Reputation
613
Top tag
Next privilege 1,000 Rep.
Create new tags
Badges
6 17
Impact
~15k people reached

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.
Dec
20
asked Normal vector for a surface: explicit vs implicit formula
Dec
12
awarded  Popular Question
Dec
11
accepted Discrepancy in counting the number of poles in complex function when refactoring