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Current time:0:00Total duration:9:29

Video transcript

so we've just computed a vector-valued partial derivative of a vector-valued function but the question is what does this mean what does this jumble of symbols actually mean in a you know more intuitive geometric setting and that has everything to do with how you visualize the function and with this specific function given that the input is two-dimensional but the output is three-dimensional and the output is more dimensions than the input it's nice to visualize it as a parametric surface and the way that I do that maybe I'm you could call this visualizing it as a transformation also because what I want to do is basically think of the TS plane think of the TS plane where all these input values live and kind of think of how that's going to map into three-dimensional space but when I do that I'm actually going to cheat a little bit rather than having a separate plane off there is the TS plane I'm going to kind of over right onto the XY plane itself and plop the TS plane down like this and this isn't the full TS plane this is actually supposed to represent just values of T that range from 0 up to 3 so each tick mark on the graph here corresponds with 1/2 so this is 1 that's 2 in then up here is 3 same with s s also ranges from 0 to 3 and the reason that I'm plopping it inside 3 dimensional space to start with kind of overwriting the XY plane with the TS plane is just to make the animating a little bit easier it's it's you could call a laziness but the benefit here is what we can do is watch each point and each one of these points you're thinking of is corresponding to some kind of TS pair an input point which is just a pair of numbers and we're going to watch each one of those points move to the corresponding output right the output is a 3 dimensional value 3 dimensional vector or point however you want to think about it and what that looks like when we animate this actually is you know each one of those points in our square of TS plane moves to the corresponding output and you end up with a certain surface and just to make it a little more concrete what's actually going on here let's focus in on just one point and we'll focus in on this point not just for the function visualization but for the partial derivative as well and the function or the the point there that I care about is going to be at one one so this point right here represents the pair of TS where each one of them is equal to one one one and you can start by predicting where you think this is going to get output and to do that you just plug it into the function this is kind of what the visualization means is we're plugging this in and for T and s we're going to plug in just one and one so that top part is going to look like 1 squared minus 1 squared which becomes 0 that middle part is going to be 1 times 1 which is 1 and then over here we're going to have 1 times 1 squared which is 1 minus 1 times 1 squared which is again 1 and you can probably see because of the symmetry there those also cancel out you get 0 which means the output corresponding with this input should be the vector 0 1 0 a vector that's of unit length pointing in the Y direction so if we look here you know this is the x axis this here is the Y axis so you would think it should be a vector that looks kind of like this unit vector in the Y direction and the point of the surface is what corresponds to the tip of that vector all right this is how we visualize parametric things you just think of the tip of the vector is kind of moving through space and and drawing out the thing in this case the thing it's drawing is a surface so if we watch that animation again and we let things play forward that dot corresponding with the input 1 1 does indeed land at the tip of that vector so at least for that value you can see that I'm not lying to you with the animation and in principle you could do that for every single point right if any given input point you kind of plug it through the function and you draw the vector in three-dimensional space as you watch this animation it'll land at the tip of that vector so now if we want to start thinking about what the partial derivative means remember this is this little DT this partial T is telling you to nudge it in the T direction so what is movement just not not even nudges but just movement in general look like in the T direction for our our little snippet of the TS plane here well the the T direction I'm saying is in this direction here where you know this this represents 1 2 3 of t-values and this line here represents the constant value for s so this will be s constantly equaling 1 which you can know because it's passing through the point 1 1 and then otherwise you're just letting T range freely and if we watch how this gets transformed under the you know under the transformation under the mapping to the parametric surface you can get a feel for what varying the input T does in the output space right so this whole pink line is basically telling you what happens if you let s constantly equal 1 but you let the variable t the input T very freely and you get a certain curve in three-dimensional space and you know if you had a different constant for s it would be another curve and maybe you can kind of see on the grid lines what shape those other curves would have and they're all in a sense you know parallel ish to this curve corresponding to s equals 1 so if instead of thinking about movement of T as a whole you start thinking about nudges this whole partial T is something where we're just imagining a tiny tiny little movement in the T direction not really that much just a tiny little move and you're it's like you're recording its value as partial T so maybe if you're being concrete you'd say partial T would be something like 0.01 and really it's going to be a limiting variable that gets smaller and smaller but I find it's kind of nice to think about an actual value like 1/100 and then if you if you let this whole thing undergo the transformation and we kind of watch watch the input point watch the line representing T that little nudge that little nudge is going to get may be stretched or squished and it's going to result in some kind of vector pointing along that curve and it'll be tangent to that curve right the the vector that tells you how you move just a tiny tiny little bit it'll be tangent in some way and that that vector that output nudge is what you're thinking of as your tiny change to the output vector that partial V and when you divide it by the tiny value right if your tiny value was 0.01 and you divide it by that it's going to become something bigger so the the actual derivative isn't going to be just some tiny little nudge that's you know hardly hardly visible but it can be that that nudge vector scaled appropriately in this case it would be divided by 1 hundred there x no 100 and it would be something that remains tangent to the curve but maybe it's pointing big and the larger it is right the the longer it is that's telling you that as you let t vary and you're kind of moving along this pink curve tiny nudges and T correspond with larger movements right the the ratio of the nudged sizes is bigger so if you are if you were to have a very long partial derivative vector that's still tangent but really goes out there that would tell you that as you vary T you're zipping along super quickly and if we just look at this you know this particular one that the zooming off of one you kind of get a feel for the curve around that point you say okay okay and that curve you're moving positively in the x-direction right you're moving to the right you're moving positively in the Z direction remove not Z sorry the Y positively in the y direction up there and the z direction is actually negative isn't it this this curve kind of goes down as far as Z is concerned so before even computing it if I were to tell you that I'm going to plug in the value 1 1/2 this partial derivative that we that we compute it in the last video you would say oh well just looking at the picture you can kind of tell the the x value is going to be something positive something greater than zero the Y value is also going to be something positive and again that's because you know the movement is to the right to positive x it's moving up so positive Y but the Z value should actually be a little bit negative right because as you look at this curve it's going down in a sense and with that being our prediction if you start plugging in 1 1 to T and s what you'll see is that you know 2 times 1 is 2 s equals 1 so that's just 1 and then over here this this looks like 1 squared minus 2 times 1 times 1 so this will be 1 minus 2 that's negative 1 so it is in fact that kind of positive positive negative pattern that you're seeing and maybe maybe even from this curve you can get a feel for why the movement in the X Direction is twice as much as the movement in the Y it's it's moving more to the right than it is up in the Y direction and again in principle you can imagine doing this not just at the point 1 1 but at any given point may be any given point along this curve or any given point along the surface and the corresponding movement the direction that you know nudge is in the T Direction thank you will give you some vector in 3 mentioning space and that's the interpretation that is the the meaning of the partial derivative of the vector valued function here and again it's it's not the tiny the actual nudge vector itself right when you know to the input and you get just a little smidgen in the output space here but it's that divided by the size of the nudge so that's why you'll get kind of normal size vectors rather than tiny vectors and in the next video I'll do kind of the same thing for what happens when you nudge in the s direction there's to get a better feel for what's going on in this example