Are Higher Dimensions Necessary or Just Imaginary Concepts?

[Mentallic]

Given the volume of a sphere V is given by $$V=\frac{4}{3}\pi r^3$$

Taking the derivative of the V wrt r gives $$\frac{dV}{dr}=4\pi r^2$$

which is the Surface Area of the sphere: $$\frac{dV}{dr}=A_s$$

Taking the derivative of A_s wrt r gives $$\frac{dA_s}{dr}=8\pi r$$
which I don't recognise as anything too significant.
It is not far off the area of a circle ($$2\pi r$$) but it's not quite.

My question is: Is there any physical significance for the rate of change of the surface area of a sphere? Since the rate of change of the volume of the sphere describes the 3-d surface area of the sphere. Then the rate of change of this should describe...? Maybe the 2-d surface area (the shape could be truncated in some odd way).

 

[n!kofeyn]

Look http://en.wikipedia.org/wiki/Sphere#Surface_area_of_a_sphere". It's a decent explanation. This also seems related to Stokes' theorem in my opinion. I know that using this, you can prove that the surface area of a unit (n-1)-sphere is n times the volume of a unit n-ball. An (n-1)-sphere is just a sphere that is in n dimensions. For example, the regular 3 dimensional sphere is a 2-sphere. They're called n-spheres because they are n dimensional manifolds, although they lie in n+1 Euclidean dimensions. A ball is just the sphere filled in. So a 3-ball is the regular 3 dimensional sphere filled in and a 2-sphere is the shell of this 3-ball. Unit just means they have radius 1. I'm sure there is a similar result for non-unit spheres involving the radius term.

 

[mathman]

Taking the derivative of the V wrt r gives

which is the Surface Area of the sphere:

Taking the derivative of As wrt r gives
which I don't recognise as anything too significant.
It is not far off the area of a circle () but it's not quite.

The surface area of the sphere is 4 times area of a circle, while the next derivative is the circumference of the circle (not area).

 

[Mentallic]

The surface area of the sphere is 4 times area of a circle, while the next derivative is the circumference of the circle (not area).

My mistake. Yes I meant to say circumference (of course, the 2nd derivative truncates the volume into 1-D)

...For example, the regular 3 dimensional sphere is a 2-sphere. They're called n-spheres because they are n dimensional manifolds, although they lie in n+1 Euclidean dimensions. ...

I find it hard to understand the importance of this and couldn't relate it to my question. Are you trying to tell me the derivative of the surface area of the 3-sphere (?) gives the circumference of a hypothetical 2-sphere? Most likely I completely missed the point.

 

[boboYO]

have a look at that wiki article (under 'Surface are of a sphere'), I've had the same question and i think it gives a nice intuitive explanation.

" It is also the derivative of the formula for the volume with respect to r because the total volume of a sphere of radius r can be thought of as the summation of the volumes of an infinite number of spherical shells of infinitesimal thickness concentrically stacked on top of one another from radius 0 to radius r. At infinitesimal thickness the discrepancy between the inner and outer surface area of any given shell is infinitesimal and the elemental volume at radius r is simply the product of the surface area at radius r and the infinitesimal thickness."

 

[Mentallic]

I have read that article and all I got out of it was a repeat of the idea I learned school of finding the surface area intuitively from first principles using the volume of the sphere. It's integration all over again.

I'm still unsure what the derivative of the surface area of a sphere wrt r is supposed to represent. Just like the derivative of the volume gives the surface area, the derivative of that gives...? Even if the mathematics is describing some impossible circle/sphere of 2/3 dimensions (?) I'm fine with that. Hypothetical ideas in maths haven't stopped me just yet

 

[n!kofeyn]

have a look at that wiki article (under 'Surface are of a sphere'), I've had the same question and i think it gives a nice intuitive explanation.

That's the article I had already linked to above.

I find it hard to understand the importance of this and couldn't relate it to my question. Are you trying to tell me the derivative of the surface area of the 3-sphere (?) gives the circumference of a hypothetical 2-sphere? Most likely I completely missed the point.

Well the part you are quoting is where I was trying to explain the definitions of terms that I used. The point was is that there is a general relation between the surface area and volume of n-dimensional spheres and balls. The significance was that they are related by using Stokes' theorem. I'm not for sure how much math you've had, but maybe look it up and see if you get anything from it. A 2-sphere is not hypothetical. As I said above it is just the regular sphere in 3-dimensions (which is the amount of dimensions we live in).

Even if the mathematics is describing some impossible circle/sphere of 2/3 dimensions (?) I'm fine with that. Hypothetical ideas in maths haven't stopped me just yet

Why is a circle or sphere of 2 or 3 dimensions some hypthothetical idea or impossible? Even the n-dimensional analogues are not hypothetical. They definitely exist, but we just can't visualize them.

 

[Mentallic]

That's the article I had already linked to above.

and I don't see any direct relation between that article and my question. It only mentions the relationship between the volume of the sphere and the surface area of the sphere. Nothing about the rate of change of the sphere which is what I'm trying to comprehend.

