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Gauss's Divergence Theorem
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0:08
Welcome back. So I'm really excited today to tell you about Gauss's divergence theorem. This is one of the most important results in all of vector calculus and it is probably the primary building block of how we derive conservation laws from physics... from science. Conservation of mass, momentum, and energy and translate into partial differential equations. So I'm going to walk you through this theorem with pictures, with words, with math. So the idea is that we have some
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some volume V and we're going to say that this has a surface S and that there is some normal vector
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at every point on the surface of this volume. And so what Gauss's theorem does is it essentially tells us if we have some fluid flow, some vector field -- I'm just going to draw some some vector field going throughout all of space like this -- and my vector field I'm going to call F-arrow. So if I have that vector field and it's carrying mass or momentum or energy through this volume, Gauss's
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divergence theorem is going to allow me to compute how much of that material is fluxing through the surface of this closed surface, or what is the rate of change of that material in this volume. So it's a really useful way of doing essentially accounting of physical quantities that we know are conserved, like mass, momentum, and energy, and using those to get quantitative expressions.
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Now one of the things that I think is really important is intuitively motivating why we want Gauss's theorem. So when I learned about this it was essentially just written down as a mathematical fact and it was proven and I think that's almost a crime not to tell you all of the cool physical applications of Gauss's divergence theorem. So I'm going to write it out in words, give you the mathematical expression, and then show you how we actually use this to compute
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partial differential equations. So this is one of the lectures I'm most excited for because I think this is going to be a tool set or a tool in your arsenal that you're going to use for the rest of your career. This is a really powerful way of thinking about the world. I mean it has Gauss's name in it -- if it's Gauss's divergence theorem, it's probably the best divergence theorem. Okay
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so let's get started. So I'm going to start in words so and I'm going to write this out somewhat slowly, and then we're going to write out the math expression. So in words, what we're saying is that the flux of a vector field -- and our vector field is F -- the flux of vector field F
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out of a closed surface -- so let's say through a closed surface -- and this closed surface is
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S... okay the perimeter of this volume is S, and so the flux of a vector field through this closed surface is equal to the integral of the divergence of that vector field throughout the entire volume. I'm going to say this again and write it. The flux of our vector field through the surface is equal
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to the integral of the divergence of our vector field -- the divergence div(F)
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over the entire volume -- and I'll just write it here over the entire enclosed
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volume V. Okay so this is a bit of a mouthful and what I'm going to do is I'm going to walk you through what this statement means, why it's important, and we're going to translate it into a mathematical expression. So maybe I will start with the mathematical expression and then we'll go back to what these words mean. So the flux of a vector field F through a surface,
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and I'm going to write this out in math, maybe I will write it in blue, so the flux of our vector field through this surface is going to be a surface integral... so I'm going to integrate over my surface S and the flux is essentially how much of F is pointing in the normal direction added up over every little surface element. So if I want to compute at every point how much of F
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is being kind of carried out of this volume -- what I do is I take my my F vector and I take my normal vector n and I *dot* them together and that tells me how much of my F component is in a direction that will actually take me out of this volume. So let's think for a minute... if my F vector field was just swirling around in this volume kind of going around in circles and it had no component that was perpendicular to the volume then nothing would be carried out
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of this volume through that vector field -- it would just be kind of recirculating but nothing would be leaving and that dot product of F if it was tangent to the normal would be zero dot product with n. And so the way we say this is the surface integral of our vector field F
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dotted with the normal integrated over over that volume, over every little area element on the perimeter of this, so this is the flux of the vector field F... this is literally the definition of what I mean by the flux: this is how much stuff is being carried out of the volume. It's this: we compute how much of F is pointing in the normal direction using the dot product, and we integrate that up over the entire surface. And that's how much stuff is leaving the volume...
