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Studying for a test? Prepare with these 11 lessons on Green's, Stokes', and the divergence theorems.
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Video transcript
Now that we know a little bit about how to construct a unit normal vector at any point in a curve--that's what we did in the last video-- I wanna start exploring and interesting expression. So the expression is: The line integral around a closed loop, and we're gonna go in the positive counter clockwise orientation of a vector field F dotted with the unit normal vector at any point on that curve ds (I'll write ds in magenta) ds. So first lets conceptualize what this is even saying and then let's manipulate it a little bit to see if we can come up with and interesting conclusion, we actually will manipulate it we'll use green's theorem and we're actually gonna come up with a 2 dimensional version of the divergence theorem which all sounds very complicated but hopefully we can get a little bit of an intuition for it as to why it actually a little bit of common sense. So, first lets just think about this. So let me draw- let me draw a coordinate plane here so we do it in white So this right over here thats our y axis that over there is our x axis let me draw ourselves my curve so my curve might look something like--I'll do it in a blue color so my curve might look something like this- my contour and its going in the positive, counterclockwise direction just like that and now we have our vector field, and just a reminder we've seen this multiple times I can- my vector field will associate a vector with any point on the x-y plane and it will-- it can be defined as some function of x and y, which I'll call that P some function of x and y times the i unit vector so it says what the i component of the vector field is for any x and y point and then what the j component, or what we multiply the j component by or the vertical component by for any x and y point so some function of x and y times i plus some other scalar function of x and y times j and so if you give me any point there will be an associated vector with it any point there is an associated vector depending on how we define this function. But this expression right over here we're taking a line integral we care specifically about the points about along this curve along this contour right over here and so lets think about what this is actually this piece right over here is actually telling us before we sum up all of these infinitesimally small pieces so if we just take f dot n so lets just think about a point on this curve so a point on this curve may be this point right over here so associated with that point there is a vector that's what the vector field does So f might look something like that right over there so that might be f at that point and we're gonna dot it with the unit normal vector at that point so the unit normal vector might look something like that that would be n hat at that point this is the vector field at that point