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Current time:0:00Total duration:22:14

Video transcript

in the last video we finished off with these two results we started off just thinking about what it means to take the partial derivative of a vector-valued function and I got to these kind of you might call them bizarre result do you know what was the whole point in getting here salan the whole point is so that I can give you the tools you need to understand what a surface integral is so let's just think about let's draw the St the St plane and then see how it gets transformed into this surface R so let's do that so let's let's say that is the T axis and let's say that that is the s axis and let's say that our our vector valued function or a position vector valued function is defined from s is between a and B a and B I'm just picking arbitrary boundaries and between T being equals C and D so if we were to so the area under question if you take any T in any s in this rectangle right here it will be mapped it will be mapped to part of that surface and if you map each of these points you will eventually get the surface R so let me draw our in three dimensions it's a surface in 3d so that is our x-axis that is our y-axis and then that is the z-axis and just as a bit of a reminder it might look something like this if you if we were to this point right here where an s is equal to a and T is equal to C remember we're gonna draw the surface indicated by the position vector function s R of s and T so at this point right here when s is a and T is C maybe it maps to I'm just you know that point right there when you when you take a and C and you put it into this thing over here you're just going to get the vector that points at that so you could it'll give you a position vector that'll point right at that position right there and then let's say that this line right here if we were to hold s constant at a and just vary and if we were to just vary T from C to D maybe that looks something like this I'm just drawing some arbitrary contour there maybe if we hold T constant at C and very s from A to B maybe that'll look something like maybe it looks something like that I don't know I'm just trying to get an I just show you an example so this point right here would correspond to that point right there when you or when you put it into the the vector valued function R you would get a vector that points to that point just like that and this point right here in purple when you evaluate R of s and T it'll give you it'll give you a vector that points right there to that point over there and we could do a couple of other points just to get an idea of what the surface looks like although I'm trying to keep things as general as possible so maybe let me do it in let's see maybe I'll do it in this bluish color this if we hold T at D and very s from A to B we're going to start here this is when T is D and s is a and when you vary it maybe you get something like that I don't know so this point right here would correspond to a vector that points to that point right there and then finally finally this line or this if we hold a set B and vary T between C and D we're gonna go from that point we're gonna go from that point to that point so it's going to look something like this oh we're gonna sorry we're gonna go from this point to that point we're holding s at be varying T from C to D maybe it looks something like that so our surface it's you know we went from this nice area on this rectangular area in the ts plane and it gets transformed into this wacky looking surface and we could even draw some other things right here let's say we have a let's say we get some arbitrary value I'm gonna pick a new color other than white or new non color and let's say if we hold s at that constant value and we vary T maybe that will look something like this maybe that will look something like oh I don't know maybe it'll look something like that so you get an idea of what this surface might look like now given this I want to think about what these quantities are and then when we visualize what these quantities are we'll be able to kind of use these results of the last video to do something that I think will be useful so let's say that we pick arbitrary s and T so this is the point this is the point let me just put my pick it right here let's say this is the point that is the point s comma T s comma T if you were to put it in if you were to put those values in here maybe it maps to and I want to make sure I'm consistent with everything I've drawn maybe it maps to this point right here maybe it maps to that point right there so this right here this point right here that is R of s and T for a particular S&T I mean I could put a little subscripts here but I want to be general I could call this a while I already used a and B I could call this x and y this would be R of x and y would map to that point right there now so that right that's that right there or that right there now let's see what happens if we take if we move just in the s direction so this is so this is we could view that as s now let us move forward by Delta by a differential by super small amount of s so this right here let's call that a s plus a super small differential in s that's right there so that's that point all right let me do that in a better color in this yellow so that point right there the point s plus my differential of s I could write Delta s but I wanted this super small change in s comma T and what is that going to get mapped to well if we apply these two points in R we're gonna get something that maybe is right over maybe is right over there and I want to be very clear this right here that is R of s plus d s comma T that's what that is that's the point when we just shift s by a super small differential this distance here you can view as D s it's a super small change in s and then when we map it or transform it or or or put that point into our vector valued function let me copy and paste in fact the original vector valued function just so we have a good image of what we're talking about this whole time what we're talking about this whole time let me put it right down there so when we talk so when we took just to be clear what's going on when we took this little blue point right here this s and T and we put it we put the s and T values here we get a vector that points to that points to that point on the surface right there when you add a D s to your s