Stokes' Theorem and Green's Theorem

[Music] welcome back so today i'm going to tell you about stokes's theorem and green's theorem which are vector calculus formulas a lot like gauss's divergence theorem so they allow us to relate surface integrals of the curl of a vector field over some surface to the uh some some equivalent contour integral around an enclosing contour okay these are really really useful uh theorems stokes's theorem and green's theorem and just like gauss's theorem can be used uh to express kind of physical quantities in partial differential equations we can also use stokes's theorem in a very similar way so gauss's theorem has to do with like volume integrals of the divergence of a vector field stokes's theorem is going to have to do with surface integrals of the curl of a vector field so i'm going to jump in okay and essentially we are looking at a surface kind of an open surface in three dimensions so the surface is given by s and we're going to assume that the surface has kind of a an edge or a perimeter that we're going to call uh partial s now this is just notation this doesn't mean we're taking the derivative of s this is just sometimes uh in kind of manifold theory if you have a manifold or a surface the boundary uh of that of that surface has is one dimension lower so if this is a two-dimensional surface this is a one-dimensional curve and sometimes we just denote it by partial s so this is purely notation and we're going to assume that you can have a tangent vector along this pink curve so at every point there is a little tangent vector that we're going to call d s arrow it's literally a vector with an x and a y component okay similarly at every patch on the surface s every little kind of patch of s there is going to be a vector in the normal direction with area given by d a and so we're going to call this vector d a vector okay important uh it is it has the magnitude of the area of that patch and it is pointing in the normal direction this is important good now i'm going to write down stokes's theorem in full generality and then very quickly i'm going to go to green's theorem because it's much easier to understand kind of how things works uh green's theorem is like a 2d flat version where instead of a surface we're just it's kind of a flat flat surface and we are working on its perimeter okay good so stokes's theorem here is i'm going to write it as the integral around this entire surface so the surface integral of the curl of some vector field and of course i should always mention there is some ambient vector field um you know kind of going through my volume we're going to call this big f and it's going to have components f1 f2 f3 just like it always does and it's kind of passing through this volume and if you like you could also think of maybe there's just a vector field on the surface of this volume maybe uh maybe this is you know the northern hemisphere and i have some vector field which is the you know atmospheric flow dynamics hurricanes and things on on that northern hemisphere or you can think of an oil you know like a little bubble a soap bubble you can see the the film um the vector field on the film so there's some vector field f that is defined on the surface it could be a three-dimensional vector field and we just are looking at how it affects the surface or it could be a vector field on the surface itself and so we're going to take the surface integral of the curl of our vector field and remember the curl of a vector field itself is a vector field so this is also a vector quantity so essentially what i'm going to do is i'm going to take my vector field i'm going to compute the curl at every little patch here every little patch i'm going to compute the curl of my vector field and i'm going to integrate those up all along the the whole surface and what i'm going to do is that curl should be pointing you know i want to see what is pointing normal to the surface so i'm going to dot that with my da vector so this is kind of like integrating around the whole surface times d area but because this is a vector field itself i'm dotting it into the normal direction okay i'll kind of give you some intuition for why that's that's we're doing this this has a direction a little bit later but anyway we're trying to keep track of all of the curl if we add up all of the curl of the vector field on this surface what does that equal sometimes i would care like maybe the integral of the curl on that surface tells me how strong the hurricane is going to be and that surface integral just like gauss's theorem allows us to take and equate volume integrals and surface integrals stokes's theorem allows us to equate this surface integral to a path or a contour integral along this pink curve and so the integral of my curl over the surface is equal to the integral around the perimeter on this pink curve of my vector field f dotted into this little tangential d s direction this little d s and so this is actually a really really nice expression this is a very easy to compute quantity i literally just walk around this perimeter and i i take my f vector maybe my vector field is pointing i don't know maybe my flow is pointing in that direction that's f and i just dot f with d s and i keep track of how much of the f vector field is kind of in this tangent direction and if i add that up around this whole curve it basically says how much of f is kind of swirling around if i am keeping track of the amount of f that's tangent how much of that f is kind of in this tangential direction and that's a measure of the curl or the circulation kind of of the vector field on that perimeter that's that's what this is and that's really remarkable that if i just keep track of how much my vector field is in this tangential direction kind of over this closed orbit again that tells me how much kind of like circulation or flow around this uh this perimeter i have that is equal to if i computed the curl at every patch on the surface and added all of those contributions up so that's what stokes's theorem is saying and it's a pretty remarkable theorem okay so i want to show you how this works in green's theorem because i think it's really a lot easier to restrict this surface to be flat and look at green's theorem so that's exactly what green's theorem is so green's theorem is essentially stokes on a flat surface s okay so what we're going to do is we're going to look at kind of a top-down version of this where now we have our our surface s we still have our perimeter partial s and essentially what we're going to do is write this expression here but uh looking at at this 2d flat surface so now my surface is flat and we're going to do the exact same thing and i'm still orienting this curve so that