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Very well let us resume our discussion on the Gauss's law and we will now apply it to
fluid mechanics and develop what is a statement of conservation of matter expressed as what
is known as equation of continuity
Now this is the Gauss's divergence theorem that we meet toward the end of the last class
what it tells less that if you take a volume integral of this quantity which is the integrant
this is what we defined as a divergence of the vector function A which is a vector point
function we first define what are vector point function is
It needs to be continuously derivable so that we can define this so the continuity is an
essential element then we defined what fluxes we have to define the flux in terms of the
scalar product of the vector field with a directed area oriented area
So we defined how to orient an area we introduce that idea constructed this surface this this
dot product this is a scalar product and then we added it up to get the surface integral
and when we did this over a closed surface which encloses a certain finite volume of
space we found that it is equal to the volume integral of a certain integrant which we called
as the divergence and then mathematically quality that we established is what we call
as the Gauss's divergence theorem
Now let us see that if this vector field A is specifically the electric intensity due
to a point charge we let us take a very simple example just to see how it works this vector
point function can be any vector field in particular we take it to be the electric intensity
of a point charge Then this is the divergence of E and on the
right hand side you got the surface integral of E dot dS and there is this dS has got a
direction indicated by the unit normal n which of course goes changes from point to point
So it is at the same point where you consider this E of r and about that point you construct
an infinitesimal surface element which will then shrink to 0 and then you add it up integrated
over the entire close surface So now let us look at the right hand side
electric intensity due to a point charge goes 1 over r square there is this usual 1 over
4 pi epsilon 0 factor there is this redial direction of the intensity vector then you
have this dot product coming over here indicated by this dot over here
Now you have got a surface element which we know is r square times a solid angle which
a subtended let say by a sphere at the center if the charge over at the center of the sphere
and then the unit normal to this sphere is nothing but the radial normal vector which
is plus E r in the spherical polar coordinate system so the right hand side is very easily
spelled out Now the evaluation of the surface integral
is very simple because this r square cancels this r square in the denominator q over 4
pi is a constant which you can pull outside the surface integration what are you left
with the 1 over epsilon 0 also comes out what do you are left with all you need to do is
to integrate the solid angle over here all the angles which is nothing but 4 pi and that
will cancel the 1 over 4 pi which is already there so the result is very simple it will
be just this charge divided by epsilon 0 the 4 pi now gets canceled off so the volume integral
of the divergence of A or divergence of E in this case is nothing but q over epsilon
0 But then you can also think of the charge
itself as the volume integral of a charge density because it could come from a finite
charge density which is sitting inside that regional space
So you can think of this as a volume integral of the charge density and then you can move
this 1 over epsilon 0 once again inside the integral and make it a part of the integrant
so you got a volume integral of rho over epsilon 0 and now you have these 2 volume integrals
the one on the extreme left which is volume integral of the divergence of E and the extreme
right which is a volume integral of rho over epsilon 0 and both of these are definite integrals
So they be in definite integrals over the same limits the corresponding integrands must
be equal and you can equate the divergence of E to be equal to rho over epsilon 0
So this is also a result of the same theorem and this is an integral form it you need to
carry out these integrations over in you know extended space so this is like a global form
because you know it needs to consider all points in space whereas when you equate the
corresponding integrands you get it in what is known as the point form so this is sometimes
called as a integral form of the Gauss's divergence theorem this is called as the deferential
form or the point form And this is the physical content or the mathematical
content of these two is not different from each other they are completely equivalent
Now this is what we have got we recognized this integrant and decided to call it as divergence
of a and offered it a notation written as del dot A I am choosing my words very carefully
the integrant of the volume integral is what we have called as a divergence of a we have
not defined divergence of a like this the reason is a definition of any quantity should
not really depend on a coordinate system So this integrant is what we have called as
the divergence of a we recognize it it is of course an appropriate correct mathematical
expression of the divergence of a in the cartesian coordinate system but this is not something
that we will admit as the definition of the divergence of a because we will like the definition
of quantity of any physical quantity to be independent of a coordinate system
