so Vx = -w y and Vy = w x

So curl(V) = cross product between nabla and V = 2 w   is a vector along the z axis

Note :
=

curl of a field force F

Consider the torque exerted by a force field on a small solid of mass m around the z axis.
The torque measures how efficient a force  F  is in spinning the object.
Newton's second law applied to rotational motion becomes:

torque / angular inertia = dw/dt = angular acceleration
to compare to translation motion :
force / mass =  dV/ dt = linear acceleration
so :
force / mass =  dV/ dt = linear acceleration
torque / angular inertia = dw/dt = angular acceleration

Since curl V = angular velocity then  curl (force/mass) = torque/angular inertia = torque per unit angular inertia.
The curl of a force field measures the torque it qpplies to a solid. It measures the spinning effect of a force field on a solid m.
So If there is no spinning then curl (force field) = 0

But before we learnt that curl ( conservative force ) = 0  So that means, conservative force can't spin objects. A conservative force
can't induce any rotational motion. So the spinning of the Earth has nothing to do with the gravitational field.

(note : conservative forces derive from a potential so force = f . force so curl 
f= 0 )  f =
-------------------------------------------
IV) Electric flux and the GAuss's law


The flux of the electric field through a close surface S (like a bag)  is : (double integral)

En is the component of E along the surface element dA. dA is a vector  perpendicular to the surface element |dA| = dA
Consider now a point charge Q. The electric field flows outward and depends only on the distance r from the charge.
Le'ts have our " bag" shaped like a sphere and let's put the charge Q inside. En = Ke Q / r2  = Q/ (4 pi eo
r2   )    
eo is a constant
8.85418782 × 10-12 m-3 kg-1 s4 A2 . eo is the electric constant. Ke is the coulomb;s constant.
Since En is the same on the surface we are integrating, the flux = En integral (dA) over the surface A = En 4Pir    
 So flux of electric field through a close surface =
En 4Pir2      
But we know En =
Q/ (4 pi eo r2   )   so flux of E through close surface around the charge Q = Q/eo
flux(E) =
Qinside/eo This is Gauss's theorem

Using this method we can show that the electric field inside a conductive sphere = 0

We can also compute the electric field produced by a charged plate infinitely large .

In that case the Gauss surface is a cylinder . The electric field flows outward through the top and the bottow,
Because of symmetry, the total flow of E through the wall of the cylinder =0.
Flux of E through clylinder = total charge / eo= ( (Q/S) x S ) / eo  where Q/S is the density of charge (charge per unit area).
So E x 2S =
( (Q/S) x S ) / eo    or E = ( Q/S) /( 2 eo)
Surprisingly, if the plate is large compared to the distance where E is probed, E is constant and does not depend on the distance.
This is true if we are probing region close to the plate. (distance small compared to size of plate).

If we bring a negatively charged plate // to the positively charged plate, we get a capacitor. Between is air or an insulator.
we can apply what we learnt above and show that the electric field outside the capacitor is 0 (except at the edges)
and equal to E =
( Q/S) /(  eo)  between the plates

This is because outside the capacitor, the negative plate field and the positive plate field cancel each other. between the fields add up.

This is not true at the edge :


A paper clip on a thread will bounce back between the plates (charge transfer ) and can probe the field. the blue lines are the equipotential lines.
(see chapter work)

Gaussian's law can also be used to show that the electric field inside a conductor is 0.  the potential is constant.

VII) curl of velocity in a vortex.
In a sink, water will start to spin faster and faster as it get closer to the vortex. Any initial spin of the flow will give the water
an angular momentum L = m R V. The anglular momentum is conserved so the water moves faster as it gets closer to the center.

The velocity of water in a vortex depends on the strength of the vortex that is on the angular momentum of the water. (initial spin that is conserved).

 
In that case: circulation of velocity of the flow of water  = 2 pi |R|  |V|
because the magnitude of the velocity V is constant
so dot product V.dr = |V|  dl   (V and dr are vectors, dl is a line element of C the circle ).
so circulation of V =   |V| 2pi  R 
(circulation of dl  =  circumference of the circle). On a circle the angular momentum :
L = |mR x V| =  m  |R| |V| = L = constant/
The velocity depends on the strength of the vortex that is depends on the angular momentum L.
So circulation of V  = 2pi L/m  ( with |V| = L / m|R|)
Because the angular momentum is conserved, the velocity of the water increases as we get closer to the vortex.
L = m |R| |V|  so |V| = L / (m |R| ) The magnitude of the velocity is inversely proportional to |R|. .

