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Current time:0:00Total duration:9:14

Video transcript

we've already explored a two-dimensional version of the divergence theorem if I have some region so this is my region right over heart right over here we'll call it R and let's call the boundary of my region let's call that C and if I have some vector field in this region so let me draw a vector field like this if I draw a vector field just like that our two-dimensional divergence theorem which we really derived from greens theorem told us that the the flux across our boundary of this region so let me write that out the flux across the boundary so the flux is essentially going to be the vector field it's going to be our vector field F dotted with the normal outward facing vector so the normal vector at any point is this outward facing vector so our vector field dotted with the normal facing vector at our at our boundary times our little chunk of the boundary times our little chunk times our little chunk of the boundary if we were to sum them all up over the entire boundary if we were sum them up we write a little bit neater if we were to sum them all up over the entire boundary that's the same thing as summing up over the entire region so let's sum up over the entire region so summing up over this entire region each little chunk of area each little chunk of area da we could call that DX dy if we wanted to if we're dealing in the XY domain right over here but each little chunk of area times the divergence of F which is really saying how much of that is that vector field pulling apart so it's times the divergence times the divergence of F and hopefully it made intuitive sense the way that I drew this vector field right over here you see everything's kind of coming out you could almost call this a source right here where the vector field seems like it's popping out of there this has positive divergence right over here and so because of this you actually see that the vector field at the boundary is actually going in the direction of the normal vector pretty close to the direction of the normal vector so it makes sense you have positive divergence and this is going to be a positive value the vector field is going for the most part in the direction of the normal vector so as the larger this is the larger that is so hopefully some intuitive sense if you had another if you had another vector field if you had another vector field so let me draw another region that looked like this so I could draw a couple of situations so one where there's very limited divergence maybe it's just a constant vector the vector field doesn't really change as you go as you go in any given direction over here you'll get positive fluxes I know the pup what the plural of flux is you'll get positive fluxes because the vector field seems to be going in Direction roughly the same direction as our normal vector but here you will get a negative flux so stuff is coming in here if you imagine your vector field is essentially some type of mass density times volume and we've thought about that before this is showing how much stuff is coming in and then stuff is coming out so your net flux will or your net flux will be close to zero stuff is coming in and stuff is coming out here you're just saying hey stuff is constantly coming out of this surface so hopefully this gives you a sense that here you have very low divergence and you have would have a low flux total aggregate flux going through your boundary here you have a high divergence and you would have a high aggregate flux I could draw another situation so this is my region R and let's say that we have negative divergence or we could even call it convergence convergences it's an actual technical term but you can imagine if the vector field is converging is converging within our within our well the divergence is going to be negative in this situation it's actually converging which is the opposite of diverging so the divergence is negative in this situation and also the flux across the boundary is going to be negative because as we see here the way I drew it across most of this boundary the vector field is going in the opposite direction it's going in the opposite direction as our normal vector at any point so hopefully this gives you a sense of why there's this connection between the divergence over the region and the flux across the boundary well now we're just going to extend this to three dimensions and it's the exact same reason if we have a and I'll define it a little bit more precisely in future videos a simple solid region so let me just draw it I'm going to try to draw it in three dimensions so let's say it looks something like that and one way to think about it is this is going to be a region that doesn't bend back on itself and if you have a region that bends back on itself and what we'll think about in multiple ways but out of all of the the volumes of three dimensions that you can imagine these are the ones that don't bend back on themselves and and there are some that you might not be able to imagine that would also not meet the case but even if you had ones that bended back or that bent back on themselves you could you could separate them out into other ones that don't so here's just a simple a simple solid region over here I'll make it look three-dimensional so maybe if you could see if it was transparent you would see it like that and then you see the front the front of it like that so it's this kind of elliptical circular blob looking thing so that could be the back of it and then if you go to the front it could be like it could look something like that so this is our simple this is our simple solid region I'll call it I'll call it well I'll call it our still but we're dealing with a three dimensional we are now dealing with a three dimensional region and now the boundary of this is no longer a line we're now in three dimensions the boundary is a surface so I'll call that s s is s is the boundary boundary of R and now let's throw on a vector field here now this is a vector field in three dimensions and now let's imagine that we actually have positive to positive divergence of our vector field within this region right over here so we have positive divergence so you can imagine that it's kind of the vector field within the region it's kind of it's a source of the vector field or the vector field is diverging out of the vector field is diverging out that's just the case I drew right over here and the other thing we want to say about vector field s it's oriented in a way that it's normal vectors outward facing so outward outward normal vector so the normal it's oriented so that the surface the normal vector is like that the other option is that you have an inward facing normal vector but we're assuming it's an outward facing n well then we just extrapolate this to three dimensions we essentially say the flux across the surface so the flux across the surface you would take your vector field you would take your vector field dotted with the normal vector at the surface dot it with the normal vector of the surface and then multiply that times a little chunk of surface so multiply that times a little chunk of surface and then sum it up along the whole surface so sum it up so it's going to be a surface integral so this is flux across the surface flux across across the surface is going to be equal to is going to be equal to if we were to sum up the divergence of F if we were to sum up across the whole volume so now if we're summing up things on every every little chunk of volume over here in three dimensions we're gonna have to take integrals along each dimension so it's going to be a triple integral so it's going to be a triple integral over the region of the divergence of F so we're going to say how much is F what is the divergence at F at each point and then multiply it times the volume of that little chunk and multiply it times the volume of that chunk the sense of how much is it totally diverging in that volume and then you sum it up that should be equal to the flux it's completely analogous to what here here we had a flux across across the line we had essentially a two-dimensional I guess we could say it's a one-dimensional boundary so flux flux across across the curve and here we have the flux across the surface here we were summing that I virgin in the region here we're summing it in the volume but it's the exact same logic if you had a vector field like this that was fairly constant going through the surface on one side you would have a negative flux on the other side you would have a positive flux and they would roughly cancel out and that makes sense because there would be no diverging going on if you had a converging vector field where it's coming in the flux would be negative because it's going in the door opposite direction of the normal vector and so the divergence then the vergence would be negative as well because essentially the vector for you would be converging so hopefully this gives you a an intuition of what that the divergence theorem is actually saying something very very very very almost common-sense or intuitive and now in the next few videos we can do some worked examples just so you feel comfortable computing or manipulating these integrals and then we'll do a couple of proof videos where we actually prove the divergence theorem