values you get a vector that points at that yellow point right there at that yellow point right there so going back to the results we got in the last in the last presentation or the last video what is this R of S Plus Delta s or R of s plus D s the differential of s T well that's that is that right there that is the vector that points to that position this right here is the vector that points to this blue position so what is the difference of those two vectors and this is a bit of you know this is just basic vector math you might remember the difference of these two vectors head to tails the difference of these two vectors is going to be this vector if you subtract this vector from that vector you're going to get that vector right there you're going to get that right there a vector that looks just like that so that's what this is equal to that vector it makes sense this blue vector plus the orange vector this blue vector right here plus the orange vector is equal to this vector makes complete sense heads to tails so that's what that represents now let's do the same thing in the t-direction let's do the same thing in the t-direction I'm running out of colors I'll have to go back to the lab I'll go back to the pink or maybe the magenta so if we have we had that s and T now if we go up a little bit in that direction let's say we go now let's say that that is T so this is the point s T plus a super small change in T that's that point right there this distance right there is DT you can view it that way what is what if you put s and T plus DT into our vector valued function where you're gonna get you're going to get a vector that maybe points to this point right here maybe points to maybe I'll draw it right here maybe it points to this point right here a vector that points right there so that will be mapped to a vector that points to that position right over there now by the same argument that we did on the s side this point or the vector that points to that that is R of s T plus DT that is the exact same thing as that right there and of course this we already saw this is the same thing is that over there so what is that vector minus this blue vector the magenta vector minus the blue vector well once again this is fairly hopefully a bit of a review of adding vectors it's going to be a vector that looks like this I'll do it in white it's gonna be a vector that looks like that this thing this thing is going to be a vector that looks just like that and you can imagine if you take the blue vector plus the white vector the blue vector plus this white vector it's going to equal this purple vector so it makes sense it's the purple vector minus the blue vector is going to be equal to this white vector so something interesting is going on here I these two this is a vector that is kind of going along the going along this parameterize surface as we changed our s by a super small amount and then this is a vector that is going along our surface if we change our T by a super small amount now you may or may not remember this and I've done several videos where I show this to you but the magnitude if I take two vectors and I take their cross product so if I take the cross product of a and B and I take the magnitude of the resulting vector remember when you take the cross product you get a third vector that is perpendicular to both of these but if you were just take the magnitude of that vector that is equal to the area area of parallelogram I always pair powers forget how many are parallelogram defined by a and B what do I mean by that well if this is vector if that is vector a and and that is vector B that's a and that is B if you were to just take the cross product of those two you're going to get a third vector you're getting a third vector that's perpendicular to both of them and kind of pop out of the page that would be a cross B but the magnitude of this so this is always going to if you just take a cross product you're gonna get a vector but then if you take the magnitude of that vector you're just saying how big is that vector how long is that vector that's going to be the area of the parallelogram defined by a and B and I've proved that in the linear algebra videos maybe I'll prove it again in this I mean it's because it's C mag it's it's it's it's well I won't go into that into detail I've done it before I don't to make this video too long so the parallelogram defined by a and B you just imagine a and then you take another kind of parallel version of a is right over there and then another parallel version of B is right over there so this is the parallelogram defined by a and B so going back to our surface example if we were to take the cross-product of this orange vector and this white vector I'm gonna get the surface area I'm gonna get the area of the parallelogram defined by these two vectors so if I take the parallel to that one it'll look something like this and then a parallel to the orange one it'll look something like that so if I take the cross-product of that and that I am going to get I am going to get the area of that parallelogram now you might say hey this is a surface you're taking a straight-up parallelogram but remember these are super small changes so you can you can imagine a surface can be broken up into super small changes in parallelograms or super so into infinitely many parallelograms and the more parallelograms you have the better approximation of the surface you're going to have and this is no different than when we first took integrals we took we and we approximately the area of curve with a bunch of rectangles the more rectangles we had the better so let's call this little change in our surface let's call this little change in our surface D Sigma for a little change for a little amount of our surface and we could even say that you know the surface area of the surface will be the infinite sum of all of these infinitely small D Sigma's and there's actually a little notation for that so surface area surface area is equal to we could integrate over the surface and the notation usually is a capital Sigma that for surface as opposed to a region or so you're integrating over the surface and you do a double integral because going in two directions right the surface is kind of a folded two-dimensional structure and you're going to take the infinite sum of all of the D Sigma's of all of the D Sigma so this would be the surface area of this so that's what a D Sigma is now we just