Problem on divergence and curl

[heman]

Hi Guys,,

i have just started to study Divergence and curl but this is not at all enetering into my mind...Pls help me out understand this...This also has Divergence and Stokes theorm ..pls help me grasp it...Thx in advance...


The Divergence Theorem and Stokes's Theorem provide the interpretation of the divergence and curl that we have given above.
The integral, over a surface S, measures the flux of v through the surface, which is proportional to the number of arrows of v that cross S.
By the divergence theorem if we take a tiny region V, the integral of div v over this region (which is the average value of div v in it times the volume of V), is the net outflux of v over the surface of V. Thus this outflux, which for V centered on the point r' is a measure of the number of v arrows originating from around the point r' is directly proportional to the average divergence of v around r'.
An exactly analogous interpretation of Stokes's Theorem on a surface S including the point r' provides our interpretation of the curl. The circulation integral of v around a small cycle encircling r can be interpreted as the difference between the path integral of v going around r' on one side and the other. By Stokes' Theorem, this is proportional to the area of the region between the paths times the average value of the component of curl v normal to S in that area. Thus Stokes' Theorem means that the average component of curl v normal to S around r' is directly proportional to the amount of path dependence of v in S produced in the neighborhood of r'.

 

[heman]

54 ppls have seen but no one willing to help...

 

[arildno]

It is somewhat unclear what you're after:
Do you want some "wordy" explanation; is that it?

 

[vincentchan]

The essay above is very clear already... if you don't point out what you don't understand more specifily, even we type a millions words here you still won't understand...
Why don't you tell us which part you don't understand so that we can help you...
usually ppls don't reply for 2 reason... unclear question and stupid question... yours is the former

 

[heman]

The essay above is very clear already... if you don't point out what you don't understand more specifily, even we type a millions words here you still won't understand...
Why don't you tell us which part you don't understand so that we can help you...
usually ppls don't reply for 2 reason... unclear question and stupid question... yours is the former

Thx actually i am not getting what does divergence and curl mean...i mean to say not their formulae but their significance ,,what does they want to speak..

 

[K.J.Healey]

The divergence is a measurement of the change in density of something, like a field in a given area.

The curl, is just the measurement of the curvature of a field.

Right?

Checkout mathworld.wolfram.com for both mathematical and a decent, if not complicated, description of what they represent.

 

[pmb_phy]

Thx actually i am not getting what does divergence and curl mean...i mean to say not their formulae but their significance ,,what does they want to speak..

I can't make sense out of that. It sounds like you quoted part of a book and without the context (there is a lot that isn't said that there which is probably made clear in diagrams and previous paragraphs and shown equations etc) and as such it gave me a headache reading it.

The divergence of a vector field is simply this - Suppose there is a vector field E. Choose a point r in that field and construct a closed surface, e.g. a sphere, around that point. Take the surface integral of the normal component of E over the closed surface and divide the result by the volume enclosed by that surface. Now let the radius of the spherical surface go to zero. That result is called the "divergence of E."

The curl is kind of like that - Take a point in space and pass a plane through it whose surface normal is parallel to the z-axis. In that plane construct a cirlce whose center is the point of interest. Take the line integral of the vector field around that circle. Let the sense of the line integral be consistent with the right hand rule with the +direction of the z-axis (if you grab the z-axis and your thumb is pointing in the +direction then the direction of the integration around the cirlce will be in the direction your fingers are curling). Now divide the result of that integral by the surface area of the circle. Take the limit and let the radius of the circle go to zero. That gives a number which is called the "z-component of the curl of the vector field E at the point." Do that with two other planes whose surface normals are in the +x and +y direction and you have the components of the curl vector.

Pete

 

[heman]

change in density...like density increase n decrease..its rate..

actually again comes basic problem..i know what is formula of curvature but not thorough what they represent..emphasis...i am checking it.thx

 

[heman]

thx Pete and Healey but what purpose they solve...in which way are they useful to us...can u explain with the help of example...

 

[vincentchan]

example? here you go...
electromagnetic, fluid dynamics, continuity equation, heat transfer...

 

[heman]

in what way ...they find their application here i agree..but how do they make the problem simple..i mean to say is there were no divergence and curl ...what could have been there //

 

[vincentchan]

if you don't use the div and curl notation, the 4 maxwell equations for EM will become a 20 variable, 20 equations monster

 

[heman]

if you don't use the div and curl notation, the 4 maxwell equations for EM will become a 20 variable, 20 equations monster

can i ask how ,,i am getting this but i read that in a static situation, the curl of the electric field is zero, and the divergence of the electric field is a multiple of the electric charge density c(x,y):

div(E(x,y)) = c(x,y) and curl(E(x,y)) = 0

how does this evolve...

