Vector fields, introduction

hello everyone so in this video I'm gonna introduce vector fields now these are concepts that come up all the time in multivariable calculus and that's probably because they come up all the time in physics you know it comes up with fluid flow with electrodynamics you see them all over the place and what a vector field is is it's pretty much a way of visualizing functions that have the same number of dimensions in their input as in their output so here I'm going to write a function that's got a two dimensional input X and y and then this output is going to be a two dimensional vector and each of the components will somehow depend on x and y I'll make the first one Y cubed minus 9y and then the second component the Y component of the output will be X cubed minus 9x I made them symmetric here looking kind of similar they don't have to be I'm just kind of a sucker for symmetry so if you imagine trying to visualize a function like this with I don't know like a graph it would be really hard because you have two dimensions in the input two dimensions in the output so you'd have to somehow visualize this thing in four dimensions so instead what we do we look only in the input space so that means we look only in the XY plane so I'll draw these coordinate axes and just mark it up this here is our x-axis this here is our y-axis and for each individual input point like let's say 1/2 so let's say we go to 1/2 I'm going to consider the vector that it outputs and attach that vector to the point so let's let's walk through an example of what I mean by that so if we actually evaluate F at 1/2 X is equal to 1 Y is equal to 2 so we plug in 2 cubed whoops 2 cubed minus 9 times 2 up here in the X component and then 1 cubed minus 9 times y 9 times 1 excuse me down in the Y component 2 cubed is 8 9 times 2 is 18 so 8 minus 18 is negative 10 negative 10 and then 1 cubed is 1 9 times 1 is 9 so 1 minus 9 is negative 8 now first imagine that this was if we just drew this vector where we count starting from the origin negative 1 2 3 4 5 6 7 8 9 10 so he's gonna have this as its x-component and the negative 8 1 2 3 4 5 6 7 we're gonna actually go off the screen it's a very very large vector so it's gonna be something here and it ends up having to go off the screen but the nice thing about vectors it doesn't matter where they start so instead we can start it here and we still want it to have that negative 10 X component in the negative 8 so when I go to 1 2 3 4 5 6 7 8 negative 8 as its Y component there so this is a really big vector and a plan with the vector field is to do this at not just 1 2 but at a whole bunch of different points and see what vector is attached to them and if we drew them all according to their size this would be a real mess there'd be markings all over the place and you'd have you know this one might have some huge vector attached to it and this one would have some huge vector attached to it and it would get really really messy but instead what we do so I'm gonna just clear up the board here we scaled them down this is common you'll scare them down so that you're kind of lying about what the vectors themselves are but you get a much better feel for what each thing corresponds to and another thing about this drawing that's not entirely faithful to the original function that we have is that all of these vectors are the same length you know I made this one I'm just kind of the same unit this one the same unit and over here they all just have the same length even though in reality the length of the vectors output by this function can be wildly different this is kind of common practice when vector fields are drawn or when some kind of software is drawing them for you so there are ways of getting around them this one way is to just use colors with your vector so I'll switch over to a different vector field here and here color is used to kind of give a hint of length so it still looks organized because all of them have the same same length but the difference is red and warmer colors are supposed to indicate this is a very long vector somehow and then blue would indicate that it's very short another thing you can do is scale them to be roughly proportional to what they should be so notice all the blue vectors scaled way down to basically be zero red vectors kind of stayed the same size and even though in reality this might be representing a function where the true vector here should be really long or the true vector here should be got a medium length it's still common for people to just shrink them down so it's a reasonable thing to view so in the next video I'm gonna talk about fluid flow a context in which vector fields come up all the time and it's also a pretty good way to get a feel for a random vector field that you look at to understand what it's all about