Vector Calculus Overview

alright thanks for watching and today I would like to give you an overview of the beautiful world of vector calculus which I like to call multi multi by variable calculus and you'll see why by the way this is completely improvised so please bear with me in case I get stuck and everything but I still think it's very important to talk about this so so far new multi variable calculus adventure u we talked about scalar functions like f of X Y Z is our X plus y plus on arc tangent of X e to the Z or something very easy and those are what are called scalar functions because what happens here you have three inputs but just one output what I mean with one output I mean it just spits up one number it doesn't spit on like three outputs for example and in that you usually write this as you know F goes from R 3 to R and again the output is a scalar because they're called center functions and now what we would like to do is to generalize this to the case where the output is r2 or r3 so in other words in vector calculus what you really do you're studying functions either from R 2 to R 2 or from R 3 to R 3 for example one can study just a function f XY equals to minus y comma X and what happens here sorry oh I've been with angular notation a second what happens here is that in the F takes two inputs as an input so two numbers as an input and spits out two outputs in other words it takes a pair of number as an input and spits out another pair of numbers this is all a nice but here's a problem well this one we could visualize at least from the graph from R 3 to R but those things it's sort of impossible to visualize you cannot visualize something from R 2 to R 2 or can be because here's a cool thing there is a way of thinking about this in the following sense instead of thinking of it as the input being two numbers and the output being two numbers think of it as follows think of it as the input being two numbers but the output is a vector and here is where vector calculus comes into play in other words what this becomes this is called a vector field because what happens is for every point in R 2 you assign a certain vector so for example let's see what happens at the point 1 0 well F of 1 0 becomes minus 1 0 sorry that F of 1 0 your X is 1 Y is 0 sorry it becomes 0 1 so at the point 1 0 F is just the vector 0 1 so you can think of it as a vector starting at this and going one unit up and you can continue what happens at the point zero one well F of zero one should be minus one zero so it should look like that and similar if you notice you can also do it at minus one zero and add zero minus one and you basically see that f becomes some sort of a rotating thing and this is what's called the vector field f of XY equals two minus y x so this is really cool because even though we cannot visualize strictly speaking functions from r2 to r2 if we think about them as vectors we can actually visualize vectors you know those kind of functions and particularly if we also think of it in terms of complex numbers this allows us to give us a way to visualize the functions from the complex numbers to the complex numbers so then is the first part if you want so that was part one what is a vector field and so in particular instead of studying you know functions from r2 to r2 or functions from R 3 to R 3 you just studying functions from R 2 that gives you a certain vector but I want to emphasize the study of vector functions it's way more than just studying each you know component separately because you could say hey you can do multivariable calculus on - Y you can do multivariable calculus on X and just put them together no no no no there is a much deeper geometric structure going on and this is what I want to convince you in the following parts for example sure you couldn't just put three functions together and get a vector field okay but there are some that are way more interesting than others mainly remember given a certain function as a scalar function you can know construct a vector called the gradient it's so creating up math which is if you want F X as Y or sometimes FZ in three variables for example if f of X Y is x squared plus y squared then the gradient of F becomes F X which is 2x and F Y which is 2y and the cool thing is notice indeed what does it do it takes X Y as an input spits out a vector so in particular it is a vector field a pretty little nice vector field you'll see why soon so in particular of course one question we could ask is is every vector field of the form if I give you for example the vector field minus y X can you write it the gradient and so the next question of course is so first of all definition capital F is nice or we call it conservative if capital F is indeed the gradient of some function f for something for example this vector field is conservative because it's the gradient of x squared plus y squared but it turns out not all of them are for example let me show you that - YF x is not of this form so non example so f of X Y is minus YX well.look supposed F is the gradient of some function little F if this is of the form gradient of F and that would be F X Y in particular by comparing the components you would get F X is minus 1 and so the X derivative of F is minus y so F because if you want the integral of minus y DX which becomes if you want minus YX for some John some junk just depending on X on the other hand if you compare the second component F y equals to X then F of XY is the integral X dy and that becomes x1 plus some junk and you see this function f is a lament for it at the same time - YX + XY and that's not possible it's not possible that you have no F being at the same X Y and minus XY so this in particular this vector field it's not conservative it's not nice and of course the next question is that how do you determine if a vector field is conservative and we'll get back to that in a couple of minutes but so that's the first thing that's you know what's a vector field is a conservative or not and of course the next part is we need to do