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Current time:0:00Total duration:7:03

Video transcript

hello hello again so in the last video I started talking about how you interpret the partial derivative of a a parametric surface function right of a function that has a two variable input in a three variable vector valued output and we typically visualize those as a surface in three-dimensional space and the whole process I was saying you you think about how a portion of the TS plane moves to that corresponding output and again I'm kind of cheating with this animation where really this isn't the TS plane right this is this is on the XY plane TS plane should just be some separate space over here and we're imagining moving that separate space over into three dimensions but that's that's harder to animate so I'm gonna I'm just not going to do it and I'm gonna instead keep things inside the XY plane here and you know we're thinking about the squares being t and s ranging H from 0 to 3 and what I said for partial derivative with respect to T is you imagine the line that represents movement in the t direction and you see how that line gets mapped as all of the points move to their corresponding output and the the partial derivative vector gives you a certain tangent vector to the curve representing that that line which corresponds to movement in the t direction and the longer that is the the faster the movement the more sensitive it is to nudge is in the T direction so in the S Direction let's say we were to take the partial derivative with respect to s so I'm kind of clear this up here also clear if this guy and if you said instead what if we were doing it with respect to s right partial derivative of V the vector valued function with respect to s and well you do something very similar you would say ok what is the line that corresponds to movement in the s direction and the way I've drawn and it's always going to be perpendicular because we're in the TS plane the t axis is perpendicular to the s plane and in this case this line represents T equals 1 right you're saying T constantly equals 1 but you're letting s vary and if you see how that line maps as you move everything from the input space over to the corresponding points in the output space that line tells you what happens as you're varying the the asked value of the input and I guess it kind of starts curving this way and then it curves very much up and kind of goes off off into the distance there and again the grid lines here really help because every every time that you see the grid lines intersect one of the lines represents movement in the T direction and the other represents movement in the s direction and for partial derivatives we we think very similarly you think of that that partial s as representing zoom on back here that partial s you think of as representing a tiny movement in the s direction just a just a little smidgen nudge somehow nudging that guy along and then the corresponding dot do you look for in the output space you say okay if we nudge the input that much and we go over to the output and and you know maybe that that tiny nudge correspond with one that's like three times bigger I don't know but it looked like it stretched things out so that tiny nudge might turn into something that's still quite small but maybe three times bigger but it's a vector what you do is you you think of that vector as being your your partial V and you scale it by whatever the size of that partial s was right so the the result that you get is a tangent vector that's not not puny not a tiny nudge but is actually a sizable tangent vector and it's it's gonna kind of correspond to the rate at which tiny chain that changes it's not just tiny changes but the rate of which changes in s cause movement in the output space so let's actually compute it for this case just kind of get some good practice computing things and if we look up here the the T value which used to be considered a variable when we were doing it with respect to T but now that T value looks like a constant so it's derivative is zero then negative s squared with respect to s has a derivative of negative two s st s looks like a variable t looks like a constant the derivative is just that constant T down here T s squared T looks like a constant s looks like a variable so two times T times s and then over here we're subtracting off s as the variable t squared looks like a constant so that constant and let's say we we plug in the value one one this this red dot corresponds to one one so what we would get here s is equal to one so that's negative two T is equal to 1 so that's 1 then 2 times 1 times 1 or let's see I'll write it 2 times 1 times 1 minus 1 squared minus 1 squared is going to correspond to 1 that's 2 minus 1 so what we would expect for the tangent vector the partial derivative vector is the X component should be negative and then the y&z components ready to be positive and if we go over and we take a look at what the movement in the you know along the curve actually is that lines up right because you're moving as you kind of zip along this curve you're moving to the left so the the X X component of the partial derivative should be negative but you're moving upwards as far as Y is concerned and you can also kind of see that the leftward movement is kind of twice as fast as the upward motion the slope favors the X direction and then as far as the Z component is concerned you are in fact moving up and maybe you could say well how do you know what you're moving are you moving you know that way or is everything switched the other way around and the benefit of animation here is we can say as as s is ranging from 0 up to 3 you know this is the increasing direction and you just keep your eye on what that direction is as we move things about and that increasing direction does kind of correspond with moving along the curve this way so you get you get a tangent vector in the other way and one kind of nice thing about this then is the two different partial partial derivative vectors that we found each one of them you could say is a tangent vector to the surface right so the one that was a partial derivative with respect to T over here's kind of goes in one direction and the other one it kind of gives you a different notion of what a what a tangent vector on the surface could be and various different you could you could have a notion of directional derivative - that kind of combines these in various ways and that'll get you all the different ways that you can be have a vector tangent to the surface and later on I'll talk about things like tangent planes if you want to express what a tangent plane is and you kind of think of that as being defined in terms of two different vectors but for now that's that's really all you need to know about partial derivatives of of parametric surfaces and the next couple videos we'll talk about what you know partial derivatives of vector valued functions can mean another context because it's not always a parametric surface and maybe you're not always thinking about a curve that could be moved along but you still want to think you know how does this input nudge correspond to an output nudge and and what's the ratio between them so with that I'll see you next video