Stokes Theorem

alright thanks for watching and congratulations for making it this far today is the last lecture of our vector calculus extravaganza and we're gonna end it with the bang by using Stokes theorem which is the quintessential vector calculus FTC theorem so what does Stokes theorem say suppose you have a surface yes so think of s as a balloon and it has a boundary curve C so maybe something and assume C you've no respects to boundary of the and the orientation of the surface by the way the way I like to think of all this is s as being the balloon and C is like this string at the end of the balloon so then what does Stokes theorem say it simply says that the double integral of the curl of F dotted with yes is equal to the line integral of F with the R in other words Stokes theorem so transforms a very complicated surface integral into a much easier line integral and vice versa it also transforms a very complicated line integral into actually an easier surface integral which is the point of view we'll will use today and I'll explain also the intuition behind it and hopefully by the end it should make sense but just one thing though the way to think about this is simply Stokes theorem tells you that the line into the double integral of the derivative F equals through the integral of F so it's like a fancy version of the fundamental theorem of calculus okay and as I said there are two applications of this first of all it allows us to calculate double integrals of curls which is pretty complicated it also allows us to simplify line integrals so in fact let's now calculate the line integral FD r where f is this very complicated vector field not only is the vector field pretty complicated but also C is pretty complicated C is the curve of intersection of the plane Z equals 2y plus 2 and the cylinder x squared plus y squared equals to 1 so let's first of all draw what this looks like so X Y Z and only one hand we have the cylinder I'm gonna use some colors right that's x squared plus y squared equals to one on the other hand we have the plane and usually those kinds of planes are easier to be drawn with intercepts so the Z intercept becomes 2 and the y intercept is minus 2 so it looks something like that except it doesn't depend on X which means we can just shift it with respect to X so Z equals 2 y plus 2 looks like that and essentially what you have to think of is simply you're taking this can this cylinder and you slice it diagonally so if you actually do that you'll see that the curve of intersection is an ellipse diagonal ellipse okay that's the first thing draw a picture see what's going on now the next thing is simply to find that s so find us and remember this picture I address your race we had this boundary curve C and the point is for s we kept it whatever we want as long as the boundary curve is the same we'll always get the same answer this is the cool thing about Stokes theorem in particular let s be the easiest thing possible in this case given the curve C that s just be the flat inside of that curve in other words what does that mean let s be the portion of the plane Z equals two y plus two over does this x squared plus y squared equals to one let s be the portion of Z equals two y plus two over the disk x squared plus y squared less than or equal to one I get the point is as long as the boundary curve is the same you still get the same answer and now would you want to use this of course Stokes theorem folks which says that the line integral of F dotted with VR which by the way it's possible to do it directly but it's a pain because you have two parametric this curve and also you have to plug in the values of F it's not fun what is more fun though is to calculate the surface integral so curl of F dotted with yes now what is the curl it's simply the determinant I'm sorry the cross product of the derivatives and the vector field do that it's essentially a number a vector which gives you the rotation or mass in each direction so curl of F that is the determinant of ijk partial or partial X partial over partial Y partial or partial Z over a vector field - 2 y XZ x + y and that becomes so the X derivative of x + y sorry so boom you take the Y derivative so partial partial 1x plus 1 minus partial over partial Z X Z then you do the second thing now partial word partial X X plus y minus partial word partial Z to Y but with the minus because of determinants so - partial over partial x x + y plus partial over partial Z - 1 and lastly cross-product partial word partial x XZ minus partial or partial Y - 1 - why okay and basically you get one minus X now minus one plus zero so minus one and lastly Z minus two and you see this is already much easier than 2 YX Z and X plus y so even though we have to do a surface integral it's a much easier surface integral ok and now remember how to surf this integrate you have to parametrize go so again this is