calculus on vector fields in particular the question is how do you integrate a vector field which is kind of a weird question because you know a vector field Isha's a bunch of vectors how do you sum up valleys on the vectors and in particular where do you sum up the values of the vectors before in single variable calculus with a function of an interval and you just sum up F on this interval but the question is what do we do in this case what's the analog of an interval but for vector fields so this is part two and for this the motivation is let's talk about functions again so scalar functions and in particular what we would like to do we would like to define the line integral of a function let's say f of X 1 in other words here's what we want to do given a curve C let's say in two dimensions and a function f maybe looks like that and crazy today we would like to sum up the values of F over this curve C in other words we would like to find the area under that fence that's bounded by C and of course this is sort of hard to do and the question is well how can how can we talk about C well look C is one dimensional right it's just a line but kind of curvy in particular what you've learned before is curves you can just parameterize them with X of T Y of T and then the question is okay that's great we parameterize them and it makes sense those curves are one-dimensional so you know parametrize them with one variable and then the problem is though we can only you know integrate over strange things here we have a curvy path but here's the nice thing and I don't know if we looked at Microsoft Paint or something like that if you have this curvy line and you zoom out let me say you zoom in like crazy and then it actually just becomes a straight line so you can think of your curve as just being this really small polygonal path and in particular the nice thing is we can sort of integrate over this small political path here how do you do this well I can zoom this part out and the question is well what is this segment a lot the length of this segment well if you think of it as follows if you increase X by a little bit and you increase Y by a little bit then by the Pythagorean theorem the length of this segment is square root of the x squared because the Y squared in particular the line integral and it's simply if you think of it as follows namely the area where the base is d s and the height is just f of X Y the line interval that's why yes well that's again the area on the defense but now we would like to use the fact also that C is parametrized with X of T and Y of T so first of all this equals to f of XY square root of DX squared plus dy squared and what this becomes is the integral from the starting point so T equals to a 2t equals to B f of X T YT ok and now the question is who would like to express DX and dy in terms of T the question is how do we do this and believe it or not the answer is use the Chen Lu I'm not even kidding that's the most important thing so in particular this is DX over DT DT and that becomes X prime of T V and same thing here dy over DT DT and this is Y prime of T V and I'm not even kidding this is the thing that makes the you'll see soon the vector calculus thing worked and in particular if you use those two formulas and you plug it into here what you end up with is in fact X prime of T squared plus y prime of T squared and then DT squared but if you take the square root you just get DT and this is the line integral of a function over a curve and again strictly speaking this part has nothing to do with vector fields that's just for any function but as I said the most important thing here is this part because this thing again it's what will make the vector field case work and now using that let's talk about line integrals of vector fields so part two so remember the first question was also you know if we wanna eat some up values of vectors okay and the question is over what do we sum them up well based on an analogy for line integrals of functions well one idea is just simply given a curve we would like to sum up values of f over that curve so suppose F is your vector field might look like crazy like that or like that the question is how can we sum up all those arrows over the whole curve this is what's called the line integral and started with dr so line integral f okay and here's the nice thing and remember we have a curve we can parameterize it by X of T Y of T well particular if you think of it in terms of vectors you can also think of it in terms of this vector you know X of T Y of T we could call it R of T namely at every point it gives us the vector X of T Y of T and here's a nice thing again it just based on this idea of using the Chen Lu well you can just do this you can just add and divide by DT and what you get is again integral from the starting point to the end point F of something dotted with r prime T DT and F of what well F at the point R of T just as before we have this thing F of T y of T and then some junk with DT that's the same thing here we plug in F at the point R of T ok and the question is what does that need so because so far it's just based on this weird trick with the chan loh but it turns out there's a nice geometric interpretation of this namely suppose you're at the point R of T and you have your vector field F of T so FM R of T well if you remember from multivariable calculus R prime of T so if you parameterize the curve with C with R of T then our prime of T is just a tangent vector at that curve tangent or Direction vector and so what do you do you have those two vectors the tangent vector and your vector field F and the point is at every point you dot F with the tangent vector and in particular what happens is this F dotted with our prime T it measures whether F goes in the same direction of the curve or not so it's just a number that measures in4 so it's a number because we're dotting that measures if F that goes in the same direction as the curve as your girlfriend the reason I'm saying this I know people they're like oh it's just