So we have you know come a long way by exploiting the cartesian geometry but we must carry this
mathematical formalism further in a manner which is independent of the coordinate system
So this is a cartesian expression but not the definition of the divergence so it is
correct in the cartesian frame of reference nothing wrong with it but let us not give
is give a this stratus of a definition it is not the signature
So let us now introduce a definition which is free from any coordinate system and if
you look at this mathematical equality it already suggest to you what the definition
should be and this mathematical equation which we now know is correct we have actually seen
that it is correct it has been arrived at by introducing the idea flux very carefully
then by adding up the flux crossing this surface which bounds a close region very carefully
And by extending this result to a region which does not depend on this rectangular parallelepiped
shape but make it applicable to an arbitrary shape and then expressing the vector in any
coordinate system the definition of the divergence of a automatically pops up because on both
sides we have integrals and we know that these are limits of a sum
So if you take a sufficiently small region of space in a sufficiently tiny volume no
matter what is shape is it does not have to a rectangular parallelepiped it does not have
to be cylinder this is not even a cylinder it has got cylindrical symmetry but it is
not a cylinder You can take any close shape this is a close
shape it is not even cylindrical it is not spherical it is got some obituary some different
shape that of a rat so you take any regional space not quite a rat it does not even have
a tail So this is a anyhow you take any region of
space and construct this flux and divided by the volume which is enclosed by that surface
take the limit delta V going to 0 and now you have got a right hand quantity which has
nothing to do with the cartesian geometry it has nothing to do with the cartesian coordinate
system Or we have exploited is the relationship between
sums limits of the sum and integrals that is all we have done regular calculus and if
you take this region to be sufficiently small take the limit delta V going to 0 we now have
what we will call as a definition of the divergence of a vector quantity this is no longer just
restricted to a cartesian geometry it is applicable to any coordinate system but then we must
find how would we express it in other coordinate systems we will learn that
So it is meaning really is total flux this is the total flux per unit volume in the limit
volume going to 0 so the divergence of a vector quantity is the total flux of that vector
field across the region which bounds a certain finite volume of space in the limit that volume
shrinks to 0 so it becomes a point function flux must be defined through the surface no
matter how small that surface is and therefore the flux is the property not just of one point
So the flux is not a point function but the divergence of a vector field is a point function
because you define it in the limit when the volume shrinks to a certain point
So the flux is a scalar quantity it is not a scalar field a scalar field is described
by a scalar point function so that it is defined for every point in space so you have to be
a little careful about these details the divergence it will describe a scalar field it is a scalar
point function defined for each point in space
So this is now our divergence theorem that the volume element of the divergence of A
is equal to the surface integral of the flux of a through the surface which encloses that
volume so very simple statement The direction of the oriented surface is always
consider to be the normal to the surface it is orthogonal to the surface and it is always
considered as the outward normal the definition of out and in there is no ambiguity in it
because this is a close region of space so the one that is pointed out over to the space
is the direction of n the unit normal
What the Gauss's divergence theorem does is it enables us to express a surface integral
in terms of a volume integral and vice versa what it physically means is that if you have
water coming out of a faucet and if you integrate over all the faucets there may be some other
holes at different places if this is a rested one there may be some holes on the top as
well that is not a one that you want use but who knows there may be some
And if you integrate over all the faucets over a certain volume region then you are
going to get the net flux which is coming out through the surface which encloses the
whole volume so it is essentially a conservation principle
Now this is what we have in the cartesian coordinate system we must ask how shall we
express the divergence of A in the cylindrical polar coordinate system some other coordinate
system let us do it So we already know how to write the gradient
in the cylindrical polar coordinate system this is something that we spends quite some
time discussing in unit 7 that was long time back once you know once up on a time but I
hope you remember it That the gradient has got a very simple expression
in this cylindrical polar coordinate system it will include the unit vectors of the cylindrical
polar coordinate systems which are e rho e phi and e z it will have partial derivatives
with respect to the polar coordinates which are rho and phi and also z in the cylindrical
polar coordinate system So all you have done is to write the gradient
over here in cylindrical polar coordinate system place the dot and written the vector
