Describing Surfaces Explicitly, Implicitly & Parametrically // Vector Calculus

in this video we're going to talk about surfaces like these ones we're going to try to describe them mathematically in implicit explicit and perhaps most importantly parametric forms now this video is part of my whole series on vector calculus the link to that is down in the description and in future videos we're going to talk about things like what is the surface area of a surface what is the surface integral analogous to the line integrals we've seen before and we're going to ask the question of well what would i do if i have a vector field combined with a surface but in this video we're just going to describe them mathematically now to be clear i'm only talking about the outside of these regions i'm not imagining any interior that is i'm sort of imagining like it's just the surface of the earth and i'm walking along the surface of the earth where i could go north south or i could go east west or a combination of those but there's really only two degrees of freedom when you're walking along the earth i'm not talking about the inside of the earth at all and so what makes a surface a surface is the fact that it can be described in some sense as having sort of two different dimensions in which you can walk despite the fact that it's this larger embedding in a three-dimensional space we'll be a bit more precise what we mean by that in a little bit now the first thing i want to do is actually go back to the study of curves we've done before because we've seen that for a curve like this this is the top half of the unit circle that there's actually three ways that we can describe this we can describe this implicitly so this is when i just give some function of x and y and set it to be a constant in this case x squared plus y squared is equal to 1. i could solve it explicitly which is when i take my formula and i try to solve it for one of the variables like y is some function of x in this case square root of 1 minus x squared and i chose specifically that we were only doing the top half of the unit circle because then i can solve it i don't have any weird plus or minus issues coming up in general though it's often not possible to start with an implicit formula and to generate just a single explicit formula for it that works for everything this in general is not possible so it's only sometimes that i would have an explicit formula for my curve and then the third and the one that's gonna be most important for us was the parametric form this is where i take some parameter t and then i specify some function in the i hat in this case it's cosine of t some function in the j hat in this case sine of t and whenever i come up with a parameterization i always have to specify the endpoints of my parameterization so in this case i'm going only half of the circle so t is between 0 and pi so that was for the specific example of a half circle if i'm going to be just a bit more general the implicit formula is just some function capital f of x and y equal to a constant the explicit is when i can solve it so that y is equal to some function lowercase f of x and then the generic parameterization is just some position function r of t which is just a function of g of t in the i hat a function h of t and the j hat and t bounded between a and b okay so let's use that base for trying to now study surfaces in three dimension so i'm thinking about a two-dimensional object embedded in three dimensions so it could be the case that i have an implicit description of this it'll be the exact same type of thing we've seen before capital f of x y now and z is equal to a constant it might be that i could solve this for z so z is given as a function of x and y explicitly that sometimes could be the pace and sometimes can't be and then most importantly is the parametric formula now i'm going to have two parameters so i'm thinking of surfaces as a two-dimensional object so there's two degrees of freedom so i have two parameters and i'll call them u and v and then because it's living in three dimension there's an i had a j-hat and a k-hat component and each of those functions depend on the parameter so it's an f of u and v in the i-hat a g of u and v in the j-hat and an h of u and v in the k-hat and then just the same as we've seen before i have to say what are the limits of my u and my v so u is going to generically go between an a and a b and v is generically going to go between a c and a d okay so let's see an example i want to talk about the cone that one way to think about this is just taking some line and rotating it around to make this cone but i have an explicit formula for it z is the square root of x squared plus y squared now whenever you have an explicit formula it's actually completely trivial to get an implicit formula as well you just subtract off so the implicit is just z minus square root of x squared plus y squared equal to zero or we haven't done anything at all here the z minus square root of x plus y squared just is the capital f of x y and z we've seen before so the point is if you have an explicit you could always get to an implicit if you start with an implicit going reversing that process and trying to get an explicit is often just not possible and even when it is possible theoretically it's often not easy to do so implicit to explicit is hard but explicit implicit is just trivial so for the parametric form i need to come up with a choice for you and a choice for v i have an x and a y already so what if i just do that why don't i set it to be the parameterization of x and y in which case the x component is just x the y component is just y and then the z component because it's described as square root of x squared plus y squared i just put in square root of x squared plus y squared k hat so this is perfectly