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Partial derivatives of vector fields

How do you intepret the partial derivatives of the function which defines a vector field? Created by Grant Sanderson.

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  • male robot hal style avatar for user Sean
    At , since v1+dv=v2, shouldn't the tip of the second pink vector touch the tip of the differential blue vector?
    (5 votes)
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    • mr pink red style avatar for user Burt Karkarot
      If they were in the vector space, yes. If the initial points of v1 and v2 were the same and dv started at the tip of v1, the tips of dv and v2 would be touching. In the xy plane, however, the initial points of the vectors are shifted to the xy coordinates where they are evaluated - v1 and v2 start at different points.
      (7 votes)
  • blobby green style avatar for user Janez Novak
    I just cannot get it what is the difference between a vector field, which is in this video, and the previous example where we had a parametrically defined plane?
    (4 votes)
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    • leaf yellow style avatar for user Heiz
      "Vector field" is just a representation for a function, just like a regular map represents Earth. We can represent the Earth with different kinds of maps: spherical - it's the most accurate (because of how the Earth is shaped) but it's less practical to inspect because you need to rotate it and manufacturing it costs more. Same for multivariable functions. If you have n inputs (variables) and m outputs as a vector:
      f(a_1, a_2, ... a_n) = (b_1, b_2, ... b_m)
      then you need (n + m) dimensional space in order to represent it accurately. If we have here n=2 and m=2 then we need 4 dimensional space to imagine how the function looks which obviously is already "impossible" as we can comprehend up to 3 dimensions. For the vector field representation we only need 2 dimensions (half less!). In general the optimal representation for a multivariable function can be categorized as follows:
      n > m (more inputs) -> regular graph
      n = m -> vector field
      m > n -> (more outputs) -> parametric (curve or surface...)
      (6 votes)
  • blobby green style avatar for user guilhem.escudero
    The mathematical formalism looks like the same as a parametric surface, so every time we do this in "real life" we need to precise "it is a vector field" or "it is a parametric surface", is not it?
    (3 votes)
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  • blobby green style avatar for user Jasmine Vulgaris
    Why do the dimensions of the input and output space need to match? Couldn't we have a vector field that associates an input in R^3 with vectors that only point along the xy-plane f(x,y,z)=[f1(x,y,z),f2(x,y,z)]?
    (2 votes)
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    • blobby green style avatar for user ahs87904142
      You are asking about a vector field in a 2D plane based upon 3D inputs. A R^2 vector will always be f=<a,b>, where a is the change along one axis and b is the change along another axis. Is it possible then that you could have some vector f=<a(x,y,z), b(x,y,z)>?

      I imagine there's nothing saying you can't... but graphically it would be difficult to interpret. If thinking of a function as a transformation, we are essentially bending and compressing every point in a 3 dimensional space into a 2 dimensional plane. So... you could map a vector field onto the input space, but what would a 2D vector f(a,b) represent in a 3D space? You'd have to set (a,b) in the (x,y) plane and have the vectors graphically drawn as <a,b,0> (or <0,a,b> or <a,0,b> et al...), but then that graphically would imply a and b as changes in their respective axes.

      If you chose to plot the vectors on a 2D plot, a vector with 3D inputs and a 2D output also may not have every point on a 2D plot in its domain, which means the vector field is not zero but undefined.
      (1 vote)
  • blobby green style avatar for user Ereje
    Why this is different to a transformation presented in parametric surface from previous videos? 'I think they are similar':
    we have two inputs (x,y) and we get two outputs (P,Q) is also a relocation of point from a plane to another different plane.
    (1 vote)
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Video transcript

- [Voiceover] So let's start thinking about partial derivatives of vector fields. So a vector field is a function. I'll just do a two dimensional example here. It's gonna 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 gonna depend somehow on the input variables. So the example I have in mind will be X times Y as 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, it's a function of X and Y. What you do is you take the partial derivative component wise. So you to each component in 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 the second component. That Y squared looks like a constant. Derivative of negative X squared with respect to X. Negative two X. 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, the important part here. How do you actually interpret this? And this has everything to do with visualizing it in some way. So the vector field, the reason we call it a vector field, is you kind of take the whole X Y 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 wanna look at? I'll say, maybe one, two. Yeah, let's do that. Let's do one, two. Which would mean you kinda go X equals one. And then Y equals two This input point. Then 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 one, and Y equals two, X times Y becomes two. Y squared minus X squared becomes two squared minus one squared, so four minus one is three. So we have this vector two, three that we want to associate with that input point. And vector fields, you just attach the two points. I'm gonna take the vector two, 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 Y component of three, something like this. So that'll be the vector and we attach it to that point. And in principal, you do this to all 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 want to squish them all onto the same page so they don't over run each other. And here, color is supposed to give a general, vague sense of relative length. So ones that are blue should thought of as much shorter than the ones that are yellow, but that doesn't really give 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 back to how we interpret partial derivatives in a lot of other contexts, what want to do is imagine this partial X here as a slight nudge in the X direction, right. So this was our original input, so 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 to the output is also going to be a vector. So what we want is to say there's going to be some other vector attached to this point, right? It's going to look very similar. Maybe it looks like, maybe it looks something like this. So something similar, but maybe a little bit different. And you want to take that difference in vector form. And I'll describe what I mean by that in just a moment, and then divide by the size of that original nudge. So 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 gonna kinda just draw a separate space over here. And be thinking of this as a place for these vectors to live. And I'm gonna them both on this plane, but I'm gonna root them each in the origin. So this first one that has components two, three, now let's give it a name, right. Let's call this guy V one, okay. So that'll be V one. And then the nudged output, the second one, I'll call V two. And let's say V two is also in this space, and I might exaggerate the difference, just so we can see it here. Let's say it was different in some way. In reality, if it's a small nudge, it'll be different in only a very small way. But let's say these were our two vectors. The difference between these guys is going to be a vector that connects the tips. And I'm gonna call that guy Partial V. And the way you can be thinking about this is to say that V one, V one, that original guy, plus, that tiny nudge, the difference between them is equal to the two, you know, the nudged output. And in terms of tip to tail with vectors, you've seen that. Kind of the green vector plus that blue vector is the same as that pink vector that connects 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 the nudge of the input?" So let's thinking of that original nudge as being, I don't know, of a size of one half, like zero point five. As the change in the X direction. Then that would mean, when you go over here and say what's that DV? That changing vector V divided by the X, you'd be dividing it by zero point five, and in principle, you'd be thinking of, that would mean that your kind of scaling this by two. As if to say, this little DV is one half 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 the, you know, in principle if this partial X change was really small, like one 100th and the output nudge was also really small, it's like one 100th, or you know something on that order, it wouldn't be specifically that. Then the DVDX, 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 scooting over. So just be concrete and actually compute this guy, let's say we were to take this partial derivative partial 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 two and then X is equal to one, so that next should be negative two. And then 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 two, negative two. So that should be something, here I'll erase the, what turns out to be, the wrong direction here. Get rid of this guy. And I guess the change would be in the direction kind of two as the X component, and then negative two, that's a negative two, 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 DVs. 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 one, two, the way that the vectors are changing, should be somehow down and to the right. The tip should move down and to the right. So if this starts highly up and to the right, then it should be getting kind of shorter, but then longer to the right. So then the V two, if I were to have drawn it more accurately here, you know, what they nudged output should look like, it would really be something that kind of like I don't know, like this, where it's getting shorter in the Y direction, but then longer in X direction, as per that blue nudging arrow. And then in the next video I'll kind of go through more examples of how you might think of this. How 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.