 

[vincentchan]

do you prefer me answering this question verbally or mathematically?

 

[heman]

thx...verbally and may be if some maths is necessary

 

[dextercioby]

if you don't use the div and curl notation, the 4 maxwell equations for EM will become a 20 variable, 20 equations monster

24 equations with 22 unknowns...Yet 22 equations are independent...

Daniel.

 

[vincentchan]

for the divergence, it came from the coulomb's law... imagine you have a box. if you see a net electric flux goes in/ come out the box, coulomb's law tells you that there is a net charge in the box.. as the #7 post said, divergence is define as the net flux goes out the box divided by its volume,
\nabla \cdot \vec{E} = \lim_{\Delta V \rightarrow 0} \frac{\int \vec{E} \cdot d \vec {S}} {\Delta V}

applied the coulomb law to the right hand side

\nabla \cdot \vec{E} = \lim_{\Delta V \rightarrow 0} \frac {q/\epsilon}{\Delta V}= \frac{\rho}{\epsilon}

the curl E = 0 came from conservation of energy... we know any line integral of a closed loop is zero for E field otherwise a charge run along the loop will gain energy every lap its finish... again, go back to the definition in post #7
(\nabla \times \vec{E}) \cdot \hat{n} = \lim_{\Delta S \rightarrow 0}\frac{\int \vec{E} \cdot d \vec{l}} {\Delta S}
n hat is the normal vector for the area delta S

the integral at the right hand side is zero because the conservation of energy, therefore..
\nabla \times \vec{E} = 0
PS. the mathematical definition for the curl makes sense here because if the E field does not have a rotational tendency, its line integral for a loop will not be zero... you really need to picture it in your mind... hard to explain

 

[dextercioby]

(\nabla \times \vec{E}) \cdot \hat{n} = \lim_{\Delta S \rightarrow 0}\frac{\int \vec{E} \cdot d \vec{l}} {\Delta S}
n hat is the normal vector for the area delta S

The integral at the right hand side is zero because the conservation of energy, therefore..
\nabla \times \vec{E} = 0

What?Where did u get that one?

Daniel.

P.S.I noticed you edited your post...

 

[vincentchan]

http://www.aw-verlag.ch/Documents/Notation of Maxwell Field Equations.PDF
22 and 22? the file here said 20/20... can you should me how do you get 22/22, and compare with my 20/20, which 2 do i missed?

 

[vincentchan]

which one?

don't argue with me which definition for div E and curl E is more fundemantal again... this definition can derive any of your difinition in 3D space... and you can really visuallize what's going on physically

yes i edited my post coz i forgot the dot n hat in the curl E

 

[dextercioby]

Identify the number of equations and the number of unknowns

\nabla\cdot \vec{D}=\rho (1)

\nabla\cdot \vec{B}=0 (2)

\nabla\times \vec{E}=-\frac{\partial \vec{B}}{\partial t} (3)

\nabla\times \vec{H}=\vec{j}+\frac{\partial \vec{D}}{\partial t} (4)

\vec{j}=\overline{\overline{\sigma}}:\vec{E} (5)

\vec{P}=\epsilon_{0}\overline{\overline{\chi}}^{(el.)}:\vec{E} (6)

\vec{M}=\mu_{0}\overline{\overline{\chi}}^{(mag.)}:\vec{H}(7)

\vec{D}=\epsilon_{0}\vec{E}+\vec{P} (8)

\vec{B}=\mu_{0}\vec{H}+\vec{M}(9)

\frac{\partial \rho}{\partial t}+\nabla\cdot\vec{j} =0 (10)

a)Count the equations.
b)Count the unknowns.
c)Count the INDEPENDENT EQUATIONS...

Daniel.

 

[vincentchan]

OK, i think i know where is the problem came from, in the maxwell's original equation... \chi is NOT a tensor...

you are right... more than 20/20

 

[dextercioby]

which one?

don't argue with me which definition for div E and curl E is more fundemantal again... this definition can derive any of your difinition in 3D space... and you can really visuallize what's going on physically

yes i edited my post coz i forgot the dot n hat in the curl E

Nope,i was asking you how did u deduce that:
(\nabla\times\vec{E})\cdot \vec{n}=0 \Rightarrow \nabla\times\vec{E}=\vec{0}

Daniel.