VI) first steps to understand Maxwell equations
1/eo = 4 pi Ke   with Ke coulombs constant = 9 109    eo is the electric constant
 Km  = 4pi K uo     with uo =  magnetic constant = 4 Pi  10-7  and Km =
10-7
Ke/Km = 9 109 m2/s2 = c2 the speed of light squared.

Circulation and flux = to describe any vector field in space / static fields


GAuss 's law
electric field lines converge to negative charges and diverge from positive charge


The flux of an  electric field through a closed surface S is proportional to the electric charge inside. No matter the size of the surface. The charge q is the source of the flux. The flux measures the number of lines crossing the surface. The total number of lines is a measure  of the strength of the field.
To map this electric field on can probe the space with a unit charge and " feel" the force. This is true for any field. If the field " expands" from a source, the flux is proportional to the function source.
 
flux = flow of water in latin.

flux through a closed surface = divergence of the vector field.


The flux of an electric field is non zero.
An electric field converges to a point but never circulates.

div (E) = q / eo
1/eo = 4 pi Ke with Ke = 9  109
eo is the electric constant. (vacuum)
Ke is the coulombs constant.
magnetic fields never diverge or converge. they go in closed curves.


The flux of the magnetic field through a closed surface = 0.
divergence compute the total number of lines leaving the surfaces out. In = out so total = 0.
 This is because a magnet has 2 poles. number of lines in = number of lines out.

______________________________________
curl of  a static electric field - curl E = 0

The line integral of an electric field E along a Path
connecting A and B = - (VB - VA )
(see above, VB-VA= difference in potential)
So the line integral of E along the path = work of E
along the path = -  difference in energy between B and A.
The line integral is independent of the path.
or E = - grad V  (E is a conservative field)

(see below.  Say the charge Q is positive. When you are moving the test charge (+)  along the horizontal @ left, the potential decreases  by dV , when you are moving the test charge in the opposite directio @ right , the potential increases but by the same amount.
Same thing @ up and @ down so the total change = 0 ) 


So if the path is closed VB=VA and curl E = 0

The circulation of the electric field along a close path C is 0.

Work done by an electric field along a closed path = 0.
you can't get the energy by going around.
-----------------------------------------------------------

Ampere's law
curl of a static magnetic field curl B = uo I
A magnetic field curls around a conducting wire.


uo is the magnetic constant = µ0 = 4π×10−7
Circulation of B is not zero. B is not a conservative field. The circulation around a current is not zero.
A magnetic field does not converge to a point.
A static magentic field curls around a current I.


     
The circulation of the magnetic field along a closed curve is proportional to the current. The current is the source of the magnetic field. The circulation does not depend on the distance from the current.The strength of  the field depends only  the current. 
 
You can compare the circulaiton of B to the circulation of water in a vortex.  the magnetic field created by a constant current is also inversely proportional to |R|:
B = 2 km  I / |R|
2km I and L /m plays the same role.
 (see above)


Green theroem :
The circulation of a vector field along a closed curve = curl of the vector field through the surface.

B is not a conservative field. The circulation = work done, you can get energy from a circulation.
Energy is a flow of water per unit volume =
0.5 density V2
Energy is a magnetic field = 0.5 uo  B2Energy is an electric field = 0.5 eo E2

NON STATIC FIELDS : ELECTROMAGNETIC Induction/Faraday's law/ LENZ law/curl E 

F = qV x B
A magnetic field acts on moving charges. (cross-product)

A moving wire in a magnetic field. The charge will experience a magnetic force because F = q V x B
A moving charge in a magnetic field experience a torque exerted by B.
A current is induced , a voltage builds up in the wire. Called electromotrice force. This is called EM induction.