the work along the curve but as I said I'm I despise the fact that they use physics in vector calculus I would like to use a more geometric description in particular suppose you have three scenarios on the one hand suppose we have this direction vector R prime of T and F actually goes along the curve so look something that would then F dotted with our prime is positive it's a good thing it's sort of F flows along the curve on the other hand what if F looks like that what if F is actually perpendicular to the curve well then because it's perpendicular and started with R prime equals to zero in other words F doesn't have any effect on the curve on the other hand suppose F points the other direction that F dotted with our prime is negative so sort of F is or hindering the curve or something think about this for example if F is like wind and you're running in the direction of C that in this case it makes the running harder because the widths would have blows against you so the point is s dotted with our prime is this number that again whether f goes or not in the same direction as c and the integral is just u something up those numbers sum of numbers so what I want to convince you is it's indeed a good analogy of summing of the values of f over the curve because strictly speaking we cannot solve all vectors that doesn't make sense but we can sum up those numbers because there are functions and we can use ordinary calculus that's already very you know impressive okay we have our vector fields we define something naught F over the vector field and of course now you can ask yourselves well now that we define this integral is there a fundamental theorem of calculus for vector fields and indeed there is so there's not only one they are four of them four fundamental theorem alkali fundamental theorems of calculus I don't know what the correct you know or [Music] something we can see a SDC for calculus so remember in one dimension the FTC of calculus test says the integral from A to B of f prime of X DX equals to F of B minus F of a and here's the cool thing we can now generalize this very easily because the multi-bear will add a lot of as prod is of course the gradient but now the question is well we have a gradient which is a vector before we didn't know how to sum it up but now we know because now we can sum up values of vectors over a curve so the correct analog would be the line integral of the weight you know that the are over some curve C suppose you parametric see with R of T then what this because so here the important thing to notice is that B is the endpoint of the interval and a is also the endpoint of the interval in this case it's the same thing namely the integral of the gradient becomes f of the endpoint minus f of the starting point and in this case if you want to write it you know more rigorously is f of R of B minus F of R away and so here comes the most important thing maybe for this part if you integrate a gradient it's actually easy to calculate the line integral because in general if you calculate the line integral it's a pain you have to parametrize you have to calculate a complicated integral none of that nonsense if you have a gradient then it's easy to calculate the line integral and in particular it doesn't really matter which curve you use so as long as if you have two end points the point is this just depends on the end points it just depends on those two points in particular no matter which curve you choose with the same start and end points you get the same answer and that's what's called path independence and in particular if you integrate a gradient its path independent and so in particular given a vector field F if you can write F as the gradient of some function then the line integral F because becomes the line integral acid queen of FDR and that becomes easy to calculate if F is the gradient of some function then this line integral is easy and not only is it easy its path independent and everything but remember we actually asked this question before right at the very beginning I said that no a vector field is nice if it is this has this form and now we can act finally see why they're called nice or conservative the vector fields because the line integral becomes very easy to calculate so in other words if F is conservative then this is true which now again now we can see why it's so nice but then the next question is well how can we know determine if a vector field is conservative well again still at this far it turns out there's a very easy criteria at least to rule out non conservative vector fields so if F let's say it's the gradient of F and now let's do it in two dimensions later we'll do it in three dimensions then again suppose you write the components as PQ that is f x and y then we get FX is P and Y is cute and now remember if everything is smooth and everything okay then by in Schwartz's theorem of Clairvaux spirit we get F XY equals to F Y X and so FX y equals to F 1 X but now FX is P so P y equals to QX and I like to call this quixotic PI an audience so look if a vector field is conservative then T y equals to QX in particular if py is not equal to QX then F is not conservative for example remember this vector field that talked about before minus 1 X if you take the white derivative of the first component you get minus 1 but if you take the X derivative of the second component you get 1 and because you're not equal you could already see that it's not conservative and in fact remember I said if it's conservative then it would the F would be equal to minus YX and also equal to X Y at the same time okay and so the question is what is next well one thing is well we found a criterion that rules out non conservative functions but then the question is if P y equals to QX doesn't imply that F is conservative and turns out yes so I like to call it an S or in German Lineman because it's a yes or no and second question is well for conservative vector fields it's easy to calculate the line integral but what do we do if F is not conservative and this