 

[dextercioby]

OK, i think i know where is the problem came from, in the maxwell's original equation... \chi is NOT a tensor...

you are right... more than 20/20

Yes,of course...In ferromagnetic & ferroelectric materials,the \chi cannot be approximated to a second rank tensor...It could be somtehing like
P_{i}=\chi_{ijk}^{(el.)}E_{j}E_{k}

(similar for the magnetic case) or even worse...(Totally nonlinear effects).

Daniel.

P.S.I hope u see that the # of equations has nothing to do with the rank of the polarisability/susceptivity tensor...

 

[vincentchan]

(\nabla \times \vec{E}) \cdot \hat{n} = \lim_{\Delta S \rightarrow 0}\frac{\int \vec{E} \cdot d \vec{l}} {\Delta S}
since n is arbitrary unit vector, the RHS vanishes iff \nabla \times \vec{E}=\vec{0} I thought it is obvious...

wanna ask you a question... you have 6 variable in \vec{E} and \vec{B}, however, E and B is related to the A field and a potential... that's mean you can reduces those 6 variable into 4... that's mean I was right, or, am I right?

 

[dextercioby]

What about the ELECTROMAGNETIC POTENTIALS??

What do they have to do with the 24 equations??Keep in mind that the electromagnetic potentials work in the case of the (electromagnetic) vacuum...Not in materials,not in a simple way,that is...

Daniel.

P.S.Unfortunately,here in Belgium i didn't bring that book with me... So basically,i'll stop here,coz i lack documentation...

 

[pmb_phy]

Thx actually i am not getting what does divergence and curl mean...i mean to say not their formulae but their significance ,,what does they want to speak..

Here is a good example and a simple example. Please see - http://www.geocities.com/physics_world/mech/mass_conservation.htm

As the URL says, its about the conservation of mass. I.e. the total quantity of mass is constant. This means that if there is a region of space in which an amount of mass is flowing out of then the total mass in that region must decrease by the amount that flowed across the surface (that's where flux comes into play). Try to follow the derivation.

the RHS vanishes iff I thought it is obvious...

It is not obvious because one does not imply the other. It could be that curl E is non-zero but simply perpendicular to n in which case the dot product is zero.

Pete

 

[jdavel]

pmb,

"It could be that curl E is non-zero but simply perpendicular to n in which case the dot product is zero."

No. n is normal to the infinitesimal surface (defined on the RHS). The orientation of that surface is arbitrary, so if the RHS vanishes, so does the curl.

 

[dextercioby]

It is not obvious because one does not imply the other. It could be that curl E is non-zero but simply perpendicular to n in which case the dot product is zero.

Pete

It needn't,Pete.it's an equality.Not an implication,simple or double...The surface is arbitrary.If u happen to find a surface whose unit vector's scalar product with the curl is zero,that's an accident.However,u need to make sure that the equality holds for every possible surface...And the only way u can do that is setting the curl to zero.

Daniel.

 

[pmb_phy]

pmb,

"It could be that curl E is non-zero but simply perpendicular to n in which case the dot product is zero."

No. n is normal to the infinitesimal surface (defined on the RHS). The orientation of that surface is arbitrary, so if the RHS vanishes, so does the curl.

Yep. You're right. Don't know what I was thinking. Rough day I guess.

Pete

 

[CronoSpark]

I can't make sense out of that. It sounds like you quoted part of a book and without the context (there is a lot that isn't said that there which is probably made clear in diagrams and previous paragraphs and shown equations etc) and as such it gave me a headache reading it.

The divergence of a vector field is simply this - Suppose there is a vector field E. Choose a point r in that field and construct a closed surface, e.g. a sphere, around that point. Take the surface integral of the normal component of E over the closed surface and divide the result by the volume enclosed by that surface. Now let the radius of the spherical surface go to zero. That result is called the "divergence of E."

The curl is kind of like that - Take a point in space and pass a plane through it whose surface normal is parallel to the z-axis. In that plane construct a cirlce whose center is the point of interest. Take the line integral of the vector field around that circle. Let the sense of the line integral be consistent with the right hand rule with the +direction of the z-axis (if you grab the z-axis and your thumb is pointing in the +direction then the direction of the integration around the cirlce will be in the direction your fingers are curling). Now divide the result of that integral by the surface area of the circle. Take the limit and let the radius of the circle go to zero. That gives a number which is called the "z-component of the curl of the vector field E at the point." Do that with two other planes whose surface normals are in the +x and +y direction and you have the components of the curl vector.