Well the part you are quoting is where I was trying to explain the definitions of terms that I used. The point was is that there is a general relation between the surface area and volume of n-dimensional spheres and balls. The significance was that they are related by using Stokes' theorem. I'm not for sure how much math you've had, but maybe look it up and see if you get anything from it.

I looked up stokes' theorem on wiki and sadly enough I haven't been exposed to such math yet. Maybe we need a summary for my feable brain to munch on?

A 2-sphere is not hypothetical. As I said above it is just the regular sphere in 3-dimensions (which is the amount of dimensions we live in).

If a 2-sphere is a regular 3-sphere (which btw, when you first said it, it confused me so I shrugged it off) then what is a 3-sphere? A 4-sphere? (i.e. some sphere we can't visualize)

Why is a circle or sphere of 2 or 3 dimensions some hypthothetical idea or impossible? Even the n-dimensional analogues are not hypothetical. They definitely exist, but we just can't visualize them.

I never believed in higher dimensions I guess. I understand how the math can work for them, by following the rules between 1,2 and 3-d objects and extending it further. Like, a 4-d cube would have 16 corners. That's fine and dandy, but I'm not going to consider an realistic object just because the math says it exists. And the opposite to realistic is imaginary so that's how I understand it.
But this is just my view on the subject. I could always be wrong (doubt it *cough*)

 

[n!kofeyn]

and I don't see any direct relation between that article and my question. It only mentions the relationship between the volume of the sphere and the surface area of the sphere. Nothing about the rate of change of the sphere which is what I'm trying to comprehend.

Oops, sorry. I misread your original question as wanting a further explanation between the relation of the derivative of the volume of a sphere and the surface area of a sphere. That's what my above response was aimed at. Sorry for the diversion.

If a 2-sphere is a regular 3-sphere (which btw, when you first said it, it confused me so I shrugged it off) then what is a 3-sphere? A 4-sphere? (i.e. some sphere we can't visualize)

I never believed in higher dimensions I guess. I understand how the math can work for them, by following the rules between 1,2 and 3-d objects and extending it further. Like, a 4-d cube would have 16 corners. That's fine and dandy, but I'm not going to consider an realistic object just because the math says it exists. And the opposite to realistic is imaginary so that's how I understand it.
But this is just my view on the subject. I could always be wrong (doubt it *cough*)

Well an n-sphere is defined as the set of points in Rⁿ⁺¹ with distance 1 from the origin. In notation
$$\mathbb{S}^n = \{x\in\mathbb{R}^{n+1} : |x|<1 \}$$
From what I know, which may be wrong, we really don't know exactly how an atom looks like (or I guess anything smaller than an atom as well), but we know they exist. We can describe their structure, but we can't visualize them exactly, only conceptually and mathematically. Do you consider atoms to be realistic? Photons? Magnetic fields? Gravity? You might enjoy reading https://www.amazon.com/dp/048627263X/?tag=pfamazon01-20.

 

[Mentallic]

Oops, sorry. I misread your original question as wanting a further explanation between the relation of the derivative of the volume of a sphere and the surface area of a sphere. That's what my above response was aimed at. Sorry for the diversion.

Ahh that's ok. At least now I'm certain I wasn't missing any hidden messages in your posts that would have lead me to an answer

Well an n-sphere is defined as the set of points in Rⁿ⁺¹ with distance 1 from the origin. In notation
$$\mathbb{S}^n = \{x\in\mathbb{R}^{n+1} : |x|<1 \}$$

ok it is defined to be that way, but why? I don't see the reasoning behind it. So a 2-sphere which I would assume is a circle (correct me if I'm wrong) is defined as a sphere on the x-y-z plane? This doesn't seem right to me...

From what I know, which may be wrong, we really don't know exactly how an atom looks like (or I guess anything smaller than an atom as well), but we know they exist. We can describe their structure, but we can't visualize them exactly, only conceptually and mathematically. Do you consider atoms to be realistic? Photons? Magnetic fields? Gravity? You might enjoy reading https://www.amazon.com/dp/048627263X/?tag=pfamazon01-20.

I consider them to be realistic in the sense that they are used by modern theories to describe the world around us. They are necessary to explain many phenomena. I'm not willing to believe that which I cannot see as much as I believe "I think, therefore I am", as theories constantly change when better ideas arise.
Now, with that thought in mind, I understand how higher dimensions are necessary to explain certain phenomena also, such as general relativity, the idea of the expanding universe etc. but I don't believe these higher dimensions are realistic. If they do actually exist (besides believing they exist because models that describe the phenomena make sense with higher dimensions), then we won't ever be able to interact with them, so in my eyes, it is an imaginary thing.

p.s. I've read a summarized version of flatland. Now, let's just wait till those insects actually do transport themselves from one point in the universe to the other, then I'll give the idea another glance