welcome to electromagnetics latest eye professor it is the lucky is going to explain you boss divergence theorem in this session so to understand Gauss divergence theorem these are my station about lines so first I will discuss basics of cost divergence theorem and after that I'll explain the basic statement which is there with Gauss divergence theorem then we will see how the proof which is there with gross divergence theorem and after that I have explained what is the physical significance which is there with Gauss divergence theorem and that I will explain by few examples so it will be more clear like how Gauss divergence theorem statement explains the position of source and sink and after that I will explain some uses which is there with Gauss divergence theorem so let us begin this session with first agenda that is basics of Gauss divergence theorem now see if you observe the basic statement of Gauss divergence theorem then that explains relationship in between volume integration and surface integration and see this is quite interesting like this theorem explains relationship in between volume integration and surface integration I will show you that statement so it will be more clear what is the relationship which is there based on cost divergence theorem see Gauss divergence theorem that is been utilized to identify the location of source and sink like say for example if I give you one system and in that system if there is a source and there is a sink then by having this theorem we can identify the location of source and sink sometimes this theorem is been utilized to identify some leakage which is there in the system so basically agenda of this theorem is to understand what are the positions of source and sink physically we can use this theorem to understand what is the range of rate of change of function with respect to food so if rate of change of function that is increasing with respect to position one can say there is a source and if rate of change of function that is B creasing with respect to position or one can say if it is negative in that case we can say that is a sink so exact location of source and sink that we can identify based on changing function with respect to position so that I will explain you in physical significance how that is there see divergence is a flux density so it explains how much flux is entering or how much flux is leaving and based on that we can identify whether that position resource or sink so if flux is entering in that case we can say that is a sink right and if flux is leaving from the pole point in that case we can say that is a source so this is what the basic thing which we do it by having divergence theorem so let us try to understand that statement and then after we will see physical significance of it so if you see the function of Gauss divergence theorem then mathematically you can see that function so that explains surface integration of function that is volume integration of divergence of that function so surface integration of function is volume integration of divergence of that function that is what Gauss divergence theorem so this explains relationship in between surface integration and volume integration so if you observe this then this is quite clear like by having this statement we can have relationship in between surface integration and volume integration and usually when you calculate this divergence that is what del dot P so that divergence explains you look that rate of change of flux right so if flux is entering in that case we can say that point is sink and if flux is leaving from the point we can say that point is source so that one we can understand based on divergence theorem and by divergence theorem we can have relationship between surface integration and volume integration now let us try to understand the proof which is there with Gauss divergence theorem now see divergence of any function that we can calculate as per divergence of function that is del dot P so dot multiplication of del operator with function is what divergence and divergence of function that even we can calculate by having limit Delta V tends to 0 surface integration of function divided by Delta V so this is what a definition of integration so if you take this del V over this side and if you integrate that then you will be finding this function that is C del V del dot P del V where del V tends to 0 and this is what integration and there is nothing in this function in terms of del V so you can say this is integration of PDS so this is practically volume integration of divergence of P so we can say this is what integration of PDS that is equals to volume integration of function P so this is what about proof of Gauss divergence theorem and that comes from the basic definition of divergence so divergence that we calculate as per Del dot P I have already explained that in my previous session how to calculate divergence but here I am showing you how we can have a proof of divergence theorem that is surface integration of function is equals to divergence of function volume integration of divergence of function right so this is what proof of cost divergence here now let us try to understand physical significance of it so it will be more clear here we have a very big tank and here we have a tube so from this tube we are pouring water in so you will be finding sea water is coming inside like this and in this tank there are a few walls like see one wall is over here one wall is over here and one one that is over here so there are three walls and out of these three walls water is coming outside you can see all right so when water is getting poured inside of this container you will be finding at this location water is continuously increasing right so if you calculate divergence at this position it will be finding that divergence is positive and this function is a function of water so here water is entering and you will be finding this is what source it is acting like a source so over here a divergence is positive so we can say this is the source at this location now here water is living to the Container so water is decreasing over here right over here even and over here the reason is you are keeping this valve open so as water is going inside of this wall and it is leaving to the container so if you calculate divergence at this position as a function of water you will be finding over here over here as well as over here that divergence of function of water that will be negative so what it means this points are acting like a sink so ultimately divergence theorem that we can use it to identify position where source and sink is been there so if divergence is positive we can say those locations are acting like a source and if divergence is negative we can say those locations are acting like a sink or one can say we can calculate flux entering or flux leaving so here we can see clearly here water is actually gathering over here so here water is increasing so we can say this point is acting like a source why water is leaving from this so we can say this points are acting like a sink see this is how physically we can understand Gauss divergence theorem let us try to understand that by one more example so here we have a positive charge you can see and here because of positive charge electric field that will leave to the charge so electric field that will be in outward direction you can see right and here we have a negative charge you can see and as this is negative charge electric field that will go inside to the charge like this you can see so definitely one thing that we can say this point charge that is a positive charge that is in that is releasing electric field and this charge is a negative charge so that is enclosing electric field right so if you calculate divergence nearer to positive charge you will be finding that divergence is positive and if you calculate divergence nearer to negative charge you will be finding that divergence is negative so we can say this charge is acting as a source and this charge is acting as a sync and that we can even identify based on electric field that is leaving to the charge an electric field that is getting an close to the charge so we can say flux is actually released from this positive charge and flux that is getting and close to the power this negative charge so based on that if you calculate divergence at any position somewhere in this vicinity nearer to positive charge divergence will be positive and nearer to negative charge divergence will be negative see this is how we can understand divergence theorem I think now you can understand that divergence theorem that we can apply to so many positions like to understand gravitational force even we can understand that by having divergence theorem where higher the mass will release gravitational field and that will attract another body right so based on that even we can apply divergence theorem to understand whether body is acting like a source or sink so let us see some uses which is there with Gauss divergence theorem so we can use this Gauss divergence theorem in mechanics we can use that to understand electromagnetic z1 so in electromagnetics we will be using this divergence theorem to understand entering flux and leaving flux as well as it will gives clear understanding regarding whether given positions are acting like a source or sink so all those things begin that we can understand by having divergence theorem we can even use this divergence theorem in slow fins like gravitational fields electric field magnetic field so that understanding even we cannot carry by having divergence here and in aerodynamics e1 we can utilize this divergence theorem so these are the broad uses which is there with divers thank you so much for watching this video I would like to get your suggestions from your side the reason is I want all those topics that has to be covered on my channel so that students 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