5 ms·

Does this relationship generalize? Hypothetically carrying further, if you take the derivative of the area (wrt radius) you get 2pi. A one dimensional circle i

Does this relationship generalize? Hypothetically carrying further, if you take the derivative of the area (wrt radius) you get 2pi. A one dimensional circle is a line segment of length 2*r right? So how is 2 pi related to a line segment? Is it in some way analogous to the way perimeter and area are for 2d or surface area and volume? I don't know the answer.

war1025 8y ago
Well in the sphere and circle cases, it works because if you are expanding size, it expands at the perimeter. In the line case, if your line is 2r, the derivative is 2. So expanding infinitesimally expands the line by 2 times that amount. So I guess it holds, but maybe isn't all that interesting.
azernik 8y ago
As the other commenter noted, the generalization to 1-dimensional hyperspheres is not super interesting. However, it generalizes perfectly to higher-dimensional hyperspheres.
baddox 8y ago
Seems to me that a one-dimensional circle would be two points on a number line separated by distance 2*r.
- iaw 8y ago
  Hopefully I can justify the reasoning behind this with the following: A circle is defined as the set of all points in a 2-dimensional space equidistant to a given center point (the distance being the circle's radius). A sphere is defined as the set of all points in a 3-dimensional space equidistant to a given center point. A line in 1-d does not analogize to this definition but a point does.
thaumasiotes 8y ago
> Does this relationship generalize? Yes, the fundamental theorem of calculus guarantees this will be true. It's the same phenomenon as that if your "volume" is the definite integral of f(t) from fixed a to variable x, then the rate of change of the volume (= the surface area) is f(x), the length of the infinitesimal sliver you're about to add to the volume. You can think of the solid sphere as an infinite stack of concentric infinitely thin hollow spheres. Each layer contributes volume to the solid sphere equal to its own area. (And this is what it means to calculate the volume of a sphere as the integral of the area of a spherical shell as the radius goes from 0 to r.)
- Someone 8y ago
  It also generalizes to any object that can be seen as an infinite stack of concentric infinitely thin objects of the same shape (that set includes all convex polyhedra) For example: - for squares, area is 4r², circumference is 8r - for cubes, volume is 8r³, area is 24r²
  - Someone 8y ago
    Correction: that rule isn’t as generic as I stated. It doesn’t apply to blocks of width 2r, breadth 2r, and height 4r, for example (volume is 16r³, area 40r²)
- amelius 8y ago
  > You can think of the solid sphere as an infinite stack of concentric infinitely thin hollow spheres. This is dangerous logic. For example a similar argument might lead you to think that a staircase in 2d between two points would have the same length as the Pythagorean distance between the two points, as you can make the steps arbitrarily small.
  - thaumasiotes 8y ago
    ...no? The same logic will tell you that you can think of a triangle as being a linear stack of infinitely thin line segments, and there, as here, you'd be completely correct. The logic you're calling "dangerous" is just the principle behind https://en.wikipedia.org/wiki/Shell_integration https://en.wikipedia.org/wiki/Shell_integration . But no similar argument will let you call the diagonal line an aggregation of little right-angled jags. A one-dimensional line is an aggregation of zero-dimensional points, not other one-dimensional lines.
  - srean 8y ago
    Its not dangerous as long one knows what one is doing. This is a poster boy of an example to illustrate the concept that if you have a sequence of functions f_i that becomes indistinguishable to the eye (being vague on purpose) from some function f, it does not mean that integral of that sequence will be close to the integral of f -- mathematical rigor does not matter except when it does
Dwolb 8y ago
Not sure the exact nuances that go into “generalizable” but please consider: To go from a radius to a circle, you can rotate a radius 360deg. To go from a circle to a sphere, you can rotate a circle on its axis 360deg. This sort of “take a shape and move it” to produce an N+1 dimensional shape is definable using calculus.

war1025 8y ago

Well in the sphere and circle cases, it works because if you are expanding size, it expands at the perimeter. In the line case, if your line is 2r, the derivative is 2. So expanding infinitesimally expands the line by 2 times that amount. So I guess it holds, but maybe isn't all that interesting.

azernik 8y ago

As the other commenter noted, the generalization to 1-dimensional hyperspheres is not super interesting. However, it generalizes perfectly to higher-dimensional hyperspheres.

baddox 8y ago

Seems to me that a one-dimensional circle would be two points on a number line separated by distance 2*r.

iaw 8y ago
Hopefully I can justify the reasoning behind this with the following: A circle is defined as the set of all points in a 2-dimensional space equidistant to a given center point (the distance being the circle's radius). A sphere is defined as the set of all points in a 3-dimensional space equidistant to a given center point. A line in 1-d does not analogize to this definition but a point does.

thaumasiotes 8y ago

> Does this relationship generalize? Yes, the fundamental theorem of calculus guarantees this will be true. It's the same phenomenon as that if your "volume" is the definite integral of f(t) from fixed a to variable x, then the rate of change of the volume (= the surface area) is f(x), the length of the infinitesimal sliver you're about to add to the volume. You can think of the solid sphere as an infinite stack of concentric infinitely thin hollow spheres. Each layer contributes volume to the solid sphere equal to its own area. (And this is what it means to calculate the volume of a sphere as the integral of the area of a spherical shell as the radius goes from 0 to r.)

Someone 8y ago
It also generalizes to any object that can be seen as an infinite stack of concentric infinitely thin objects of the same shape (that set includes all convex polyhedra) For example: - for squares, area is 4r², circumference is 8r - for cubes, volume is 8r³, area is 24r²
- Someone 8y ago
  Correction: that rule isn’t as generic as I stated. It doesn’t apply to blocks of width 2r, breadth 2r, and height 4r, for example (volume is 16r³, area 40r²)
amelius 8y ago
> You can think of the solid sphere as an infinite stack of concentric infinitely thin hollow spheres. This is dangerous logic. For example a similar argument might lead you to think that a staircase in 2d between two points would have the same length as the Pythagorean distance between the two points, as you can make the steps arbitrarily small.
- thaumasiotes 8y ago
  ...no? The same logic will tell you that you can think of a triangle as being a linear stack of infinitely thin line segments, and there, as here, you'd be completely correct. The logic you're calling "dangerous" is just the principle behind https://en.wikipedia.org/wiki/Shell_integration https://en.wikipedia.org/wiki/Shell_integration . But no similar argument will let you call the diagonal line an aggregation of little right-angled jags. A one-dimensional line is an aggregation of zero-dimensional points, not other one-dimensional lines.
- srean 8y ago
  Its not dangerous as long one knows what one is doing. This is a poster boy of an example to illustrate the concept that if you have a sequence of functions f_i that becomes indistinguishable to the eye (being vague on purpose) from some function f, it does not mean that integral of that sequence will be close to the integral of f -- mathematical rigor does not matter except when it does

Dwolb 8y ago

Not sure the exact nuances that go into “generalizable” but please consider: To go from a radius to a circle, you can rotate a radius 360deg. To go from a circle to a sphere, you can rotate a circle on its axis 360deg. This sort of “take a shape and move it” to produce an N+1 dimensional shape is definable using calculus.