Parametrizing Surfaces, Surface Area, and Surface Integrals: Part 1

welcome to 16.4 16.4 we're gonna parameterize surfaces we're also going to talk about computing surface areas and surface integrals so our goals for today we're going to talk about how to construct parametric equations of surfaces we're also going to compute and describe some tools that we need to be able to compress describe surface area these tools are our tangent vector in the U Direction our tangent vector in the V direction and a normal vector and then finally we're going to compute some surface areas and surface integrals using these parametric equations and sub surfaces so before as a start let's start with a review we've seen parametric equations before in the context of a line so let's say that we have so I'll say recall we have a line with Direction D and so that's gonna be D 1 D 2 D 3 it has a three component vector that's the direction it's in and let's say that the line also has some point and we're gonna call the point a B C that's the point that goes through this line and we constructed equations for these lines typically we call them L of T and it would be equal to just like point-slope form it's the point ABC plus T times oh I should actually be meticulous and represent these as vector outputs I always you know I little bit and I apologize so we have our point ABC our initial position essentially we can think of it sort of as our T intercept when T equals zero and then our direction vector V 1 D 2 D 3 multiplied by T because that means that we start at our point ABC and we can stretch and shrink our T vector out in whatever direction we want to be able to trace out the line in this direction D I'm going to point out that we can simplify this a little bit I don't know if you'll call it simplifying but if I distribute this through and do some addition I can think of the first component of my line as being a plus TD 1 just by adding the first components and then the second component would be B plus TD 2 and the third component is 3 C plus T D 3 where I'm thinking of this equation for my X component this equation for my Y component and this equation for my Z component the reason why I want to point this out because this is exactly analogous to what happens when we talk about parametrizing services so by definition our parameterize surface typically we represent with a capital P I'm not sure why we have some capital P and instead of so in the last example our line was parameterised just by one parameter T and it gave us a one-dimensional thing it gave us a lot now we're going to consider parameterizations with two input variables and our outputs in this case are going to be it's going to be a vector in r3 so it's going to have three component functions a component function that's a variable of UV and the X a variable of UV and the Y and what variables UV in the Z so notice that my inputs I have two different inputs using these I could think of these as points in r2 and the outputs that I get are a set of vectors which are outputs in r3 let's see a quick example of something that looks like this let's say that I have a plane a plane is a type of surface the surface is something that's of two dimensions that's out in three-dimensional space so let's say I have the plane 3x plus 2y minus Z equals four and instead of writing it in terms of XY z-- and Z's I want to be able to write it as a parametric form why yeah it's sort of for the sake of illustration because planes we already know how to work with like this and there are other shapes that we don't know how to work with as easily so considering this plane how old would I be able to parametrize this I need to come up with a UNM D variable and I need to be able to represent each of the components as U and V variables I'll show you my trick this is a magic trick magic trick maybe it's not that magic but if it's ever the case that I can set Z or any of the other variables as a function of x and y that it means that my parametric surface can be parameterised really really easily so let's go ahead and solve for Z and we see that when we solve for Z we can get that Z is equal to 3x plus 2y minus 4 so the magic trick is when I have to decide what my using these are - I'm gonna make it as easy as possible and I'm just gonna set my view exactly equal to X and might be exactly equal to Y because then I know that my Z in this case maybe I should flip-flop the order but my Z is equal to 3 times X which we're calling u plus 2 times y which we're calling V minus 4 and it means that I can use these three equations as the equations that go into my parametric function so in this case my parametric function for this plane the first component function were saying that X is just the boring function U Y is the morning function V and Z's the morning function 3 ok Carolyn that was a lot of work and we aren't quite sure what the payoff is just yet this sort of looks like something that would be is more complicated than what we started with before let's see another example so for our next example let's consider this cone let's consider the cone where Z squared is equal to x squared plus y squared alternately I actually I only want the positive part of it so I could rewrite it like this z is equal to the square root of x squared plus y squared how do we want to parameterize this well we can do the same magic trick that we did before I'm gonna let my X be equal to u my Y be equal to B and in that case it means that my Z is just going to be equal to the square root of U squared plus B squared and it means that my final parameter ish parameterization of this V of UV is going to be equal to my X component function which is just u my Y component function which is just e in my Z component function which is the square root of u squared plus V squared and that's that however if I actually want to deal with this function this is messy to integrate this is hard to deal with so instead of leaving it like this I'm going to give us an alternate parameterization we're gonna think of this somehow differently for my alternate parameterization instead of assigning my use of these exactly to my X and my Y values I'm gonna assign them two things that you might think are a little crazy I'm gonna decide I want my use to be equal to this radius value and I'm gonna let my fees be equal to theta just like we would in cylindrical coordinates so these are the things that I'm thinking in my head I still have to translate these things in terms of XY z-- and Z's so when setting up my V function in this case I still am looking for a way to express my x value in terms of use and Vees so if I pick a random point on this code I need to be able to tell what is the x value at that point right and it turns out that that x value just by the geometry of this I project this down whatever this x value is maybe you don't like this picture but I claim that the x value in this case it's going to correspond exactly to R times the cosine of theta and that's exactly because we're using our assignment to be consistent with our cylindrical coordinate system so in this case that means that because our x coordinate is our cosine theta and I'm letting my R be equal to u my X component function is going to be u cosine of B similarly my Y component function is going to be U sine of V and the reason why this is sort of nice is the fact that now I don't have to have my Z in terms of square roots in this case because of this very particular cone that we chose we know that x squared plus y squared is R squared and so Z is exactly equal to R another way that we can do that is just to assess visually we know that this slope of the cone is exactly equal to one so if my R is equal to one my Z this height right here would also be able to that my Heights are equal to the radiuses whoops and in this case we're letting you represent our so I'm pointing out that this is an alternate way to parameterize this particular cone and it's a parameterization that now might allow us to be able to do integrals a little more easily than if we had square roots trapped up in here

Channel: Multivariable Calculus

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Length: 11min 3sec (663 seconds)

Published: Tue Nov 26 2013