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this is basic accounting... this is the only way stuff can leave the volume. Unless we believe that stuff can be created or destroyed inside the volume. Okay, so if our quantity like mass, momentum, or energy can't be created or destroyed and it's conserved then this is a basic accounting of how much stuff leaves through that surface s and so we call this the flux
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through s the flux through s and this is like a definition of the flux and what Gauss's divergence theorem says that this flux... this might actually be kind of a pain to compute maybe I don't want to do the surface integral what this is equal to is a volume integral so we'll do a triple volume integral over the entire volume of the divergence of F of
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div dot F dv ... and this is extremely useful. What this says is that these two quantities are completely equivalent ways of computing the flux through some volume and this is for an arbitrary volume, it doesn't matter what size V is... it could be a skinny little rod it could be a sphere it could be a little hockey puck shaped thing it doesn't matter what this volume is this is always true as long as F is a continuous vector field, if it doesn't have sharp abrupt
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changes or shock waves then this is going to be true... this is Gauss's divergence theorem. And this is really important, so I'm actually going to box this off. This is super important, this is Gauss's divergence theorem We'll put some stars next to it just so that we remember that this is really really important. Okay so I'm going to restate why this is true. I'm going to show you some intuition of why
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this expression should equal this expression and then we're going to use this to actually compute some real conservation laws and translate those into partial differential equations. Things like the mass continuity equation or the Navier-Stokes equation. So why do we think that this volume integral of the divergence of F should equal the amount of stuff being carried out of F. So I'm going to draw another little picture
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here... I'm going to try my best to duplicate this so that we can play around.
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So that's roughly my volume and what we're going to do, is we're going to chop this up into a number of little boxes... so I'm going to chop this up into some boxes and I'm not a fantastic artist, so we're just going to use a few boxes here and what we're going to do is look at the divergence in each of these boxes. So in your imagination, imagine that this is a really fine graph paper with millions or trillions
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of boxes and you could take the limit as all of those boxes become infinitesimally small that's what we're going to do, and we're going to look at the divergence at every little grid box in this graph paper representation, so at every little point here, what I'm going to do is I'm going to draw a representation of the divergence, and what we're going to have is that maybe I have some positive divergence here and some positive divergence here
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let's say my flow is kind of expanding and some positive divergence, okay I'm just going to draw this. I'm not going to draw any of these negative because it's a little harder to visualize and I've got this guy here. Now the idea is, and this is again requiring that my vector field F is continuous to do the divergence theorem... to kind of take this limit as all of these grid cells become infinitesimally small and the argument I'm about to to
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give you is the following... and I really like this picture it just looks super cool so these pink sources, the way I'm drawing them here, these are the divergence fields locally that are induced by this this flow F so this F might be doing something complicated you know there's some complicated physics, Navier Stokes fluid flow, whatever and this is the local divergence at every little point. Now what we observe is that if I really do
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assume that this this vector field is continuous and I make this grid infinitesimally fine then these neighboring boxes, the positive upward component of this flux and the negative downward component of this flux should cancel each other out, because if i make my boxes small enough the flow is basically identical in neighboring boxes and so the upward component of this pushing on the downward component of this cancel out. So you actually get this kind of cancellation of
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all of the inner boxes every box that shares a wall... those two neighboring forces are going to cancel each other out and the only thing that you're going to be left with is this perimeter of all of the divergences that don't have a neighboring grid cell, so all of these these little boxes that are pointing out this is the only thing that's left when I integrate over this volume. Okay so i'm going to say this again
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when i integrate all of my divergences over this volume when i integrate all of my divergence on this volume all of the inner grid cells kind of cancel each other out because of the continuity of f because f is a continuous vector field and i'm making these these boxes infinitesimally small the you know the right vector and the left vector on neighboring grid cells cancel each other out and the up vector and the down vector on neighboring grid cells cancel each other out
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and so we only have this thin perimeter right on the surface of our volume that don't have uh divergence vectors that are being cancelled out and you'll notice that if i integrate these along that looks a heck of a lot like my flux through through the volume so this is a very cartoony sketch of why gauss's divergence theorem is true it's because i can take i can integrate this flow kind of dotted with the normal along the surface or i can compute its divergence