it moves in the i believe counterclockwise orientation okay that conforms to the right-hand rule so we consider this um kind of the positive direction along that that perimeter okay good um now green's theorem is is exactly stokes's theorem restricted to uh to a flat surface and we can write it down pretty simply so i'm going to write this down and we're going to say now that our our f here f in 2d just has two components f1 and f2 and any curl that i have if my vector field is doing some you know maybe it's doing some swirly things like this i have some some kind of like circulation components you know doing some circulation then uh the curl is going to be pointing out of the board so any positive curl is going to be in the kind of k or z out of the board direction okay so that's why it also helps to think about this in 2d because all of the curl is in the kind of out of plane third component and so green's theorem or stokes's theorem on this 2d slice essentially simplifies to this surface integral so this integral over s of the curl of my vector field now the curl of a 2d vector field is just partial f2 [Music] partial x minus partial f1 partial y and now if i think about it every little area patch the normal is already in the z direction so i just have to multiply this by the little d d area patch here which is dx times dy okay so this is a simpler expression for this this integral of the curl here i'm just integrating up the curl of this 2d vector field over this entire surface and green's theorem says that this is equal to again it's equal to my vector field f dotted into the tangential direction into this little tangent direction so if i take um you know if i have this little d s vector and i have my f vector i just go along the perimeter adding up how much what is f dotted with d s okay and so the nice thing is is that this is uh an integral around the perimeter of f dotted into the tangential direction but because i'm already moving around uh uh this this curve that's that's parameterizing the perimeter this is essentially just f one d x plus f two d y okay so i literally take um how much of of f how much of f1 of the f1 component is in the uh dx direction of this little sorry i'm getting ahead of myself let me slow down this little ds vector d s vector is equal to d x d y it tells me at every little point here is my vector moving more in the dx direction or the d y direction it tells me what the the direction of my curve is dx dy those are the dx and dy here okay and we integrate all of those little dx's and dys along this the surface so this is green's theorem and again it tells us that i can basically just walk around along the perimeter of this vector field adding up this quantity just literally taking f1 dotted into the dx component of this tangent vector plus f2 dotted into the d y component of this tangent vector and if i add up that number around this entire perimeter it equals the sum of all of the curl of my vector field on the entire surface surface integral so again this is a lot like gauss's divergence theorem where we're going from a big integral you know over a volume down to a smaller integral over a surface area here we're going from a big integral on a surface down to a smaller integral on the perimeter of that surface and the same thing here so here's a couple things i want to do i want to give you some physical intuition for why stokes's theorem and green's theorem are true and then i'll tell you some cool examples of these okay so maybe the first thing i'll do is i'll just draw this picture again and we're gonna do the same kind of pillbox uh decomposition that we did before for uh for gauss's theorem so what we're going to do is break this up into a number of little boxes so i have my kind of kidney bean uh shape here for for the the surface and we're going to break it up into a bunch of little boxes now i've only drawn you know 12 boxes here but in what we're going to do is essentially imagine we take the limit as these boxes get infinitesimally small and infinitesimally close together so this is going to be a really really finely resolved graph paper kind of uh representing this this area and so what we can do is essentially look at the curl in every little box here so the little local curl in every little box is kind of a little vector field that's going around in a circle this is a little positive curl uh representation here and the next thing is going to have a little curl these are all going to have you know a little bit of curl and i'm just going to draw these and just like gauss's divergence theorem you can already start to see what's about to happen is that the inner cells the curl from the neighbors are going to cancel each other out and so if i integrate all of these curls over the entire area all of the inner kind of components of the curl are going to cancel out to zero and i'm only going to be left with the component that's kind of in this tangent direction around the perimeter and so that's the real idea here maybe i don't have to draw the whole thing but i think you get the the idea hopefully that just like um gauss's divergence theorem if we have these infinitesimal little uh little boxes and if we assume that our vector field is sufficiently smooth that these are continuously varying from cell to cell so if f is a smooth vector field and most of the vector fields i think about are smooth fluid flows are smooth typically except for shock waves the flow over a soap film is smooth and continuous and so that basically means that these different uh kind of components on these different edges are canceling each other out this little box is swirling a tiny little vortex this little box is swirling a tiny little vortex and the up component and down component cancel each other out on every single inner wall here and so all that's left are the little components that are kind of in this tangent direction here that are kind of aligned with the tangent of the perimeter and so hopefully that gives you at least some intuition for why the surface area of the curl is equal to a path integral of the vector field dotted into the tangential direction the tangent component the the component of the vector field that is tangent to uh to this enclosing pink perimeter okay so that is uh at least the cartoon of why green's theorem is true and stokes's theorem is just kind of a more general case of of green's theorem where you now take this uh this two-dimensional surface and you imagine that there is some three-dimensional uh surface over this perimeter and the same basic argument applies if i if i grid this up into little boxes and i look at all the little curl components all of the inner kind of inner walls are going to cancel out and the only component that's