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Main content
Current time:0:00Total duration:5:10

Video transcript

so in the last video I talked about vector fields in the context of two dimensions and here I'd like to do the same but for three dimensions so a three dimensional vector field is given by a function a certain multivariable function it has a three dimensional input given with coordinates XY and Z and then a three dimensional vector output that has expressions that are somehow dependent on XY and Z I'll just put dots in here for now but we'll fill this in with an example in just a moment and the way that this works just like with the two dimensional vector field you're going to choose a sample of various points in three-dimensional space and for each one of those points you consider what the output of the function is and that's going to be some three dimensional vector and you draw that vector off of the point itself so to start off let's take a very simple example one where the vector that outputs is actually just a constant so in this case I'll make that constant the vector 1 0 0 so what this vector is is just got a unit length in the X direction so this is the x axis so all of the vectors are going to end up looking something like this where it's a vector that has length 1 in the X direction and when we do this at every possible point well not every possible point but a sample of a whole bunch of points whoops we get a vector field that looks like this and then at any given point in space you get one of these little blue vectors and all of them are the same they're just copies of each other each pointing with unit length in the X direction so as vector fields go this is relatively boring but we can make it a little bit more exciting if we make the input start to depend somehow on the actual input so what I'll do is start I'll just make the input y0 0 so they're still just going to point in the x direction but now it's going to depend on the Y value so let's think for a second before I change change the image what that's going to mean the y-axis is this one here so now the z axis is pointing straight in our face that's the Y so as Y increases value 2 you know 1 2 3 the length of these vectors are going to increase it's going to be a stronger vector in the X direction a very strongly dr. India direction and if Y is negative these vectors are going to point in the opposite direction so let's see what that looks like there we go so in this in this vector field color and length are used to indicate how the magnitude of the vector so red vectors are very long blue vectors are pretty short and at zero we don't even see any because those are vectors with zero lengths and just like with two-dimensional vector fields when you draw them you lie a little bit this one should have a length of one right because when Y is equal to one this should have a unit length but it's made really really small and this one up here where Y is you know five or six should be a really long vector but we're lying a little bit because if we actually drew them to scale it would really clutter up the image so a couple things to notice about this one since the output doesn't depend on X or Z if you move in the X direction which is back and forth here the vectors don't change and if you move in the Z direction which is up-and-down the vectors also don't change they only change as you move in the Y direction okay so this is we're starting to get a feel for how the output can depend on the input now let's do something a little bit different let's say that all three of the components of the input depend on x y&z but I'm just going to make it kind of an identity function at a given point XYZ you output the vector itself XYZ so let's think about what this would actually mean and let's say you've got a given point some point floating off in space what is the output vector for that well the point has a certain X component a certain Y component and a Z component and the vector that corresponds to XYZ is going to be the one from the origin to that point itself let me just draw that here from the origin to the point itself and because of how we do vector fields you move this so that instead of stemming from the origin it actually stems from the point but what the main thing to take away here is it's going to point directly away from the origin and the farther away the point is the longer this vector will be so with that let's take a look at the vector field itself we go so again you kind of lie when you draw these like the vectors these red guys that are out at the end they should be really long because this vector should be as long as that point is away from the origin but to give a cleaner vector field you scale things down and notice the blue ones that are close to the center here are actually really really short guys and all of these are pointing directly away from the origin and this is one of those vector fields that is actually pretty a good a good one to have a strong intuition of because it comes up now and then thinking about what the identity function looks like as a vector field itself in the next video I'll talk through another example that's a little bit more complicated than this and can hopefully give an even stronger feel for how the output can depend on x y&z