leads us to what's called Green's theorem which is the second FTC for line integrals so part for Queen spirit and what is that so first of all it only goes for close curves one where the starting point equals to the ending point so suppose you have a closed curve C and the nice thing is once you have a closed curve you actually have a region inside then it turns out that the line integral the line integral is still possible to evaluate but I just want to emphasize something look if F is conservative so suppose F is the gradient of some function then the line integral of F would be the line integral of the gradient of that ok that would be f of the endpoint minus F of the starting point but look if you have if you have a starting point here then the end point would be the starting point in particular this would be zero so for for conservative vector fields this is not a question the line integral is zero that's what this is more interesting for non conservative vector fields and it turns out it's the following and let me maybe motivate this with calculus in calculus you say that the integral F equals to the double integral of F prime so here we want to see the line integral F is the double integral of some derivative of F and the cool thing is the answer is precisely quixotic ions and so does I need to go that is a double integral of quixotic my aunt and let me motivate this a little bit so this number it measures what's called the microscopic rotation of s and let me give you again a typical example where mass is minus YX in that case quixotic diane's that's two and in fact if you look at the picture I've drawn the s looks like that in particular notice it rotates so in fact what this says is that the rotation of F is two on the other hand if you take a conservative vector field so f of X Y is X Y which looks something like that but every point is just expands then quick sonic PI ends are zero and calculate this because Y X minus x1 that is zero and in particular this dozen extent a that off its just expands at every point so in particular this number I like to call it a rotation number or something it's a number and it locally says how your vector field rotates and I like to call bright rods with little hurricanes so suppose you're a curve C then quick saw deep ions or like little hurricanes they measure how your vector field rotates QX / Q quick saute pans and what does green theorem said it says that if you sum up those little hurricanes those little rotations you actually get the global rotation of F over C so one interpretation of the line integral is that it measures the global circulation I can start for using physics term would think of it in terms of math to do physics okay in other words if you sum up those little rotations on earth you actually get the actual rotation of that or if you some of these little hurricanes you get a big hurricane which this shirt should make sense right and that's why in some sense greens theorem make sense I took a very useful theorem that allows us to calculate line integrals of non conservative vector fields and it also allows us to answer questions we asked at the beginning I said that well if P y equals to Q X so if P Y is not equal sir I said if every sponsor then dy equals to QX another question is what is p1 equals to Q X then what quixotic diane's r0 and so by greens theorem f dot dr equals to the double integral of 0 DX e 1 and that is that means so that means that the line integral over every closed curve is 0 and it turns out that this implies that f is conservative so it turns out if you're the line integral over every closed curve is 0 turns out it is conservative and essentially the reason is first of all if the line integral over every closed curve is 0 it implies path independence because let's say it is path independent and you want to calculate F over a closed curve then sure you can either go around everything and that gives you the line integral of F or you could just do nothing stay at this point then of course the line integral over that point is 0 but because its path independent you actually get it it's also equal to the line integral F over the curve that's one thing and conversely if the line integral over every closed curve is 0 its patent dependent simply because if you have two paths with the same starting and the ending point you can consider the following path you just go for C and then - C prime then because it's closed the line integral is zero and therefore basically this line integral minus this line integral equals to 0 and you get that they're equal so this property implies path independence and it turns out path independence implies conservative vector fields because basically if you have in this case you can just define so suppose F has this property that its path independent and then you can just define F to be an anti derivative of capital F which if you just think of as the line integral from any point a B to any point XY F started with the art so a be 2xy and this of course makes no sense if it's time dependent but because we can choose any path we just define it as the line integral any path from a B to X Y and then using some little calculations you can show that indeed if the gradient of F equals to capital F then F is conservative okay that was a parenthesis so what I want to say is if F is conservative and py equals to Q X and if P y equals to Q X then F is conservative little technical detail you need to make sure that there are no holes to apply greens theorem if there are holes and it's a little problem this is not necessary matrix alright that's very good so we don't greens theorem which is a FTC two-dimensions and of course now the question is what do we do in three dimensions so which leads us to another topic which are called surface integrals so I think part five because you see before in two dimensions and even three dimensions we had lines but now the new feature is that in three dimensions we also have surfaces and so suppose you have a surface s and because s is two-dimensional now