fine from the perspective of have you found a parametrization but it's actually not that useful what i really am going to want to do is come up with parameterizations that nicely capture the symmetries involved in whatever the surface was and there is some symmetry there's this sort of circular symmetry to a cone that is just sort of ignored by this choice of parameterization and this is going to be a real problem for us later we actually want to do things like compute the surface area of this and this parameterization can have some really messy square roots and integrals it's going to be a whole mess so i'm in fact going to get rid of the implicit forms i had before because it was sort of trivial and then i'll also get rid of that canonical parameterization that you could always do if you have an explicit formula but might not be very helpful and we're gonna try to come up with a better one so first thing i'm gonna put up a sort of cylindrical coordinates if you will on this picture basically what i do is i take a point of the cone and i project it down to the xy plane then i get some vector going from the origin out to the spot where it projects down and that has a length of r so r is defined to be that particular length and then there's also an angle of theta that starts at the positive x-axis and rotates counterclockwise until it hits that line the line we've labeled of having length r because my explicit formula was z equal to square root of x squared plus y squared it could have been anything that's what i've chosen this happens to be the exact same thing as r r is defined when you're talking about polar coordinates as the square root of x squared plus y squared this is just r so the point is if you know the theta and you know the r value you know exactly where you are on the surface because theta and the r is enough to describe a point down on the plane and then if you want to know how high up you should be what the z value is well whatever your r was that's what the height is as well z is equal to r and so we only need a theta and an r and as a result i can say my parametrization is the position vector r vector there's two r's here one with a vector hat and one without so the position vector is a function of the two parameters r and theta and well we do exactly what we've done any number of times we've talked about polar coordinates in the past we've gone and said well the i hat is our cosine the j hat is our sine but what we're adding now is the k hat component which is the height which would normally be z but since that is equal to r it's plus r k hat and so if you know the r and the theta we can describe where you are on this cone no this isn't slight contrast to what we might have done when playing with cylindrical coordinates in the past where we would have had a theta and an r and a z all three of them floating around and we would have had a triple integral that used all three of those variables if we were computing say the volume of a region in cylindrical coordinates that's three-dimensional here we're trying to describe a two-dimensional surface and so basically what you do is you're choosing two of the three dimensions r and theta which is enough to describe everything because we have that hook that relationship that z was equal to r and so a third variable isn't needed to describe this here theta and r is enough now i have to be careful here because i've been a little bit coy with you thus far i said it's a cone but which specific cone because what is the height of that top off where it stops graphing it i actually told it to stop graphing at z less than or equal to 3 so i'm going to put that up there z is less than equal to 3 is a restriction on my formula so it's only that portion of the infinite comb that it would be if you didn't have the restriction okay so how does this translate into r values and theta values and the restrictions on those well z is equal to r z is clearly positive here it's a positive square root and so what we have is that r is between 0 and 3 as well and then theta is one revolution around it's the entire revolution so theta between zero and two pi and d whenever we try to write out a parametrization we additionally want to describe the boundaries on the parameters in this case on the r and the theta the final way i'm going to think about this is okay let me just look at my cone i want to imagine this as some sort of transformation there is a two dimensional space that has axes r and theta and in that r theta coordinate system the region is very simple it's just a rectangle on the r values going between the 0 and the 3 and the theta value between the 0 and the 2 pi it's a simple region in that coordinate system and then what a parametrization really is is a transformation from the simple region that i have the r theta two-dimensional region to this more complicated two-dimensional region that is the cone embedded in three dimensions the parameterization is just a way to map between this simple space and the r theta plane to this more complicated situation and because i've chosen natural parameters that reflect the underlying geometry we're able to really simplify the process and we'll see how that works in a little more detail coming up in future videos so for now if you enjoyed the video please do give it a like if you have any questions about this video please leave them down in the comments below i try to get to as many questions as i possibly can and we'll do some more math in the next video

Channel: Dr. Trefor Bazett

Views: 9,513

Rating: 4.9940119 out of 5

Keywords: Solution, Example, math, Surface, Integral, multivariable calculus, vector calculus, implicit, explicit, parametric, sphere, cone, torus, parabola, Surface Area, function

Id: jZRqCfi5_Uo

Length: 11min 4sec (664 seconds)

Published: Mon Nov 23 2020