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everywhere throughout the volume and just add up all of those divergences the ones in the middle will cancel each other out and it will leave kind of something that's equivalent to this integral on the perimeter so that's a really useful in my mind kind of cartoon of why this is why this is true okay so let's uh the way i talk about this uh when i'm teaching this in class this is essentially just accounting this is uh you know being counting of some quantity being flown
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through a vector field and so as long as you can count things you know this these arguments have to be true the stuff coming into a volume has to equal the stuff coming out of a volume unless you have sources you know that are kind of creating things or destroying things inside the volume
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okay good and there's a lot of ways you can compute with this so for example if i want to compute the volume of an irregular shape i can create a vector field that has divergence divergence 1 and i can integrate that vector field all along the perimeter of my volume i'm going to say that again if i want to compute the volume of an irregular object
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all i have to do is create a vector field that has divergence one and then i integrate that vector field along the perimeter or along the surface of my volume dotted in the normal direction that will equal the volume of that object so that's kind of a neat application of gauss's theorem but that's a little esoteric what i really really really want to show you is how we can use this to derive partial differential equations so let me just say gauss's divergence theorem
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for pdes for partial differential equations and this is really exciting okay so all of our modern physics all of uh kind of you know the textbook equations that you're you're used to seeing the navy or stokes equations you know maxwell's equations almost all of our physical laws boil down to a conservation of some quantity in some volume using gauss's theorem to derive a partial differential equation so it's an extremely extremely powerful way of mathematically
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quantifying uh physics and that's what i'm going to show you right now so we're going to look at gauss's theorem for the conservation of mass in a volume okay so we're going to specifically look at
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mass conservation mass conservation uh in a volume and we're basically assuming that we have some fluid flow you know uh maybe we're in the gulf of mexico maybe this is the flow in the ocean and maybe i have some oil spill at some point in the in the gulf of mexico maybe that's what this this uh this volume is or maybe there's some some spill here and we know that that is a conserved quantity there's not you know more or less um you know at least in the short term as it flows
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through and so we want to somehow quantify this conservation uh of mass in our volume
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so here's what we're going to do we're going to say that our total mass let me see i want to write that here so our total mass in the volume equals the triple integral over v of the density rho
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dv okay so i'll just say rho is the density
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so the density of whatever my fluid is if i integrate that whole you know that density over the entire volume i will get the mass in that volume
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and so the rate of change of that mass and the volume is equal to how much of that flow that mass is being fluxed out of that volume through this surface integral so that's what i want to write down this is the the basic conservation i want to write down so the so d dt
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of this integral which is rho dv integrated over the volume this is the total this is the rate of change of mass in my volume so maybe mass is being carried out of my volume maybe mass is coming into my volume and it's happening because this vector field this fluid flow is either carrying mass in or carrying mass out and so this rate of change equals the only way that that mass can enter or leave this volume is through it being carried out through the boundary
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by this flow f so this would equal minus the integral through the surface
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of my you know my mass that's being carried out through the normal direction d s and i have this minus sign because essentially if i have a positive flow carrying mass out of my volume if this f dot n is a positive thing carrying flow out then that is a negative rate of change of mass inside the volume so the rate of change of mass inside the volume is equal to minus how much of it is leaving the volume and again this is the only way mass can leave the volume
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is that if it's carried out through this flow kind of normal to uh to that volume okay so this is this is the basic setup this is again just a counting this is just counting the mass in my volume here and counting how much mass enters and leaves my volume here good
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but what we want these are are this is a triple integral and this is a double integral and what we want is to somehow you know write these in the same triple integral formulation so i want to convert this into a triple volume integral and i'm going to use gauss's divergence theorem to do that so we're going to turn this into a triple integral by using gauss's divergence theorem
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and the way we do it is as follows so we're going to say
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this quantity here is equal to minus the triple volume integral of the divergence of this quantity right so the the surface integral of some flux dotted with a normal is equal to the volume integral of the divergence of that vector field so this is divergence of rho
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f dv and remember rho my density is just a scalar it's just a number at every point in space and f is a vector field it's my flow that's carrying this mass in and out
