 Today in this topic we will see the divergence theorem, myself Piyusha Shedgar, Electronics and Telecommunication Department, Walchen Institute of Technology, Sola. These are the learning outcomes for this session. At the end of this session students will be able to define divergence, they will be able to state and explain the divergence theorem and again they will be able to solve the problems depend on this divergence theorem. These are the contents we will cover in today's session. So before going to start, divergence theorem, you can pause video here and you can recall what is the Dell. So as you know that Dell operator is used in the last PPT's, so you can pause here and recall that what is Dell. Yes, the Dell is one of the operator which is defined by, it is nothing but vector differential operator. It is represented by the symbol nabla, so it is represented by this Dell symbol, so Dell is used to take the differentiation, partial differentiation with respect to each axis. So let us start with the divergence, divergence of a vector, any vector at any given point suppose the point P is defined is nothing but the outward flux per unit volume. So as the volume shrinks about that point P. So divergence of a vector V written as dive of V, so meaning of this dive of V is nothing but the divergence of V vector represents the scalar quantity. So when you are taking the divergence of any vector, so it results in scalar quantity. So this equation defined by this divergence of V, so divergence is nothing but you are taking the dot product of this V vector, so Dell dot of V is nothing but the divergence. So divergence of V vector is defined with this equation, dow Vx by dow x plus dow Vy by dow y plus dow Vz by dow z, where Dell operator is defined by this partial differentiation with respect to x, y and z and whereas V is one of the vector defined by Vx A bar x plus Vy A bar y plus Vz A bar z. So remember that this V vector is defined in Cartesian coordinate system. So what is this physical significance of the divergence? So divergence of a vector field is the extent to which the vector field flux behaves like a source at a given point. So it is used to measure the outgoingness of the flux, a point at which the flux is outgoing is defined with the positive divergence and it is often called as a source of that field whereas a point at which the flux is directed inward has negative divergence and is often called as a sink of that field. So if the greater the flux of the field through the small surface enclosing a given point then the value of the divergence is greater at that particular point whereas a point at which if you are considering the zero flux in that case the divergence obtained is equal to 0. So the zero divergence is enclosed by that surface. So meaning of this is nothing but outgoing flux is equal to the incoming flux in that case the divergence is equal to 0. So continuing with the physical significance of the divergence, the divergence of the vector field for that consider one of the example of the fluid or liquid or gas flow. So moving gas has the velocity, a speed as well as the direction. So the velocity of a gas forms a vector field. So if this gas is heated it will expand. So if it expand causes a neat motion of a gas particle outward in all directions. So there will be the outward flux of the gas through the surface. So the velocity field will have the positive divergence everywhere in this case as we have seen in the previous slide. Now suppose the gas is cooled it will contract. So more room for gas particles in any volume so the external pressure of the fluid will cause a net flow of a gas volume inward through any closed surface and therefore the velocity field has negative divergence everywhere when the gas is cooled. In contrast in an unheated gas with a constant density the gas may be moving but the volume rate of a gas flowing into any closed surface must equal to the volume rate flowing out. So here you can say that the net flux passing through this whatever is the volume is considered of that fluid is equal to 0. Thus the gas velocity has the 0 divergence everywhere in this case. So let us discuss about the properties of the divergence. So first property of the divergence a field which has 0 divergence everywhere is called the solenoidal field. A field which has the 0 curl, curl means nothing but when you are considering the cross operation with respect to any vector with the del operator. So del cross any vector is called as a curl. So a field which has a 0 curl everywhere is called a irrotational field. The divergence of a vector function is always a scalar function. Fourth property the divergence of a scalar field does not make any sense. Divergence of the curl of any vector must be equal to 0. So when you are taking the del cross any vector operation so that operation is nothing but the curl operation and when you are taking the divergence of that curl operation of any vector field it results in 0. So del dot del cross of a is nothing but any vector is equal to 0. So recall the Gauss law, Gauss law states that the total electric flux passing through any closed surface is equal to the total charge enclosed by that surface. So this can be defined in another way that the electric flux out of a closed surface is equal to the charge enclosed divided by the permittivity. Permittivity defined with the letter epsilon. The electric flux in an area is defined as the electric field multiplied by the area of the surface projected in a plane and perpendicular to the field. So according to the Gauss law statement mathematically it can be represented with psi is equal to integration of d psi equal to integration d bar dot ds bar whereas d bar dot ds bar is considered with respect to the surface it can be equated to the total electric flux. So as you know the relation between the d bar and e bar it given by this equation d bar equal to epsilon not e bar. So by putting this equation you can form the equation in terms of the e bar or you can write this equation in terms of e bar as psi is equal to epsilon not integration e bar dot ds bar and according to the definition of a Gauss law the ratio of this psi is can be represented with q enclosed by the surface divided by epsilon. So whereas this can be nothing but the total flux within a closed surface whereas q enclosed by epsilon not is proportional to the enclosed charge psi is proportional to the charge. If the charge is not placed at the origin that means it is placed either any point is defined with the reference point point p suppose. So in general the charge can be obtained in terms of the volume charge density. Volume charge density is denoted with this rho v. So consider this is the figure for the Gauss law. So charge is suppose the not placed at the origin this is nothing but the d bar dot ds bar is given by the charge value or it is equated to the flux passing through this surface. d bar is the flux density ds bar is the differential area. So the q is represented in form of the volume suppose the volume is considered. So q equal to integration rho v dv and according to the Gauss law then you can equate these two equations with respect to surface and with respect to volume. So q is defined with d bar dot ds bar integration is with respect to surface and whereas the volume rho v dv. So this is nothing but total flux crossing the closed surface whereas this is the total charge enclosed by closed surface. So q is equal to represented with this d bar dot ds bar equal to integration rho v dv. So this is like consider this is the first equation. So according to the point form of Gauss law rho v is defined by del dot d bar. So if you are putting this rho v equal to del dot d bar in this equation you are getting this equation as d bar dot ds bar is equal to del dot d bar dv integration with respect to surface for d bar dot ds bar whereas the integration is with respect to volume for del dot d bar dv. So this equation is nothing but it is called as the divergence theorem. So this relation is also called as the Gauss theorem. So how to write the statement for this theorem? The integral of the normal component of a vector function over a closed surface s equals the integral of the divergence of that vector throughout the volume enclosed by the surface. How to define the divergence in different coordinate system? So divergence as you know that the divergence is nothing but the dot operation. So when you are taking the dot product with any vector and del operator is used for that. So del dot a bar is equal to dow ax by dow x plus dow ay by dow y plus dow az by dow z. So remember that here the vector is considered as a a bar, a bar can be defined with a bar equal to ax a bar x plus ay a bar y plus az a bar z and this is nothing but the Cartesian coordinate system. Next define the divergence in cylindrical coordinate system. So cylindrical coordinate system having the coordinates are r, phi and z. So del dot a bar for cylindrical coordinate system is given by 1 upon r dow by dow r ar plus 1 upon r dow phi by dow phi plus dow az by dow z. Similarly the divergence can be defined in spherical coordinate system as del dot a bar is equal to 1 upon r square dow by dow r r square ar plus 1 upon r sine theta dow by dow theta a theta sine theta plus 1 upon r sine theta dow a phi by dow phi. These are the references for this session. Thank you.