figure it out we just said well that D Sigma can be represented that value that area of that little part of the surface of that parallelogram can be represented as a cross product of those two vectors so let me write here this is all it's not rigorous mathematics the whole point here is to give you the intuition of what a surface integral is all about so we can write we can write that D Sigma is equal to the cross product the orange vector and the white vector the orange vector is this but we could also write it like this this was the result from the last video I'll write it in orange so the partial the partial of R with respect to running out of space with respect to s D s and it's going well this thing was going to be the the magnitude of the cross product the cross product the profs product by itself will just give you a vector and that's going to be useful when we start doing vector valued surface integrals but just think about it this way so this orange vector is the same thing as that and we're gonna take the cross product of that with this white vector this white vector is the same thing as that which we saw which is the same thing as this the partial of R with respect to T DT the partial of R with respect to T DT and we saw if we take the magnitude of that that's going to be equal to that's going to be equal to our little small change in area or the area of this little parallelogram over here now you may or may not remember that if you take these so let's just be clear this and this these are vectors right when you take the partial derivative of a vector-valued function you're still getting a vector this d s this is a number that's a number and that's a number and you might remember when we in the linear algebra or whenever you first saw taking cross products taking the cross product of some scalar multiple you can take the scalars out so if we take this number and that number we essentially factor them out of the cross product this is going to be equal to the cross the magnitude of the cross product of the partial of R with respect to s crossed with crossed with the partial of R with respect to T and then all of that times these two guys over here times D s and DT so I wrote this here hey maybe our surface area if we were to take the sum of all of these little D Sigma's but there's no obvious way to evaluate that but we know that all of the D Sigma's they're the same thing as if you take all of the DS's and all of the DTS if you take all of the DS's all of the DT so this is a d s times the DT right a D s times the DT d s times the DT is right there if we multiply this times the partial derivatives of the of the cross or the cross product of the partial derivatives this is going to this times this is going to give us this area so if we summed up all of this times this or this times this if we sum them up over this entire region we will get all of the parallelograms in this region we will get the surface area so we can write I know this is all a little bit convoluted and if you need to kind of ponder a little bit surface integrals at least in my head are one of the hardest things to really visualize but it'll all hopefully make sense so we can say that this thing right over here the sum of all of the little parallelograms on our surface or the surface area is going to be equal to instead of taking the sum over the surface let's take the sum of all the DS times DTS over this region right here and of course we're also gonna have to take this cross product in here and we know how to do that that's a double integral so we're gonna take the double integral over this we could call it this region or this area right here that area is the same thing as that whole area right over there of this thing I'll just write it in I'll just write it in yellow of the cross product of the partial of R with respect to s and the partial of R with respect to T DS and DT and so you literally just you know take and it seems very convoluted now how you're going to actually evaluate it but we were able to express this thing called a surface or well this is a very simple surface integral in something that we can actually calculate and in the next few videos where I'm going to show you examples of actually calculating it now this right here will only give you the surface area but what if at every point here so over here what we've done in both of these expressions is we're just figuring out the surface area of each of these parallelograms and then adding them all up that's what we're doing but what if associated with each of those little parallelograms we had some value where that value is defined by some third function by some third function f of X Y Z so every parallelogram it's super small it's around a point it's you could kind of say it's maybe the center of it it doesn't have to be the center and maybe the center of it is some point in three-dimensional space and if you use some other function f f of XY and Z you'll get the value of that point and we want to do is figure out what happens what happens if for every one of those parallelograms we were to multiply it times the value of the function at that point so we could write it this way so this is where you can imagine the function is just one we're just multiplying each of the parallelograms by one but we could imagine we're multiplying each of the little parallelograms by f of x y and z d sigma and it's going to be the exact same thing where this is each of the little parallelogram so we're just going to multiply it by f of XY and z there so we're going to integrate it over the area over that region of f of x y and z and then times the magnitude of the partial of R with respect to s crossed with the partial of R with respect to T D s DT and of course we're integrating with respect to s and T hopefully we can express this function in terms of s and T and we should be able to because we have a parametrizations air wherever we see an X there it's really X as a function of s and T Y is a function of s and T Z is a function of s and T and this might look super convoluted and hard and and the visualizations for this of why you'd want to do this it has applications in physics it's a little hard to visualize it's easier just to visualize this straight up the straight up surface area what we're gonna see in the next few videos that it's a little hairy to calculate these problems but they're not too hard to do that you just kind of have to stick with them