A again in the cylindrical polar coordinate system so you resolve it in 3 components along
e rho then along e phi and along e z that components being A rho A phi and A z but these
are point function so each component of A is a point function and in general each component
may depend not just on the subscript that it is referring to but all the 3 coordinates
So A rho will be a function not only of rho but also a phi and z likewise e phi will be
a function not just a phi but also of rho and z and A z will be a function not just
of z but also of rho and phi because these are point functions so they change from point
to point Now here I had learned how to get the divergence
in the cylindrical polar coordinate system all we have done is to written it but what
we have in front of us are two operations In what we have written here there are two
operations which are involved one is vector algebra because there is a dot sitting over
here the other operation is calculus because we also have to take derivatives
So there are 2 different mathematical features that we must simultaneously respect and whatever
we do must not be in conflict of vector algebra and it must not be in conflict with the calculus
and our mechanisms mathematical mechanisms to take the differentials and the derivatives
And the reason we have to be careful with in the cylindrical polar coordinate system
which we did not have to do in the cartesian coordinate system is because the unit vectors
of the cartesian coordinate systems which are e x e y e z they are not changing from
point to point in space But the unit vectors of the polar coordinates
they do change from point to point well not with respect to all the points because e z
of the cylindrical polar coordinate systems is of course a constant vector it does not
change with respect to anything and e rho and e phi change only with phi and not with
rho we have done this quite carefully when we introduce these coordinate systems that
was also once up on a time So the unit vectors e rho and e phi change
with the azimuthal angle phi but not with rho so whenever you take derivatives and this
is where you take the derivatives derivative with respect to rho not just a derivative
but this is the partial derivative which means that when you take derivative with respect
to rho you hold the other 2 coordinates phi and z constant
You take the derivative with respect to phi over here holding rho and z constant so whenever
you take the partial derivatives you must take partial derivatives not just of A rho
which obviously depends on all of these but also of e rho which depends at least on phi
if not on rho and z So the partial derivative with respect to
phi of e rho must be considered explicitly so whenever you take partial derivatives you
have to treat this as a product of 2 functions one is a vector function e rho the second
is a scalar function A rho And treat it as a product of 2 functions so
that you get the first function times of derivative of the second plus the derivative of the second
function times the first function that usual rule of taking the derivative of a product
of 2 functions will come into play So let us do this carefully there are two
operations so I do is carefully now all I have done here is to write it term by term
the first term is e rho del by del rho then I have this dot and then I have the rest of
the vector so I am still not carried out any of the two operations I haven't carried out
either of the two operations I have only separated the 3 terms so that we do one step at a time
Very carefully one step at a time so we have separated the 3 terms one is this this is
the second this is the third you have got the dot placed over here and then you have
got the vector a expressed in cylindrical polar coordinates on the right of this dot
And now you carry you respect the fact that you have to take the derivatives not just
of the components but also of the unit vectors so before you take the scalar product this
derivative must be carried out this process of differential there are two operations the
vector algebra and the differential calculus or the differential vector calculus if you
like the components a rho a phi a z of course depend
on all the 3 coordinates but the unit vectors also depend at least on phi f not on others
in spherical polar coordinates the dependence is a little larger but in this case there
is no dependence on rho and z So here we go so you must first take the derivative
this operation which is del by del rho let just look at the first term look at the first
term you have to take partial derivative with respect to rho of this entire thing so I move
this derivative operator del by del rho to the right of this dot
So then I can carry out the derivation of the product of e rho and A rho with reference
to rho with respect to rho holding phi and z I do the same with the second term which
is 1 over rho del by del phi and then I operate I think of this as a deferential operator
which is a derivative operator this must operate on the right hand side and this right hand
side has got 3 terms each of which is to be seen as a product of a scalar function and
a vector function of the point So and keep a track of where the differential
operators are coming so this deferential operator which were setting to the left of this dot
has now moved to the right of this dot likewise this differential operator has move to the
right of this dot it really does not matter where the 1 over rho comes because that is
just a multiplying factor here so it does not matter and then same thing with the z
