Stokes Theorem

all right let's get stoked for stoke's theorem in particular today let's calculate the surface integral of the curl of f where f is a vector field x z y squared x y and s is the part of the paraboloid z equals 1 minus x squared minus y squared above the x y plane oriented upwards so first let's draw a picture of this surface as i said what it looks like it's just a paraboloid so think 1 minus x squared so it might look something like that and with the upward orientation meaning that the normal vectors face upwards now there's two ways of evaluating this the silly way would be to first calculate the curl of f and then evaluate the surface integral of that but when that's way too much work luckily there's a theorem that allows us to simplify this tremendously called stokes theorem and without further ado here it is all it says is that in order to calculate the surface integral of the curl of f dotted yes so think double integral of a derivative all you have to do is just calculate the line integral of f over the boundary of this surface and the boundary curve in this case is this curve here so literally instead of calculating a surface integral of a curl all you need to do is calculate the line integral of the original vector field which usually it's way easier to do especially if you're like me and you forgot how to calculate surface integrals now as i said let's first figure out the boundary curve and how do you figure this out well remember we said s was the paraboloid z equals 1 minus x squared minus y squared but above the x y plane which has equation z equals zero so just set 1 minus x squared minus y squared equal to 0. and see what we get so 1 minus x squared minus y squared equals 0. so x squared plus y squared equals 1. so c is actually the unit circle x squared plus y squared equals 1 in the x y plane now the question is how do we parameterize c so let's see how to parametrize c because we need that for uh our line integral and here you have to be very careful because the orientations need to match namely we know already that s is oriented upwards and now for c we have two choices is it clockwise or is it counterclockwise well and as i said make sure the orientations match and the way you can check for this is simply as follows if you're a person walking on this curve make sure that when you're walking your surface which you can think of a mountain is to the left and in and in particular he's a nice mnemonic it's always walk l and left so here for instance which orientation for c do we use well if we go counterclockwise then in fact the mountain is to your left so on your left hand whereas if you go counterclockwise then you have a problem because then the mountain would be to your right so in fact now we just choose the counterclockwise orientation for c so the usual one which is simply x of t equals cosine t y of t equals sine of t and z of t because it's in the plane z equals zero z of t equals zero and of course you can actually check that because if you go this way your y-coordinate becomes bigger and your x-coordinate becomes smaller so be careful sometimes there are some tricky problems that fool you with this all right very good and now that we parameterize c cloud okay c we're almost done because now we can just use stoke's theorem which tells us again this horrible integral becomes much much easier so again what do we have we have by stoke's theorem the double integral of the curl so the surface integral of the curl equals to the usual integral where so f the line integral of f d r where again f was a vector field let's see x z y squared and x y and remember c was parameterized just as follows x of t equals cosine t y of t equals sine of t and then z of t equals 0 and again t from 0 to 2 pi so how do you calculate a line integral remember it's the integral from 0 to 2 pi of f of r of t so just f but you plug in x of t y of t z of t and dotted with the derivative so x prime y prime z prime dt and so let's calculate this this becomes the integral from 0 to 2 pi again x of t becomes cosine t z of t becomes 0. so again that's your x z and then y squared becomes sine squared of t that's your y squared and then x y just becomes cosine t sine of t so that's x y and you dotted with the derivative of that so you dotted with cosine prime which is minus sine so again think x prime and then sine prime which is cosine y prime and lastly zero prime which is zero so let's see prime d t and now this looks like a huge mess but the nice thing is when you dot those things so again with you dot those two vectors you actually get bunch of zeros so in fact what you get this simplifies tremendously this becomes simply integral from 0 to 2 pi 0 times something so just 0 sine squared of t times cosine t plus cosine t sine t times 0 plus 0 dt and so in the end all you need to do is just evaluate sine squared times cosine and you're left with integral from 0 to t sine squared of the cosine of t dt and again an antiderivative which was 1 3 sine cubed of 3 t from 0 to 2 pi but if you evaluate this both at 0 and 2 pi you get 0. so in the end the value of the surface integral of the curl is simply 0 which is very neat

Channel: Dr Peyam

Views: 6,431

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Id: -fYbBSiqvUw

Length: 8min 40sec (520 seconds)

Published: Sun Jan 10 2021