your surface s ok and it turns out you're slightly easier s is the graph of a function and there is a different formula for it but I hate that formula so let's just do it from principles again what is s is the part of the plane Z equals 2 y plus 2 over the disk x squared plus y squared less than or equal to 1 so because it's the graph it's easier just to choose parameterization our XY or XY is here so our X is basically X y equals to Y so parameterize so X equals 2x y equals 2y and what is Z is just Y plus two other words R of X Y is X Y y plus two subject to x squared plus y squared less than equal to one that's your parametrizations the second thing is simply finding the normal vector so our X is 1 0-0 our Y is 0 1 1 cross it o to cross products in one problem how exciting so ijk 1 0 0 0 1 1 if you do that you get that the 0 1 1 so minus 1 1 you just want to think you always have to make sure that the normal vector again respects the bound the orientation of your curve so for your surface because notice by convention C is counterclockwise and then if you use the right hand rule you notice that the normal vector has to point upwards so make sure whatever normal vector you have in this case points upwards and indeed it does because the last component is positive so good and lastly now we have everything that's calculate our surface integral so double integral as curl of F dotted with D s that double integral of D o remember the curl was 1 minus X minus 1 Z minus 2 so X is 2 X minus 1 is minus 1 but remember Z is y plus 2 so we get y plus 2 minus 2 dotted with your normal vector 0 minus 1 1 and again DX dy those two cancel huh and then if you got it you get 1 minus x times 0 minus 1 times minus 1 which is 1 and y times 1 which is y so the point is in the end it's much easier to calculate that integral and so you get the integral over D of 1 plus y DX dy how much easier is that all we have to do is just calculate this double integral and for this remember D in this case is just a disk of radius 1 so it's a perfect time to use polar coordinates see if this works sort of integral from 0 to 1 0 to 2 pi 1 plus R sine theta Rd Rd theta that's equal integral from 0 to 2 pi integral from 0 to 1 of are plus R squared sine theta the Rd theta now for this part well are doesn't depend on theta so we have 2 pi and then if you calculate the anti do it's R squared over 2 from 0 to 1 and here the interesting thing is the integral from 0 to 2 pi of sine of theta just becomes 0 so I'll give it energy this goes this goes down and you left with 2 pi times 1/2 which is PI again so you see well I'm gonna see no messy calculations no messy calculations really it's just a bit long that's all but again the alternative would be really to calculate the line integral which is pretty messy ass but not least before we end this let me give you some intuition behind this because I'm claiming that souks theorem is just a surface version of greens theorem remember what greens theorem said it says that read the line integral F that's in two dimensions FDR equals to the double integral of quixotic pius x dy and the interpretation of this was again the global circulation of Fr on C is just the sum of those little circulations QX minus py so in other words if you sum up those many Americans you should get the big breaking it turns out we have the same thing here suppose you have a surface so we have a surface and the boundary curve is C then what does Stokes theorem say it says that the global circulation around that surface equals to the double integral of curl of F dotted with yes and what is the curl it's again it's a vector which measures the again this mini circulation may be the rotation of F in the XY XZ and Y Z directions so it's still the same picture you have those mini circulations here there with those mini circulation vectors and Stokes theorem says that if you sum up those mini circulations on the surface you get the global circulation on the curve so it is really unlike a surface version of greens theorem Queens theorem works for this flat 2d surface versus Stokes theorem works for this curly surface if you like it's the same thing but both are very useful you so easy calculate line integrals except greens theorists for two dimensions Stokes theorems for three in higher dimensions alright and this officially ends a vector calculus extravaganza I hope you're the good time and if you're taking my class good luck on the final I'm counting on you and yeah so if you like more math they want to see more extravaganza please make sure to subscribe to my channel thank you very much

Channel: Dr Peyam

Views: 26,944

Keywords: stokes, stoke's, stoke, theorem, vector calculus, curl, surface integral, line integral, multivariable, calculus, math, peyam, dr. peyam, dr peyam, ftc, fundamental theorem of calculus

Id: bDILtddFKxw

Length: 18min 14sec (1094 seconds)

Published: Fri Dec 07 2018