Faraday's law

A moving loop in a magnetic field will cause charges to circulate. A voltage build up. The total flux of the magnetic field through of the surface A = sum of the components of B perpendicular to the surface B. Nds. The current flows as it was driven by a voltage. The voltage depends on the rate of change of the flux.  It does not matter if it is the magnetic field that is changing or if it is the loop that is moving, as long as the flux changes. (number of lines going through the loop).



As the loop moves in the magnetic field, the flux of the magnetic field varies. Increases or decreases. The voltage induced  is =  - rate of change of the flux = -  dflux/dt
Lenz law

As a current is induced by the motion of the loop in the magnetic field, the induced current creates another field of its own. The effect of this field is to oppose the building up of the induced currents. THis is called Lenz law. You can't get more energy that you put in.



consequence of Lenz law:
job of a coil  in a circuit = opposes change of current.
voltage induced = -L di/dt


In a circuit with alternative current, a inductor ( a coil) opposes the building up of current.  (like a mass opposes acceleration, a coil opposes changes in current). The inductor is characterized by its inductance L..
In a circuit with alternative current, the alternative current is responsible for the changing of flux. It produces a changing magnetic field in the coils
(instead of a changing area in a stationary magnetic field).
In that case , dflux/dt = dflux/di . di/dt
and L = dflux/di = constant.
so voltage induced = -L di/dt
curl E = dV = - d flux/dt


Now sthe circulation of the electric field along the closed path is not zero anymore. The charges are driven by a voltage = circulation of the electric field around a loop = - dflux/dt
E curls around a changing magnetic field

The circulation of E along a closed path = - d/dt (flux of B through the surface). because either the magnetic field is changing or the surface.
There is a way to make an electric field circulate after all = by changing the magnetic field.


EM waves / displacement term

displacement current.
the magnetic field curl around a changing electric field. So in space, where there is no current, magnetic field can be produced if there is a changing electric field.

Maxwell used a capacitor to explain how a changing electric flux in space can create a magnetic circulation.

If a closed membrane go through a capacitor, Gauss;s law tells us that there is an electric flux through the membrane = q/eo with q the charge on the plate inside the membrane.
The charges build up on the plates so there is a current I = dq/dt
and the charges vary with time. see next picture:

differentiate both side.

If the charges build up on the plates and change with time, there is a changing electric flux  through surface
= 1/eo  I
or I = eo d(flux of E) / dt
Maxwell called this term the displacement current.

If there is a (displacement) current going through the surface enclosed by the path then there is a magnetic circulation.
So circulation of B is due not only to a constant current Io but also to a change in the electric flux.

This is the last equation of Maxwell.
The circulation of a magnetic field along a closed path = uo ( the constant electric current going through the loop + displacement current).
The displacement current is created by the changing electric flux through the surface enclosed by the path.

this explain how EM waves propagate in space where no current can circulate.

The displacement current = eo . the rate of change of the electric flux through the surface.


wave in the intensity of electric  field implies the existence of a changing magnetic flux. that means a wave in the magnetic field.

Consider an alternative current in an antenna. A changing B curls the current. Then E curls around B. Then E curls around B....
.
electric field and magnetic field can generate each other.  This how an EM wave can propagate in space.



B curls around the aternative current and E curls around B and B around E in space.. the waves moves in space.


 

Maxwell equations unified  electricity and magnetism . his equation descibes how  the magnetic field and the electric field effects are inter winded.
Maxwell equation : (2 triple integral, over volume = 2 divergences and 2 line integrals around surface = 2 curls).


or:



*Let's take the first equation. The electric field is conservative. It derives from a potential such as E =
grad(f) f is the electric potential.

The first equation gives: flux of E through a closed  surface S = total charge inside the surface divided by a constant.
This equations describes how an electric charge produces a conservative  electric field E out of a surface S.
The field expands radially out of the charge,
The charge is a " source" of flowing electric field .
We can write the equation as : div (E) = density of charge / constant. (divergence theorem). This is Gauss 's law.

