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In the last video we figured out how to construct a unit normal vector to a surface and so now we can

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use that back in are original surface interval to try to simplify a little bit or at least give us a clue

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of how we can calculate these things and also think about different ways to represent this type of surface

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interval so if we just substitute what we came up with are normal vector are unit normal vector

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right here we will get we will get so once again its the surface interval of F . and F . all of this

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business right over here and im gonna write it all in white so it dosent take me too much time so the

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partial of r with respect to you crossed with the partial of r with respect to v over the magnitude

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of the same thing. Partial of r with respect to you crossed with the partial of r with respect to v.

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And now we've played with ds a lot we know that another way to write ds and I gave the intuition hopefully

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for that. In several videos ago when we first explored what a surface interval was all about we know

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that ds, we know that ds can be represented as the magnitude partial of r with respect to you crossed with the

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partial of r with respect to v du dv and obviously the du dv can be written as dv du it can be written

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as da a little chunk of area in the uv plain or in the uv domain. And actually since now this intervals

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in terms of uv were no longer taking a surface interval were taking a double interval over the uv domain.

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so you can say kind of a region in uv so this is ill say r and say thats a region in in the uv plain

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that were now thinking about. Their is probably a huge or their should be or im guessing their is a huge

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simplification that's popping out at you right now were dividing by the magnitude of the cross product of these two vectors

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and then were multiplying by the by the magnitude of the cross product of these two vectors these are

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the scalier quantities you divide by something and multiply by something well thats just the same thing

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as multiplying or dividing by 1. So these two characters cancel out and are intergrals simplifys two

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the double integral over that region the corresponding region of the uv plain. f of are vector field

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f dotted with this cross product this is gonna give us a vector right over here

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thats gonna give us a vector and it gives us

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actually a normal vector then we divide by its magnitude and it gives you a unit normal vector

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so this your gonna take the back product of f with

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r the partial of r with respect to you

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crossed with the partial of r with respect to v

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du dv du let me scroll over to the right a little bit

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du dv and we'll see in a few videos that this is how essentually

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how we go about calculating these things if you have a primarization

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you can get everything in terms of a double interval in terms of uv this way now

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the last thing I wanna do is explore another way taht you'll see a surface intergal like this written