The Surface Area formula for Parametric Surfaces // Vector Calculus

In this video we derive the formula to compute surface area given some surface described parametrically. Thus if you have a parametric description, all you need to do is plug it into this formula. The derivation works by looking at a tiny section of surface area, and approximating this with a little tangential parallelogram whose area can be computed by the length of the cross product of r_u Delta u and r_v Delta b, the partial derivatives of the position vector with respect to the two parameters u and v. Thus the integral is effectively just summing up these little surface areas and becomes a double integral of the length of that cross product. We will see a concrete example of this in the next video in the vector calculus playlit.

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I was not understanding anything in college lectures this playlist has made me comfortable in vector calculus. Your 10-minute videos are better than an hour of lectures.

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[bendavis]

Wow, I got that 'aha' moment when you introduced the connection between cross products and areas that I wasn't thinking about for some reason. These equations seem arbitrary at first, but inevitable at the end. Thanks for your helpful videos

[nolapickering]

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[j.o.]

I'm assuming this makes sense for a curved area, since r_u X r_v gives direction, while du*dv gives magnitude. The direction would change as the area curves and we're fine. Question to self: what does it mean when the figure is in 2D, but lives in 3D? My guess is that you can have a flat sheet, like a piece of paper, that's curved in some way or another. The thing itself is 2D, yet it exists in 3D. Because it is 2D, we can find a parameterization that uses u and v (and not a third variable), while it still applies to 3d space since u and v are variables in f, g and h. I'm guessing that if we're doing this for something with a volume (like a stack of paper), we'd have to do computations in the 4th dimension with u, v and w.

[briandwi]

What an amazingly brilliant idea. Fantastic.

[sriraghavt]

Superb, nice refresher. One particular question that has always bothered me is how does the approximation of transformed infinitesimal surface to a parallelogram make sense. Let’s say the surface is contorted protein, will it’s surface infinitesimal also be a parallelogram?

[PriyanshuSingh-uodr]

okay so I have a doubt, since at 5:11 you are multiplying r_u and r_v with a scaling factor, but for scaling, shouldn't the vector be a unit vector? and so the cross product will also be unit vector and the magnitude will be 1 it's just doesn't make sense

[filomenamendes]

Thank you. This was incredibly helpful.

[moon]

Thank you for this video!
I have a question about the part in 4:25. I always thought that the derivative with respect to the parameter pointed in the direction of the length with the respective parameter. Why is the partial of "u" in the direction of constant "v" and vice versa?