Surface Integrals // Formulas & Applications // Vector Calculus

in this video we're going to talk about surface intervals this is an extension of surface area which we've talked about previously indeed this video was part of an entire playlist on vector calculus and the link to that is down in the description now the generic notation i'll use for just surface area is the double interval over a surface d sigma the sigma here is thought of as a little element of surface area so i'm just saying add up all the little elements of surface area this is a great definition and it's much like how previously we would often talk about the arc length parameter ds which was great for definitions but wasn't how we typically computed things so in our previous video we've talked about how to compute this in a few different cases first there was parametric if your surface would describe parametrically some position function r in terms of u and v then your surface area could be computed by taking the double integral of your integrand being the length of the cross product our u cross rv du udv and so the combination of the cross product and the du dv is our little element of surface area when it's described perimetrically alternatively it might be described implicitly with some big function capital f of x y z equal to c a level surface and when that was the case we've previously seen the surface area could be written as the double integral of the length of the gradient of f divided by the absolute value of the gradient of f dotted with the vector p which was often k hat but could be i hat or j hat as well and then finally we saw that when it was described explicitly z is a function of x y it could be described as a double integral of square root of f x squared plus f y squared plus one so the point is three different ways to describe a surface three different methods to compute out the surface area of that surface okay so that's what we did in the past now let's go to the concept of a surface integral and the idea here is i have some new function i'll call it capital g of x y and z and this is the function that is defined on that surface so at any point on the surface there is a height if you will the value of this function g of x y and z it's a little harder to visualize this as a height now because your surface might live in three dimensions so the graph of this would be a four-dimensional object but regardless i want you to think at any point along the surface there's some value of the function then a surface integral is adding up the values of that function along each little element of surface area that is this the double integral along the surface of the function g d sigma this is exactly analogous to the concept of a line interval which would be the integral of some function d s and then i can return to my three different ways of describing it and two things have changed first instead of surface area i'm writing s i for surface integral everywhere and then i've gone and i've put a g into the integrand every spot so this is double integral of g times whatever the d sigma is for the three different presentations the g's look a little bit different and it's just to reflect the type of computation i'm doing so when it's done parametrically the three different components are written in terms of the parameters f of u and v g of u and v and h of u and v when it's implicit you can just write it in terms of g of x y and z as we had before that's fine and when it's explicit well since the z is f x y i'll write g as x y and f x y in for the third component regardless it's just putting a g into each of these different places okay so let's see a couple applications first application is the mass of some surface some thin shell for example i've drawn this cone surface and i want you to imagine that at some spots this material that's used to make that shell is thicker than in other spots for example if you're making a pottery bowl that's spinning on a wheel some points you squeeze in and the wall is thinner some spots it's a little bit thicker so you could have sort of a variable density a variable thickness as you go along so then if you had that so i'll let delta of x y and z be the mass density of my thin shell then the total mass would be adding up the density times the areas so in other words it would be a surface integral that is the double integral over the surface of integrating this density function the sigma that is for a little element of surface area a second example i'm going to use is actually involving a type of averaging so i want to imagine that there's a temperature at any point on the surface of the earth imagine that's a sphere of radius a i'm going to assign a particular temperature it's kind of like saying you have a height above the earth except instead of height it's a temperature above every point on the earth and then i just made up some sort of quasi-realistic temperature function it's really not that good but i'm basically just saying that if phi was zero or pi so the north and south pole the temperature would be minus 20 if c was pi over 2 aka along the equator then the temperature would be minus 20 plus 50 which is 30. okay sort of works i guess anyways if i want to figure out the average temperature well what's the formula for any average it's you add everything up and then you divide out by the number right if i wanted to compute the average grade in the course i'd add up all the scores divided by the number of students so the average temperature here is i'm going to add up all of the temperatures on all the little elements of surface area and then divide out by the total surface area so the 1 over 4 pi a squared is be dividing out by the total surface area and then i've got this surface integral where i'm adding up all the temperatures the integral of the temperature function d sigma okay so now i want to compute that out so i guess i should put in the actual temperature function i have and then this is described parametrically and imagining my surface here in the phi theta parameters and so i know what they are in spherical coordinates the ph goes between 0 and pi and the theta between 0 and 2 pi and then the most important part here that's been added is this a squared sine phi that corresponds to r phi cross r theta and we computed that before as in in a previous video we computed the surface area of a sphere we could parametrically describe it in exactly the same way and we'd come up with this formula that we need to do when talking about surface area parametrically it's the rp cross r theta that exact thing comes up in surface integrals as well and so since i've already computed it i'll just assert a few was a squared sine phi now this is just a double integral i suppose i can go and try to clean it up a little bit here by just expanding everything out and then for the integrand well okay minus 20 sine phi nothing happened there but i do have a sine squared and so i get to use the trig identity of one half one minus cosine of two phi at that spot regardless after that trig identity is really just sort of born and computational so the final result i'll give is this notice there's an a squared on the top and the bottom cancel the a squared it's a number about 18 degrees celsius so in my made up uh temperature function for the surface of the earth the average temperature on the earth is 18 degrees celsius it's worth perhaps noting that the minimum temperature in my model was minus 20 and the maximum temperature in my model was plus 30 and the average temperature of 18 is like way way way closer to the 30 than it is to the minus 20. so why might that be but if you think about the earth yes it's minus 20 at the poles but the surface area around the poles is pretty small whereas the surface area around the equator is actually really biggest it's proportionately larger surface area in the values of fee that are nearby pi over two than are nearby the zero or the pi and so that's why the average sort of gets weighted a little bit gets weighted more towards the top end than to the bottom end regardless this is just an example using the parametric formula to come up with a surface interval uh we can do the exact same thing for the implicit or the explicit as well regardless i hope you enjoyed this video please do give it a like if you did if you have any questions about this video leave them down in the comments below and we'll do some more math in the next video

Channel: Dr. Trefor Bazett

Views: 98,049

Keywords: Solution, Example, math, surface integral, surface area, formula, application, parametric, explicit, implicit, vector calculus, calc 3, calc 4, multivariable calculus, total mass, mass density, average, temperature, surface of earth

Id: 0VNhvwnqoTo

Length: 8min 18sec (498 seconds)

Published: Sun Dec 06 2020