we can parameterize it with two variables are UV before we are the line which we parameterize with one variable and also before we have you know the direct tangent line which you know directed by our file of T in this case what we have is a tangent plane and the first question is of course how do we find the equation of this tangent plane well notice there are two vectors in that plane one which is our u so partial R over partial U the other one which is our V which is partial R over partial V and here comes the most important thing in my opinion about this before we use the tangent line you know which was directed by our prime of T so for clients you use a direction vector now in order to find the tangent plane the equation you use what's called a normal vector and I call it n hat which is ru cross RV so we have this new character that comes into play which is n half and again this is so important the analog of the R prime of T for lines is the normal vector for planes so what the direction vector is two lines the normal vector is two a plane and then from that you can in theory find the equation of the tangent play and everything but this is not why we're here today there is a separate video on that if you want the next thing is simply to define what's called the surface integral and just like for line integrals of functions where we summed up F over a line this time we can sum on F over a plane so again this is if you wanted three dimensions so this is an XYZ and assume somehow in the fourth dimension you have a function that just floats on top of your surface impossible to visualize but think of it as follows against suppose you have a point XYZ we have f of X Y Z but s is two-dimensional so let's just use our parametrizations or UV and an F of our UV then again the line integral the surface integral it does it measures the volume under F and over the surface and how do we find this basically base times height here the height is f of our UV and what is the base so beforehand it's all about the base but the base so before our base was just this little segment D s now what we want is a two-dimensional segment which in this case is just a little parallelogram so D s capital s bigger D s well how can we find this parallelogram simply by using this idea of the tangent plane so suppose you have a point are you V then consider the following parallelogram the parallelogram spanned by ru and RV maybe it's this thing here right the problem is it's just it's a bit too big so we would like to scale it so instead of finding ruv we have our U multiplied by an on little number d u + RV x little number DV and we get this tinier parallelogram yes and the question is how do you find the area of such a parallelogram it turns out you learn in multivariable calculus that is just given by the length of this vector cross this vector so the s equals to or UD u cross RV EB and you take the length of that and you can just pull out the D U and DV and you get ru cross RV d very good so what do we have the base is this the height is that and then you just - no base times height and simply f of ruv times length of ru cross R so we define a surface integral f of XY Z the S to be simply the double integral of f of r UV then d s is simply ru cross RV d UV and again it's height times base and what do we integrate it on namely just the domain of U and V and we call that D and so this is how to surface integrate a function and now the question is for how do we generalize two vector fields so and you see it's not quite the same but it's kind of similar so so kind of wasted your time with this but now it's done important because what was the most important thing from before it's the idea of a normal vector so before maybe as an analogy the line integral of FDR that was the integral from A to B F of R of T dotted with R prime of T and I get the idea was we had a curve C and with the direction vector R prime of T and all we did with dot F with our prime of T and so this quality measures how in what direction and faces the curve well it turns out it's the same thing here except not quite so suppose now you have a surface s and you know let's say a point r UV then on the one hand we are f after at that point so f of r UV and look remember what i said the most important thing of a line is the direction vector well here the most important thing about the plane is the normal vector and have so before we dotted F with the direction vector because we had lines now because we have a plane we don't f with the normal vector I started with n hat D okay so it is a bit weird but again the reason this makes this work is because well for curse we can describe it with one vector R prime of T plates we can also describe them with one vector and hot that's what we can do this otherwise the problem is which direction for s T you pick do you pick up are you do you pick RV you pick something else no we need something more systematic and the systematic thing is at hat and if you do mathematics then this works because the co dimension of a plane is one so we can sort of take the plane fill it up with one vector to get all of our three that's why we can even define those surface integrals and again it's sort of the same idea you want to sum up numbers but this time this number measures how far as points away from your surface so for example if you have the surface here and then the normal vector here and as sort of flows out of the surface then F dotted with and hot is positive and think of it and guess in terms of physical like think is the fluid this just flows out of your surface suppose in this case F flows into your surface maybe it goes like this then F dotted with and half is negative so F flows in think of something flowing into a ball and lastly if F is tangent to your surface then F dotted with and hat equals to 0 so as neither flows in and/or flows out and what you're doing essentially you're summing up those numbers over is over you and me sorry that should be over D just a domain um so this is from a number that measures