the derivative with respect to z
So now what we have to do is to carry out the deferential calculus first and then we
will do the scalar product which is the vector algebra that is something that we will do
in the end now when you do the calculus you have to remember that the derivative of the
unit vector e rho with phi is e phi the derivative of e phi with respect to phi is minus e rho
so this we have done we will use this results and we will need this when we take the derivatives
of e rho and e phi with respect to this deferential operator del over del phi
So if you do this term by term and I will let you work it out as an exercise it is a
simple one then the result for the divergence of A is del over del rho a rho 1 over rho
A rho rho phi z 1 over rho del by del phi a phi and then del over del phi del z A z
so there are really 4 terms 1 2 3 and 4 that you must take into count
So I hope that the procedure to obtain the explicit expression of the divergence in the
cylindrical polar coordinate system is clear see here arrived that it in 2 different steps
first we establish the Gauss's divergence theorem in the coordinate system then we suggested
that the result must be independent of the coordinate system and when we expressed it
in a manner which is independent of the coordinate system we had the left hand side of the Gauss's
divergence theorem to be given by a volume integrand the integrand of which we defined
as the divergence We recognize a form of the divergence in the
cartesian coordinate system and then we asked what must be its form in the cylindrical polar
coordinate system and to get that form we carried out the 2 mathematical processes which
are involved one is calculus the differential calculus the second is the vector algebra
in the correct sequence so that whatever is derivable with respect to the independent
degrees of freedom is properly accounted for by using a calculus after which we use the
vector algebra so the result is rather simple
In this spherical polar coordinate system the same procedure has to be adopted details
are for you to work out at home staying awake not watching a movie but enjoying calculus
And you have met these results earlier actually it is more fun than many movies which are
so boring but some are good but only some anyhow so you have these relations which we
have established already and using this you can get the expression for the divergence
of A in this spherical polar coordinate system there is absolutely no need to by heart this
expression because it is so easily derivable So I am particularly alerting those of you
who have a very sharp memory because you will mug it up very fast and you will remember
the expression and then when you if you ever have to go for an exam you would have spent
what you call as night out and then you will be so tired at the time of the exam then you
will miss the 1 over r sin theta you will forget about it so no matter how sharp your
memory is please do not by heart it it will take you only 2 or 3 minutes to derive it
from first principles it is very easy
Let us take up some examples of the divergence so whenever the divergence of vector field
is 0 that vector field is said to be solenoidal because that is the kind of thing you have
for a magnetic field in a solenoidal so when the divergence of a vector field is 0 it is
called a solenoidal so it is just a name of the divergence name of a vector field when
its divergence is 0 the magnetic field is a good example of this
If you have a some other field here is an example of another a vector field and you
can determine its divergence by using the cartesian expression and find it is divergence
is constant it is a scalar it is a constant number it is a same no matter where you find
it
And now we can ask if there is any net accumulation of the flux in a volume element this is the
question that we had raised earlier on but now we are ready with the answer even if sources
and sinks were present Our interest is in determining not just what
will happen if there will be any net accumulation of the flux in a volume element but especially
so when the volume element shrinks to a point so that we can talk about point functions
we can talk about properties of a given point in space
So we consider you know mass or charge density either rho m or rho c which is crossing a
certain cross sectional area at a certain rate we have defined this quantity called
density times velocity I introduce this quantity earlier or it could be a charge density so
rho c which will have the dimension of charge per unit area per unit time so QL to the minus
2 T to the minus 1 and the quantity we have introduced is a product of density and velocity
what it represents is the amount of mass or charge crossing unit area in unit time
Essentially it is density time's velocity and it is called as current density vector
that is what it is called whenever you consider the product of density time's velocity it
could be the mass density it could be the charge density it does not matter
It is called as a current density vector which is a vector point function because for each
point in space you have a density and for each point of space you have got a velocity
defined and this is the velocity defined in the continuum limit of fluids
And accordingly this is the mass current density vector or a charge current density vector
and in general sources and sinks may be present in the region
What we have is a conservation principle because if you look at the net charge current density