going to be left when i integrate over this entire blue surface area is the part of the vector field that is tangent to this pink perimeter curve so again the integral of the curl of f over this entire surface is going to equal the integral over the little contour the perimeter of my vector field dotted into the tangent direction just the tangential component and this is really cool if we think about you know let's say we're talking about planet earth and so we have um you know we're going to look at the the northern hemisphere here and let's say i have a big um you know i'm not sure i can draw a big hurricane but let's say i have you know some some big hurricane forming in the northern hemisphere if i added up all of the curl of every little patch on this northern hemisphere that would equal to the amount of the vector field integrated along this perimeter dotted into the tangent direction and so you can think about it if this if this surface if this vector field is swirling you know around in the surface there will be a signature of that swirling on this pink perimeter here and we'll be able to compute kind of how much circulation or vorticity there is in this surface just by computing what is the accumulated amount of circulation on this uh this perimeter here so it's a really powerful idea and this is going to be useful uh for things like fluid flows where we have circulation and rotation so this is actually really useful when we think about um airfoil theory so if i have a wing i might want to compute kind of how much uh what is the circulation of flow kind of over that wing and that that will be relevant for computing the lift over the wing things like that so you know curl and stokes's theorem are really important when we think about aerodynamic surfaces and fluid flows over aerodynamic surfaces hurricanes things like that and and this is going to be how remember divergence theorem is how we encoded conservation of mass conservation of momentum into navy or stokes equations stokes's theorem is how we're going to encode conservation of angular momentum so all of this curl is kind of measuring how much angular momentum i have and if i add up all of that angular momentum there's a signature of it in the perimeter of that surface so really really powerful idea here and one last thing i'll point out and this is just something neat i i remember one of my favorite calculus teachers in high school taught us this and i think it's a really cool idea you can also use green's theorem or stokes's theorem to compute the area of a tract of land so back before gps back before you could just kind of use satellites and gps to compute the area of an irregular shape of land or use a computer program surveyors would actually have to walk around the boundaries of of someone's property so maybe i have a farm here and i have another farm here and i have another piece of land here and maybe there's a river you know that's going uh you know on the boundary of these how do i compute the area you know how many acres or how many hectares of land are inside this area if i want to sell a parcel of land and it's gotten a regular boundary well it turns out you can use stokes's theorem for this in a really clever way so let's say i pick you know one of these areas here s one of these irregular shapes s then the area of s is equal to one half times the integral around its perimeter around the perimeter of s of x d y minus y d x and i'll tell you why this is true so this is true uh and so what i can literally do is i can walk around i can go you know i bring an apple and a nice book and i'm walking around this perimeter and i literally count you know how many times i step in the d y direction versus how many times i step in the dx direction as i go along this perimeter and i add them up in this formula so i literally as i walk around this perimeter i compute this quantity and when i come back to my starting point i will have swept out the area of s now why is this true this is true specifically because our vector field here this corresponds to a vector f that equals minus y x and so if i think about it the curl of this vector field so the curl of f is equal to partial of f2 with respect to x that's one minus partial of f one with respect to y that's minus minus one so this curl equals two so if i do a surface integral over s of the curl of my vector field d a that should equal two times my area two times uh this area let's call it a okay but i don't want to actually break this up into a bunch of little surface areas i don't want to actually parcel this out to measure the area so we're going to use stokes's theorem or equivalently green's theorem because this is a flat surface to instead integrate around the perimeter so this is equivalent to the same integral around the perimeter of my vector field f dotted in the perimeter tangent direction and again f dotted into the primitive the perimeter tangent is minus y in the dx direction plus x in the dy direction that's what this quantity is and so if i just integrate this quantity around the perimeter it's the same as two times the area of the parcel so that's why i divide by two here so long story short if you want to measure the area of an irregular shape you can just walk around the perimeter and compute this quantity along the perimeter of the surface and and you can then get the area so one of my favorite examples of this is you can compute the area of pretty interesting shapes like these hyper hypocycloids just by kind of parametrizing this and walking around the perimeter computing this quantity okay so that's stokes's theorem and green's theorem it essentially allows us to quantify conservation of angular momentum or curl or vorticity in these kind of rotational flows in a similar way to gauss's theorem for uh kind of you know quantifying conservation of mass using divergence um it turns out mathematically there is kind of a higher dimensional and n-dimensional generalization of stokes's theorem and gauss's theorem and you can almost kind of unify those it's a little bit outside of it's a little bit more sophisticated than what i can show you with kind of simple vector calculus it's more uh doing calculus on n-dimensional manifolds but it's really really interesting um interesting kind of concept that you can actually generalize this to n-dimensional surfaces okay good so stokes's theorem green's theorem in the next few lectures we will likely be talking about things like potential flow and which vector fields have rotation which vector fields are irrotational and things like that all right thank you

Channel: Steve Brunton

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Id: LqNqqidw2mg

Length: 23min 54sec (1434 seconds)

Published: Fri May 20 2022