the what's called the flux it's how much it flows in or flows out and just one little remark you can indeed write this as you know in terms of in terms of D s so first of all this is f of ruv dotted with n hat dudv and that's double integral over D F of ruv dotted with ru cross RV DB and notice the similarity you dotting F with some sort of derivative here you dotted f with an R prime here u dotted with ru cross RV and just one little thing so that's the same thing as f of ruv dotted with ru cross RV over length of ru cross RV times length of ru cross RV dudv this we said well that's just the length of this area of this parallelogram so that is D s and what you left with because you're integrating over the surface it's indeed double integral s f dotted with so that was a normal vector divided by the length of the normal vector but whenever you divide a vector by its length you get what's called a unit normal vector so this is called n it's unit again you have your surface as your bank vector field F and the N is just what's called the unit normal vector to surface so and that also really explains how is something of the values of F over the surface and so this becomes your surface integral as an ottoman yes equals to the surface integral of the function f dotted with n TS which relates the two topics we've talked about before all right very good so now we have a clear idea hopefully of how to sum up the values of a vector field over a surface and now of course the question is are there any STC's for that and in particular the question is can we generalize greens theorem or you know to three dimensions but screens theorem is awesome it allows us to calculate surface in the houses to calculate for example line integrals of non conservative vector field okay well let's see what you which part you're on let's say we're in part six this will lead us to what's called Stokes theorem there is a very natural analogy of a Green's theorem but we have to first we'll define a new operation of vector fields because the question is for greens theorem we have quixotic diane's what is the analog in three dimensions well not a problem let's again ask ourselves if a vector field is conservative then what do we have so suppose if F is conservative so suppose F is the creating of some function and now in 3d then PQR FX FY FZ in particular we have you know P equals to FX Q equals to FY r equals to FC then let's just apply the clear watch steering trick again f of X y equals to f of Y X f of X D equals to f of Z is and F of y Y Z equals to F of Z Y so F X y equals to F of Y X which gives us you know quixotic Mayans P y equals to Q X so QX minus P y equals to 0 f of X is f of Z X that gives us f of X Z equals to F of Z X so P Z is easy peasy equals 2 power X so if you want rx minus P Z equals to 0 and lastly of severity of pizza that's almost lunchtime so okay f Y Z so f of Y Z equals to F of Z Y and that becomes I guess Q Z equals to R 1 so something like R Y minus Q Z equals to 0 so alpha vector field is conservative then those three identities hold and the question is is there some quality that combines all those three things and yes it's what's called the curl so and we define it as follows defiance of the curl of F is simply the finest follows is the gradient cross F and that's i JK partial over partial X partial over partial Y partial over partial Z P Q R and that just becomes if you want R Y minus Q Z minus R X plus P Z so P Z minus rx and QX minus P 1 which is precisely more or less the three identities we have our Y minus QZ our X minus P Z so in the other direction and Q X minus py and in particular if f is conservative then you see that the curl is zero so if the curl is not zero we can already say that it's not conservative okay and what does it measure well I would like to think of it really the analog of quixotic pie and what this is is another vector that measures the micro rotation of F but in you know in three dimensions you may ask why are there three numbers and that's because they're really three planes here at stake F X Y Z suppose your vector field looks like that or something then your gradient me you curl maybe look like that but you see what happens is that you know queue it quixotic pans measures if you want the rotation in the XY plane on the other hand Y PZ - yeah P Z minus rx it's a rotation in the XZ plane and lastly you have this what's called a ry minus QZ it's the rotation in the Y's evening so you see because you have those three served from axis of rotations you do have a vector with three components so for example if you have this vector field here XYZ and maybe looks like that except you see it's a constant side when one I say it's like it's like constant it doesn't depend on Mencia yeah it doesn't depend on X at all it's always the same thing with X I think it's maybe something zero minus ey or something then look in the front in the XY plane there is no rotation at all so something Q X minus py that should be zero and then in this plane is also no rotation so PZ minus our x equals to zero but in the Y Z plane is a big rotation so our Y minus QZ is positive and in particular for this vector field if you calculate the curl the curl should be two zero zero because in our Y minus Q Z is positive which explains how it rotates in the Y Z plane and this direction you just get it from this right-hand rule from physics yeah your thumb points this way that's what is this thing so it's really the the rotation perpendicular to this x axis so it does measure to my crew rotation and because it measures the micro rotation there should be an analogue of Stokes theorem which now I can finally say that it's super super neat to Stokes theorem I like to remind you of greens theorem it's just an F dotted with the R equals to the double integral d of QX minus py and again the interpretation was if you have a cur closed curve C then the global circulation of f is the sum of micro rotations well in this case it's the same thing so this is little hurricane's suppose you have a surface like