Pete

Very nicely defined. I can visualize it precisely with your definition... just have a few questions that I would like to have clarified.

I am just wondering for the divergence, what is the purpose of dividing the closed surface by the volume... is it to find an average outward projection within each point of the enclosed surface?

And simularly for the curl... just the average projection of the xy vector components in the xy plane for the z-component of the curl?

Thanks

-CronoSpark

 

[heman]

I can't make sense out of that. It sounds like you quoted part of a book and without the context (there is a lot that isn't said that there which is probably made clear in diagrams and previous paragraphs and shown equations etc) and as such it gave me a headache reading it.

The divergence of a vector field is simply this - Suppose there is a vector field E. Choose a point r in that field and construct a closed surface, e.g. a sphere, around that point. Take the surface integral of the normal component of E over the closed surface and divide the result by the volume enclosed by that surface. Now let the radius of the spherical surface go to zero. That result is called the "divergence of E."

The curl is kind of like that - Take a point in space and pass a plane through it whose surface normal is parallel to the z-axis. In that plane construct a cirlce whose center is the point of interest. Take the line integral of the vector field around that circle. Let the sense of the line integral be consistent with the right hand rule with the +direction of the z-axis (if you grab the z-axis and your thumb is pointing in the +direction then the direction of the integration around the cirlce will be in the direction your fingers are curling). Now divide the result of that integral by the surface area of the circle. Take the limit and let the radius of the circle go to zero. That gives a number which is called the "z-component of the curl of the vector field E at the point." Do that with two other planes whose surface normals are in the +x and +y direction and you have the components of the curl vector.

Pete

..but still 1 thing in case of divergence how to decide the normal compnent becoz we have considered an apherical surface and this was from urs integral point of view..can u explain tthat from differntial point of view too...the previous one was well explained .thx very much..

 

[dextercioby]

It's coming from here
\oint\oint_{\Sigma} \vec{E}\cdot \vec{n} \ dS

You see,that "vec{n}" is the NORMAL UNIT VECTOR EXTERIOR TO THE CLOSED (ARBITRARY,YET ORIENTABLE) SURFACE SIGMA.Taking the scalar product of this unit vector with the vector field,the normal component of the vector field is the only that stands,because the tangential one (which forms a right angle with the (exterior) normal to the surface)) does not give any contribution.

That scalar product is essential...

Daniel.

 

[heman]

but how do we really decide the surface...on which we do surface integration..and that normal is perpendicular to it..

 

[dextercioby]

I believe i inserted the word ARBITRARY...Orientable (as to be able define the vector field of unit vectors normal to the surface in each point of the surface) and closed.Nothing more.

The fact that this surface (denoted by me with \Sigma) is ARBITRARY is very useful...

Daniel.

 

[heman]

ahh is it...we take an spherical surface..take any arbitrary planar surface and take its normal and then take the dot product with the field and compute ...thx clear now..

 

[vincentchan]

not only sphere, you can take a cube, a cylinder or anythinig you what...

 

[heman]

ok...and how will be explain divergence and curl if we take from differential point of view//

 

[dextercioby]

not only sphere, you can take a cube, a cylinder or anythinig you what...

Does the word SMOOTH mean anything to you (mathematically speaking,of course)??How would you define the exterior normal in the 8 corners of the cube and on its 12 sides??

Daniel.

 

[dextercioby]

ok...and how will be explain divergence and curl if we take from differential point of view//

In this case,"explanation"----------->"definition"...You define the divergence as the contracted tensor product between the derivative operator nabla and the vector field.And the curl as the cross product between the same diff.operator and the vector field...

Daniel.

 

[vincentchan]

Does the word SMOOTH mean anything to you (mathematically speaking,of course)??How would you define the exterior normal in the 8 corners of the cube and on its 12 sides??

what is the total surface area for the 8 corners and 12 sides?
do I have to show you the proof of \nabla \cdot \vec{F} = \frac{\partial F_{x}}{\partial x} + \frac{\partial F_{y}}{\partial y} + \frac{\partial F_{z}}{\partial z} when I apply \nabla \cdot \vec{F} = \lim_{\Delta V \rightarrow 0} \frac{\int \vec{F} \cdot d \vec {S}} {\Delta V} to a cube?

