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Current time:0:00Total duration:8:34

Video transcript

so let's start thinking about partial derivatives of vector fields so a vector field is a function I'll do I'll just do a two-dimensional example here there's going to be something that has a two-dimensional input and then the output has the same number of dimensions that's the important part and each of these components in the output is going to depend somehow on the input variables so the example have in mind will be x times y is that first component and then y squared minus x squared as that second component and you can compute the partial derivative of a guy like this right you'll take the partial derivative with respect to one of the input variables I'll choose X it's always a nice one to start with partial derivative with respect to X and if we were to actually compute it in this case it's another function of x and y what you do is you take the partial derivative component-wise so you go to each component and the first one you say okay X looks like a variable Y looks like a constant the derivative will just be that constant and then the partial derivative of this second component that Y squared looks like a constant derivative of negative x squared with respect to X negative 2x so analytically if you know how to take a partial derivative you already know how to take a partial derivative of vector valued functions and hence vector fields but the fun part that the important part here how do you actually interpret this and this has everything to do with visualizing it in some way so the the vector field the reason we call it a vector field is you kind of take the whole XY plane and you're gonna fill it with vectors and concretely what I mean by that you'll take a given input what's an input you want to look at like um I'll say maybe 1 2 or yeah let's do that let's do 1 2 now which would mean you kind of go x equals 1 and then y equals 2 this input point and we want to associate that with the output vector in some way and so let's just compute what it should equal so when we plug in x equals 1 and y equals 2x times y becomes 2 y squared minus x squared becomes 2 squared minus 1 squared so 4 minus 1 is 3 so we have this vector 2 3 that we want to associate with that input point and vector fields you just attach the two points you just I'm going to take the vector 2 three and attach it to this guy so we should have an X component of two and then a Y component of three so it's going to end up looking something like let's see so a lot of component of three something like this so that'll be the vector and we attach it to that point and in principle you do this to all of the different points and if you did what you'd get would be something like this and remember when we represent these especially with computers it tends to lie where each represented vector is much much shorter than it should be in reality but you just want to squish them all on to the same page so they don't overrun each other and here color is supposed to give a general vague sense of relative length so ones that are blue should be thought of as much shorter than the ones that are yellow but that doesn't really give like a specific thought for how long they should be but for partial derivatives we actually care a lot about the specifics and if you think if you think back to how we interpret partial derivatives in a lot of other contexts what we want to do is imagine this partial X here as a slight nudge in the X Direction right so this this was our original input and you might imagine just nudging it a little bit and the size of that nudge as a number would be your partial X so then the question is what's the resulting change to the output and because the output is a vector the change in the output is also going to be a vector it's what we want is to say there's gonna be some other vector attached to this point right it's gonna look very similar maybe it looks like maybe it looks something like this so something similar but maybe a little bit different and you you want to take that that difference in vector form and I'll describe what I mean by that in just a moment and then divide it by the size of that original nudg so to be to be much more specific about what I mean here if you're comparing two different vectors and they're rooted in two different spots I think a good way to start is to just move them to a new space where they're rooted in the same spot so in this case I'm just gonna kind of draw a separate space over here and be thinking of this as a place for these vectors to live and I'm gonna I'm gonna put them both on this plane but I'm gonna route them each in the origin so this first one that has components 2 3 let's give it a name right let's call this guy v1 ok so that'll be V 1 and then the nudged output the the second one I'll call v2 vitu and let's say v2 is also in the space and I might exaggerate the difference just so that we can see it here let's say let's say it was different in some way but in reality if it's a small nudge it'll be different in only a very small way now but let's say let's say these were our two vectors the difference the difference between these guys is going to be a vector that connects the tips and I'm going to call that guy like partial V and the way that you can be thinking about this is to say that V 1 V 1 that original guy Plus that that tiny nudge the difference between them is equal to V 2 you know the nudged output and in terms of tip to tail with vectors either you're saying that kind of the green vector plus that blue vector is the same as that that pink vector that connects the the tail of the original one to the tip of the new one so when we're thinking of a partial derivative you're basically saying hey what happens if we take this the nudge the size of the nudge of the output and then we divide it by that nudge of the input so let's say you were thinking of that original not just being I don't know like of a size 1/2 like 0.5 as the the change in the X Direction then that would mean when you go over here and you say what's that DV that changed changing vector V divided by DX you'd be dividing it by 0.5 and in principle you'd be thinking of that would mean that you're kind of scaling this by 2 as if to say this little DV is 1/2 of of some other vector and that other vector is what the partial derivative is so this other vector here the full blue guy would be DV you know scaled down or scaled up however you want to think about it by that partial X and that's what makes it such that you know in principle if this partial X change was really small was like 1/100 and the output and I'd also was really small as like 1/100 or you know something on that order it wouldn't be specifically that then the DV DX that change would still be a normal sized vector and the direction that it points is still kind of an indication of the direction that this green vector should change as you're as you're scooting over so just to be concrete and you know actually compute this guy let's say we were to take this partial derivative partial of V with respect to X and evaluate it at that point one two that we're just dealing with one two what that would mean Y is equal to two so that first component is 2 and then X is equal to 1 so that next one should be negative 2 and we can I guess we can see just how wrong my drawing was to start here I was just kind of guessing what the pink vector would be but I guess it changes in the direction of 2 negative 2 so that should be something you're all erase the what turns out to be the wrong direction here yeah but to do get rid of this guy and I guess the change should be in the direction kind of 2 as the X component and then negative 2 that's a negative 2 as the Y component so the derivative vector should look something like this which means all corresponding little DV nudges will be slight changes will be slight changes on that so these will be your your DV is something in that direction and what that means in our vector field then as you move in the X direction and consider the various vectors attached to each point as you kind of passing through the point 1 2 the way that the vectors are changing should somebody somehow you know down into the right the tip should move down into the right so if this starts you know highly up and to the right then it should be getting kind of shorter but then longer to the right so the v2 if I were to have drawn it more accurately here you know what that nudged output should look like it would really be something that's kind of like I don't know like this where it's getting shorter in the Y direction but then longer in the X direction as per that blue nudging arrow and in the next video I'll kind of go go through more examples of how you might think of this how do you think of it in terms of what each component means which becomes very important for later topics like divergence and curl and I'll see you next video