*** Farady's law (induction) The third equation is non intuitive. The line integral of E along a close path (or circulation)  is non zero if there
is a changing (with time)  magnetic field (produced for example by an alternative current) that flow through the
surface enclosed by the closed path.
.If there is no changing magnetic field, the line integral of E along a close line should be zero.
This is because E derives from a potential and is conservative. curl (conservative field) = change in potential = 0
along a close surface.  The work done = change in energy = 0 since no net work is done. (if there is no changing  magnetic field)
However, if there is a changing magnetic field inside that closed path  (changing flux of B through surface enclosed by closed path),
then  there is  an induced voltage ( = -  d (flux of B through surface) / dt) . This is the EM law of induction.
and the curl(E) is no longer zero. cicrulation of E along the closed path = induced voltage.
or (Stoke's theorem) double integral of curl (E) =  - d/dt(flux of B through surface enclosed but the curve C).
This is Faraday's law of induction.

** The second equation is div (B) = 0 (or flux of B through a closed surface). Because a magnet has 2 poles the flux of B through a close surface = 0.
The fux of a magnetic field through a closed surface = 0 because there are as many lines in than lines out.
This is because a magnet has 2 poles. This is Guauss's law.

****Last equation. Ampere's law.  If there is no changing electric field,  the circulation of a magnetic field (or curl)
is equal to the constant current flowing through the path C. B curls around a stati current. This current Io gives rise to the circulation of B.
This was Ampere's Law. Maxwell added a second term that says that  a changing electric field flux through a path C
should also create a magnetic circulation. (since a changing magnetic flux creates an electric circulation).
That means B  also curls around a changing electric field.
This term is called the displacement current. It explains how EM magnetic waves can propagate in space
where is no current for magnetic field to be generated.
moving charge = changing electric filed  = circulation of B = changing magnetic field flux = circulation of E = changing electric flux ...
1/eo = 4 pi Ke   with Ke coulombs constant = 9 109    eo is the electric constant
 Km  = 4pi K uo     with uo =  magnetic constant = 4 Pi  10-7  and Km =
10-7
Ke/Km = 9 109 m2/s2 = c2 the speed of light squared.
  c = 1
√ μ0ε0





VI- KEPLER equations

Here is the derivation of :
1) If the orbit is an ellipse , the motion is in the plane (the angular momentum stays along the z axis)
the angular momentum per unit mass = J 
2) the motion is a conic (based on newton's second law F = ma and F = - G M Mo / r2 )
r = J2/GMo / (1+ecos 
q)  , this is the polar equation of an conic.  e is the eccentricity
3) The angular momentum is constant
4) the area swept per unit time stays constant.
5) The total energy of the object determines the type of conic the object follows




6) The polar equation for a conic is r= ep /( 1 +/- ecos(
Ɵ) )  
The origin is placed at F1. (one of the foci).
The + or - tells you on which side of the conic is the directrix. If the directrix is @right, for an ellipse, then it is a +
if the directrix is @ left of the ellipse, its a - . Same idea for hyperbola and parabola.
or:


Polar ellipses with F2 on the given axis and  F1 = (0,0)

positive x-axis:  
r p
1-ecos( q)
                positive y-axis:  
r p
1-esin( q)
negative x-axis:  
r p
1+ecos( q)
 negative y-axis:  
r p
1+esin( q)


The cosine/sine tells us about the orientation of the conic. For an ellipse with the major axis along the X-axis, it is a cosinus.

Kepler's laws derived

NOTES FROM CLASS: multivariable calculus


picture to show the direction u along which the rate of change is computed (see summary)/This is a paraboloid z = f(x,y)
the yellow vector u gives the direction u along which the rate of change of f is computed.
Note that the vector u is drawn in the plan x,y.

double integrals
finding volume, switching order of integration

changing variable/ polar coordinates / jacobians

applications of double integrals : mass, center of mass, average, weighted average, moment of inertia, charge.

vector fields

vector fields and line integrals

gradient field / conservative field

how to test for conservativeness
/ how to find the potential

Curl and Green theorem


flux in 2D = flux crossing a curve C
Green theorem for flux



Flux of water through a surface A during a time t  = vtA


vector field in 3D
flux in 3D / how to find the normal vector to the surface


divergence theorem Gauss-Green theorem

The Green theorem for work becomes the Stoke's theroem in 3D.
see summary for formula. The curve C and the surface have to be oriented accordingly.


orientation of curve and surface

 











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