EDIT...heman...differential form? you mean you want me to show \nabla \cdot \vec{F} = \lim_{\Delta V \rightarrow 0} \frac{\int \vec{F} \cdot d \vec {S}} {\Delta V} =\frac{\partial F_{x}}{\partial x} + \frac{\partial F_{y}}{\partial y} + \frac{\partial F_{z}}{\partial z}?

 

[heman]

i was asking for the geometrical interpretation...

 

[pmb_phy]

Does the word SMOOTH mean anything to you (mathematically speaking,of course)??How would you define the exterior normal in the 8 corners of the cube and on its 12 sides??

Daniel.

You don't. To find the flux you evaluate over the flat surfaces of the sides of the cube.

Go ahead. Give it a try. Do exactly this. You'll find that the answer you get is

\nabla \bullet E = \frac{\partial E_x}{\partial x} i + \frac{\partial E_y}{\partial y} j + \frac{\partial E_z}{\partial z} k

In fact this is how this expression is usually derived, i.e. by using a cube. I'm surprised that you've never seen this derivation ... or have you? If not then see div grad curl and all that - 3rd ed, by H.M. Shey, pages 36-40. Most Calculus texts do this same derivation as I recall.

Pete

 

[heman]

what is the total surface area for the 8 corners and 12 sides?
do I have to show you the proof of \nabla \cdot \vec{F} = \frac{\partial F_{x}}{\partial x} + \frac{\partial F_{y}}{\partial y} + \frac{\partial F_{z}}{\partial z} when I apply \nabla \cdot \vec{F} = \lim_{\Delta V \rightarrow 0} \frac{\int \vec{F} \cdot d \vec {S}} {\Delta V} to a cube?

EDIT...heman...differential form? you mean you want me to show \nabla \cdot \vec{F} = \lim_{\Delta V \rightarrow 0} \frac{\int \vec{F} \cdot d \vec {S}} {\Delta V} =\frac{\partial F_{x}}{\partial x} + \frac{\partial F_{y}}{\partial y} + \frac{\partial F_{z}}{\partial z}?

yeah ..and also pls explain geometrically..

 

[pmb_phy]

yeah ..and also pls explain geometrically..

That is a geometrical explanation.

Pete

 

[heman]

That is a geometrical explanation.

Pete

i mean to say ..doesn't we explain in any specific way like we did by view of integral calculus and take the surface do integral and apply limit...and it will be nice to see how does that come..

 

[dextercioby]

You don't. To find the flux you evaluate over the flat surfaces of the sides of the cube.

Go ahead. Give it a try. Do exactly this. You'll find that the answer you get is

\nabla \bullet E = \frac{\partial E_x}{\partial x} i + \frac{\partial E_y}{\partial y} j + \frac{\partial E_z}{\partial z} k

In fact this is how this expression is usually derived, i.e. by using a cube. I'm surprised that you've never seen this derivation ... or have you? If not then see div grad curl and all that - 3rd ed, by H.M. Shey, pages 36-40. Most Calculus texts do this same derivation as I recall.

Pete

I've seen it...In my 10-th grade and i still remember it.However,it defines the flux through the LATERAL SURFACES OF THE CUBE/RECTANGULAR PARALLELIPIPED.And yet my problem is unswered.In the corners and on the sides u cannot define normals to the surface... The flux through these 0 and 1-D manifolds is still zero,but the normal cannot be defined...

Daniel.

 

[vincentchan]

even the normal @ the corner is undefine... you can still take the divergence of a cube... so why are you bring it up here?

i mean to say ..doesn't we explain in any specific way like we did by view of integral calculus and take the surface do integral and apply limit...and it will be nice to see how does that come..

the maths is straight forward and easy, and most vector calculus textbook has it... no one will waste his time to type it here, why don't you google a little bit and see if you have luck

 

[heman]

okay...its also not good to ask every minute thing..thx very much

 

[pmb_phy]

I've seen it...In my 10-th grade and i still remember it.However,it defines the flux through the LATERAL SURFACES OF THE CUBE/RECTANGULAR PARALLELIPIPED.And yet my problem is unswered.In the corners and on the sides u cannot define normals to the surface... The flux through these 0 and 1-D manifolds is still zero,but the normal cannot be defined...

Daniel.

Perhaps you can find your answer in a question. Ask yourself how you can calculate the flux through the entire surface of the cube and yet you can't define it. If you can answer that question to your own satisfation then you have the answer to your question.

Pete