hesson
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 Nov 5 awarded Popular Question Oct 28 awarded Yearling Oct 27 awarded Notable Question Feb 24 awarded Notable Question Oct 28 awarded Yearling Sep 24 awarded Autobiographer Jul 2 awarded Curious May 11 awarded Popular Question Apr 27 revised Is every integrable function on the real line with compact support also square integrable? improved title format Apr 27 suggested approved edit on Is every integrable function on the real line with compact support also square integrable? Mar 28 awarded Notable Question Feb 25 awarded Popular Question Feb 9 awarded Citizen Patrol Dec 21 accepted Normal vector for a surface: explicit vs implicit formula Dec 21 comment Normal vector for a surface: explicit vs implicit formula Thanks, that makes sense, I thought the normal they used was weird because the z-component was always 1, and that's not true for a hemisphere, but if you warp it into a planar disk then it makes sense. I looked at the way they derived the formula and I understand it now because they changed the area element - thanks again! Dec 20 comment Normal vector for a surface: explicit vs implicit formula I posted the original problem - they get $8\pi$ using the line integral, and then they get the same answer using the surface integral. It's still beyond me why it works for the normal vector that they used if it's not a unit vector. For the surface integral, the page is cut off, but they eliminate the term with the $sin$ function and the integrand becomes $2rdrd\theta$ - the integration also gives $8\pi$. Using the other normal vector, I get the dot product to be $y + z$ and I get the integral to be $16\pi / 3$ (wolfr.am/1bUdi6R). Dec 20 revised Normal vector for a surface: explicit vs implicit formula added 79 characters in body Dec 20 comment Normal vector for a surface: explicit vs implicit formula I posted a picture from the textbook so you can see how its used. The result that the textbook gets is $8\pi$ which agrees with the result from the line integral which can be calculated easily. Using the other form of the normal vector seems to yield a different result for me for the surface integral. Dec 20 revised Normal vector for a surface: explicit vs implicit formula added 75 characters in body Dec 20 comment Normal vector for a surface: explicit vs implicit formula But as I understand Stokes' theorem which relates a surface integral to the line integral of its boundary, the normal vector of the surface to integrate must be a unit vector, but the textbook used the normal vector derived from the explicit formula which is not a unit vector. So is the textbook wrong? When I use the other normal vector to do the surface integral, I'm off by a factor from the textbook's answer.