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Current time:0:00Total duration:2:18

Video transcript

hello everyone its what I'd like to do here and in the following few videos is talk about how you take the partial derivative of vector valued functions so the kind of thing I have in mind there will be a function with a multiply variable input so this specific example have a two variable input T and s you could think of that as a two-dimensional space as the input are just two separate numbers and its output will be three dimensional the first component T squared minus s squared the Y component will be s times T and that Z component will be T times s squared minus s times t squared minus s times T squared and the way that you compute a partial derivative of a guy like this is actually relatively straightforward it's it's if you were to just guess what it might mean um you probably guess right it will look like partial of V with respect to one of its input variables and I'll choose T with respect to T and you just do a component wise which means you look at each component and you do with the partial derivative to that because each component is just a normal scalar valued function so you go up to the top one and you say T squared looks like a variable as far as T is concerned and it's derivative is 2t but s squared looks like a constant so it's derivative is zero s times T when s is s looks like a constant and when T looks like a variable has a derivative of s then T times s squared when T is the variable and s is the constant it just looks like that constant which is s squared minus s times T squared so now derivative of T squared is 2t and that constant s stays in so that's 2 times s times T and that's how you compute it probably relatively straightforward um the way you do it with respect to s is very similar but where this gets fun and where this gets cool is how you interpret the partial derivative right how you interpret this value that we just found and what that means depends a lot on how you actually visualize the function so what I'll go ahead and do in the next video and in the next few ones is talk about visualizing this function it'll be as a parametric surface in three-dimensional space that's why I've got my graph or program out here and I think you'll find there's actually a very satisfying understanding of what this value means