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Current time:0:00Total duration:6:21

so I have here a very complicated function it's got a two-dimensional input to different coordinates to its input and then a three-dimensional output specifically it's a three dimensional vector and each one of these some expression that's bunch of cosines and sines that depends on the two input coordinates and in the last video we talked about how to visualize functions that have a single input a single parameter like T and then a two-dimensional vector output so some kind of expression of T and another expression of T and this is sort of the three dimensional analog of that so what we're going to do we're just going to visualize things in the output space and we're going to try to think of all the possible points that could be outputs so for example let's just start off simple let's get a feel for this function by evaluating it at a simple simple pair of points so let's say we evaluate this function f at T equals zero I think will probably be pretty simple and then s is equal to pi so let's think about what this would be we go up when we say okay T of 0 cosine of 0 is 1 so this whole thing is going to be 1 same with this one and sine of 0 is 0 so this over here is going to be 0 and this is also going to be 0 now cosine of PI is negative 1 so this here is going to be negative 1 this one here is also going to be negative 1 and then sine of Pi just like sine of 0 is 0 so this whole thing actually ends up simplifying quite a bit so that the top is 3 times 1 plus negative 1 1 times negative 1 is negative 1 and we get 2 then we have 3 times 0 plus 0 so the Y component is just 0 and then the Z component is also 0 so what that would mean is that this output is going to be the point that's 2 along the x axis and there's nothing else to it it's just 2 along the x axis so go ahead and well remove the graph about at that point there so that's what would correspond to this one particular input zero and PI and you know you could do this with a whole bunch and you might add a couple other points based on other inputs that you find but this will take forever to start to get a feel of the function as a whole and another thing you can do is say okay maybe rather than thinking of evaluating at a particular point imagine one of the inputs was constant so let's imagine that s stayed constant at pi okay but then we let T range freely so that means we're going to have some kind of different different output here and we're going to let T just be some kind of variable while the output is PI so what that means is we keep all of these this negative one negative one and zero for what sine of pi is but the output now it's going to be three cosine of T cosine of T plus negative one times the cosine of T so this can be minus cosine of T the next part it's going to still be three sine of T this is no longer zero I should probably erase let's go those guys actually so we're no longer evaluating I won t was zero so three times sine of T that's just still the function that we're dealing with three sine of T and then minus one times sine of T so minus sine of T keep drawing it in green just to be consistent and then the bottom stays at zero and this whole thing actually simplifies three cosine t minus cosine T that's just two cosine T and then same deal for the other one it's going to be two sine of T so this whole thing actually simplifies down to this so this is again when we're letting s stay constant and T ranges freely and when you do that what you're going to end up getting is a circle that you draw on you can maybe see why it's a circle because you have this cosine sine pattern it's a circle with radius two and it should make sense that it runs through that first point that we evaluated so that's what happens if you let just one of the variables run but now let's do the same thing but think instead of what happens is s varies and T stays constant I encourage you to work it out for yourself I'll go ahead and just kind of draw it because I kind of want to give the intuition here so in that case you're going to get a circle that looks like this so again I encourage you to try to think through for the same reasons imagine that you let s run freely keep T constant at zero why is it that you would get a circle that looks like this and in fact if you if you let both T and s run freely a very nice way to visualize that is to imagine that this circle which represents s running freely sweeps throughout space as you start to let T run freely and what you're going to end up getting when you do that is this shape that goes like this and this is a doughnut we we have a fancy word for this in mathematics we call it a torus but it turns out the function here is a fancy way of drawing the torus and in another video I'm going to go through in more detail if you're just given the tourist how you could find this function how you can kind of get the intuitive feel for that and in that it'll involve going through in a bit more detail why when you sweep the circle out it gets the torus just so and what the relationship between the thread circle and the blue circle is but here I just kind of want to give an intuition for what parametric surfaces are all about how it's a way of visualizing something that has a two a two dimensional input and a three dimensional output