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so now i have an apples to apples uh kind of comparison before it was apples and oranges i had a volume integral and a flux surface integral now i have a volume integral for the rate of change in time of my mass equal to this kind of spatial divergence of rho times my vector field f and so what's really neat about this is that i can move this over to the left hand side and now i can put everything under the same triple volume integral and i'm going to move my ddt inside of here
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okay and and this is really going to be a partial of rho rho is a function uh let me write this down
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so my density uh rho is a function of space let's say x y z but also of time okay this density might be changing i might have a compressible fluid you know if i'm flying a jet the air is compressing and the density is changing as that wing or the nose or whatever is hitting the air okay so sometimes this can change in space and time it can vary and so what i'm going to eventually get when i bring this all under the integral is that i have a triple volume integral
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of uh partial rho partial t the partial derivative of rho with respect to time
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now this minus when i bring it over becomes a plus plus the divergence of rho times my fluid velocity field all of this dv equals zero so this is pretty special okay this uh equation here essentially accounts for all of the change of mass due to uh like the mass changing in time things fluxing through the surface and things like that and when you add that all up it has to equal zero this is a statement of conservation of mass
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now here is one of my favorite parts of this whole story of how you use gauss's divergence theorem to derive partial differential equations is that this is true this is true for all volumes
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for all volumes v this is true if i had a big volume v a stretched out volume v it's also true if i take my v and i take the limit as it becomes a teeny tiny tiny little delta v this is true at every little patch uh in space if i shrink my volume down to that little patch in space this is still true and the only way that this integral can equal zero for all volumes is if the thing inside
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of it equals zero everywhere and so this is how we derive our partial differential equation this thing inside has to be equal to zero for this volume integral to equal zero on all volumes and so then we finally get partial rho partial t plus the divergence of rho times my vector field f equals zero and this is called the mass continuity equation mass continuity
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continuity equation and it's one of the fundamental equations in fluid mechanics this is the equation that quantifies conservation of mass in a flowing fluid in the air when i'm speaking you know air is moving but it's still the mass is conserved this room mass is not being created or destroyed and so this equation is true everywhere in in space and in time okay even if i have a jet moving through even if it's a compressible fluid flow this is still true um it is only not true
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when uh when my vector field f is sufficiently or my density rho is sufficiently non-continuous if it has a really sharp step change in space or in time then this these derivatives don't actually make sense they would go to infinity and i can only represent kind of the integral effect in a control volume that's large enough to include that discontinuity okay so what i'm going to do i'm going to take a step back and summarize and i think in the next video i'm probably going to dig
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a little bit deeper into this mass conservation equation and show you what happens for example if i have an incompressible flow if my density is a constant then we can get specific kind of we can derive specific quantities that have to be true for this fluid flow f okay but let's talk about the whole pipeline here if i have physics physics is generally written in terms of conservation laws we have conservation of mass conservation of momentum conservation of energy newton's second
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law is essentially a statement of conservation of mass uh sorry conservation of momentum okay
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you know the rate of change of momentum is equal to the forces applied to the system
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we can encode those conservation laws through integral equations like this we literally count how much stuff we have in a volume and how much it changes in time and we say that the only way that that can that the amount of stuff in that volume can change that mass or momentum or energy is if it enters or leaves through the surface of that volume so i have a volume integral for the rate of
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change of stuff equals a surface integral for how much stuff enters or leaves through the boundary and gauss's theorem what gauss's theorem allows us to do is turn that surface integral into a volume integral so now i can kind of use apples to apples i can i can add up everything under one common integral and i now have this partial derivative expression that has to equal zero everywhere over
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all volumes and again if f and if the quantity i'm adding up if my mass and my momentum or my energy and my fluid flow are continuous smoothly varying in space and time then this is true for all volumes and i can get rid of the volume integral and i get my partial differential equation so this is how most of the pdes that you're used to from classical physics are derived is by writing down
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a conservation law and then using gauss's theorem to kind of simplify um that that conservation law into a single expression under a common volume integral okay now again that's this relies on some continuity that's why it's called the continuity equation because this relies on you know rho and f being continuously varying they can't be really really sharp shock waves in the system so if there's shock waves in the system other interesting things will happen and i'll show you
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that in a little bit okay awesome thank you and stay tuned for for kind of the next video on this
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