The length of the vectors' output by this functionĬan be wildly different. I made this one 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 And another thing about this drawing that's not entirely faithful to the original function that we have is that all of these 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. Gonna clear up the board here we scale them down, this is common you'll scale them down and Huge vector attached to it and it would get really really messy. Huge vector attached to it and this one would have some There'd be markings all over the place and this one might have some Here and we still want it to have that negative ten X component and the negative eight, negative one, two, three, four, five, six, seven, eight, negative eight as its Y component there and a plan with the vector field is to do this at not just one,two but at a whole bunch of different points and see what vectors attach to them and if we drew them allĪccording to their size this would be a real mess. But the nice thing about vectors it doesn't matter where they start so instead we can start it One, two, three, four, five, six, seven, we're gonnaĪctually go off the screen its a very very large vector so its gonna be something here and it ends up having This as its X component and then negative eight, Seven, eight, nine, 10, so its going to have Negative one, two, three, four, five, six, Now first imagine that this was if we just drew this vector where we count starting from the origin, Nine times one is nine so one minus nine is negative eight. Two cubed is eight nine times two is 18 so eight minus 18 is negative 10 negative 10 and then one cubed is one, So lets walk through anĮxample of what I mean by that so if we actually evaluate F at one,two X is equal to one Y is equal to two so we plug in two cubed whoops, two cubed minus nine times two up here in the X component and then one cubed minus nine times Y nine times one, excuse me down in the Y component. Vector that it outputs and attach that vector to the point. This here's our X axis this here's our Y axis and for each individual input point like lets say one,two so lets say we go to one,two I'm gonna consider the So I'll draw these coordinate axes and just mark it up, Visualize a function like this with a graph it would be really hard because you have twoĭimensions in the input two dimensions in the output so you'd have to somehow visualize this thing in four dimensions. Looking kind of similar they don't have to be, I'm The Y component of the output will be X cubed minus nine X. Y cubed minus nine Y and then the second component, So here I'm gonna write a function that's got a two dimensional input X and Y, and then its output is going to be a two dimensional vector and each of the components That have the same number of dimensions in their Is its pretty much a way of visualizing functions It comes up with fluidįlow, with electrodynamics, you see them all over the place. That come up all the time in multi variable calculus,Īnd that's probably because they come up all the time in physics. Everyone, so in this video I'm gonna introduce vector fields.
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