which is flowing out of a close region of space charge per unit area integrated over
area will give you the charge And the charge flowing per unit time will
give you the current right d q by d t is the current so when you are integrating the current
density vector over the surface you will essentially get the current because of physical quantity
which the current density vector is which we have seen is the quantity of either mass
or charge crossing per unit area per unit time a quantity which is defined per unit
area integrated over the whole area will give you the current itself will this is the charge
per unit area so it will give you the net charge per unit time will give you the current
And therefore when you integrate this flux of the current density vector you are going
get the current which is oozing out of that region we have already establish the Gauss's
divergence theorem I have written it not for the obituary vector field a but now specifically
for the current density vector we know are we know are talking about specific physical
quantity which is either the mass current density or the charge current density and
our formalism develops in parallel applicable to both
So this current which is coming out which is the rate of change of charge rate of flow
of charge it will be del q by del t right it must come with the minus sign because there
is a conservation principle it has to come at the cost of the charge which is inside
the charge which is coming out has to be at the cost of the charge which is inside you
are not creating any charge or you are not destroying any charge in the absence of any
source of sink right So whenever you are not creating any charge
whenever you are not destroying any charge then the net flux which is coming out of a
closed volume which is given by the surface integral of this flux must be exactly equal
to minus del q by del t this is the simple conservation principle and this conservation
principle now that you have you understand what this minus sign is telling us then you
also see that this charge itself can be written as the volume integral of the charge density
And now the integration is over special coordinates the differentiation is over time so these
are independent of each other so I can carry them out carry out these 2 process one is
differentiation with respect to time and integration with respect to space independently of each
other and in any order that are like because here as are completely independent so I move
the derivative operator del by del t inside the integral sign
And now if you see this relation I have got a volume integral of the rate of change of
density now what this allows me to do is to interpret the net current which is oozing
out of their region in terms of this volume integral of the time derivative of the density
so we can write this a little more clearly now
These results we have got but this charge now the total charge we write as the volume
integral of the charge density move this time derivative operator inside the integration
sign because these 2 processes can be carried out independently of each other and now you
have got a volume integral over here which is equal to the surface integral but this
surface integral is equal to this volume integral over here
And now if you look at these two volume integrals this one over here and these this volume integral
these are definite integrals over essentially the same volume of space so the corresponding
integrands must be equal in other words del dot J must be equal to minus del rho by del
t the corresponding integrands must be equal So we have got a mathematical equality now
which is that the divergence of J is equal to the negative partial derivative partial
time derivative of the density now you can write this term on the left hand side and
you get an equation which is called as the equation of continuity the del dot J plus
del rho by del t is equal to 0 And the entire derivation is based on a conservation
principle so the equation of continuity expresses a conservation principle but you can express
it either in this integral form or in this differential form this is a point form because
it corresponds to properties at a given point here it includes integration over various
elements surface elements on the right hand side volume elements on the left hand side
so these required extended features whereas in this form when you equate the appropriate
integrands of definite integrals you have expressed it as point functions
So this sums it up you have got the equation of continuity what it tells us is that if there is no creation
or destruction of matter in the absence of any source or the sink that density within
a region of space can change only by having matter flow into the region or out of it
And it can do so only across the surface so if you carry out the integration of the flux
across the surface you get the net quantity which has oozed out or converged in so this
is the divergence theorem this is the mathematically very regular statement of the conservation
principle we will take a break here if there are any questions or comments I will be happy
to take them
Otherwise we take a break here and in the next class we will establish the equation
of fluid motion which is the equation of motion for a fluid this will require us to consider
a very rigoures description of a fluid in what is called as the eulerian description
or the lagrangian description of the fluid so we will discuss this in the next class
So that is the topic for the next class and at this point I will take a break if there
is any question I will be happy to take