that with boundary curve C then the line integral F dotted with the R is the double integral of the curl of F and again I want to convince you that it's the same thing so this is in three dimension I want to convince you it's really the same thing as greens theorem in three dimensions because again the line integral F dotted with the art is the global circulation of F over C and this thing is just the sum of micro rotations if you think of like little hurricanes inside see if you sum up to sum of hurricanes you indeed get the hurricane around C and it was the same thing here if you sum up the little hurricanes in D you get the big hurricane around C and so in some sense you're two ways of thinking about it you can either think of greens theorem as a flat version of Stokes theorem or you can think of Stokes theorem as a curved version of greens theorem so think of it like this is like a curved carpet version of Stokes theorem where's grease theorem is just a flat tile or something so it is indeed you know nice generalization and it also explains that before we had that if F is conservative then the curl is zero and in particular if the curl is zero we do have the converse now then the line integral F dotted with the R is the surface integral of the curl of F dotted with DX but if this is zero then we get that the line integral over C is zero and I forgot to mention of course is for any closed curve otherwise this wouldn't make sense and therefore we get that the line integral of F over every any closed curve is zero so by the other fun fact and mentioned with greens theorem we get that F is conservative so conservative is similar is equivalent to curl of F is zero which is nice it gives us a clear criteria of showing whether a vector field is conservative all right so great we had greens theorem we had this generalization but it turns out there's another nice generalization which I think it's maybe easier to use so and it might be a bit more it's definitely easier to use I think it's not as intuitive us it's so cool it's called the divergence theorem so part seven divergence theorem or I like to call it the Gauss screen theorem because anything that has to do with Gauss's awesome so divergence theorem and here's the idea so what does greens theorem say greens theorem relates a line integral with a double integral so that was Green in 2d well in 3d the element of a line integral is a surface integral well and integral should be a triple integral and it turns out that that I virgins the herbal eats those two and remember Stokes theorem relates those two so it's the kind of neat thing not only that you also have a fool PD he relates the double integral with that your point ago foodini just says that if you integrate a double integral you get a triple point there also that's not what why are you here today so what would like to say is somehow that the Dom integral F dotted with the S equals to the triple integral of something and what is that something so first of all X has to be a closed surface so it's just made up or a one-piece and if you think of Nasser a plastic bag where air doesn't go out or anything that s is like no skirt surface what's nice about closed surfaces is that there's an inside and interior here so if the double integral F dotted with D s is the triple integral something we need something that with the rivet piece of f it turns out that's what's called the divergence and the divergence is as follows if n is P Q R and the divergence is just PX plus qy plus RZ you take the first derivative of the first component the second derivative of the second component and the third derivative of the third component and if you know linear algebra it's the trace of the derivative matrix that's what we choose this one precisely and then the divergence theorem stresses surface integral F equals the triple integral of the divergence of F and just one little thing so what does that mean physically so the divergence is sort of the expansion of think over like a gas and it that's expanding in other words that seek the typical example if F equals to X Y Z then the divergence of F is xx dos equis YY plus Z Z 1 plus 1 plus 1 and that's three so in this case the vector field F in three dimensions it looks maybe something that and indeed F is expanding so the divergence is explains that expansion of earth and basically what does the divergence theorem say it says that it is so long the little expansions of earth if you sum up the little divergences of Earth you should get to the global expansion of Earth over the surface and it makes sense cuz remember how we define surface integrals we defined it as how much goes out of the surface so you see it's a net net flux is how much goes out of the surface and indeed in this picture it makes sense if you sum up those little micro things then you get the macro expansion of yourself of your vector field and this is also very useful to calculate surface integrals because sure the problem is like with Stokes theorem is you need your vector field to be a curl this works for any vector field so that's nice all right and this officially ends the vector calculus extravaganza you'll see it's quite something and I hope you enjoy this and you know if you're taking vector calculus good luck it's very very exciting and so if you like this and you wanna see format please Nick please subscribe to my channel thank you very much

Channel: Dr Peyam

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Keywords: vector calculus, multivariable calculus, multivariable, vector, line, integrals, line integrals, vector fields, conservative, gradient, ftc, calculus, fundamental theorem for line integrals, fundamental theorem of line integrals, ftc line integral, ftc line int, curl, divergence, green, green's theorem, stokes, stokes' theorem, stokes', irrotational, gauss, gauss-green theorem, theorem, math, mathematics, dr peyam, dr peyam overview, overview, stewart, chapter 16, math 2e, 2e

Id: ISqwmuGkOXQ

Length: 72min 16sec (4336 seconds)

Published: Wed Dec 12 2018