How do I manipulate infinitesimals? How do I know what assumption is valid?

Oct 2008
104
0
How do I manipulate small finite increments? How do I know what assumption is valid?

Hello,
I know certain things about small finite increments, like how the general formula for dy/dx is derived, but I have no idea how to manipulate the infinitesimals. The assumptions made seems arbitrary to me, how do I know that it is ok to assume one thing, but wrong to assume in case of others? I mean I can understand that this follows from that assumption and etc, but what made the assumptions valid I have no idea. Can you please explain to me the concept of delta x, or give me links to some resources so that I can understand at least well enough to get sums like these done from scratch: (Nod) (Nod)


 

Attachments

Last edited:

CaptainBlack

MHF Hall of Fame
Nov 2005
14,972
5,271
someplace
Hello,
I know certain things about infinitesimals, like how the general formula for dy/dx is derived, but I have no idea how to manipulate the infinitesimals. The assumptions made seems arbitrary to me, how do I know that it is ok to assume one thing, but wrong to assume in case of others? I mean I can understand that this follows from that assumption and etc, but what made the assumptions valid I have no idea. Can you please explain to me the concept of delta x, or give me links to some resources so that I can understand at least well enough to get sums like these done from scratch: (Nod) (Nod)
There are no infinitesimals in your attachments there are small finite increments (that is what \(\displaystyle \delta x\) denotes) which are subject to all the usual rules of arithmetic and common algebra and there are limiting processes converting sums with finite increments to integrals as increments become arbitrary small.

CB
 
  • Like
Reactions: ssadi
Oct 2008
104
0
There are no infinitesimals in your attachments there are small finite increments (that is what \(\displaystyle \delta x\) denotes) which are subject to all the usual rules of arithmetic and common algebra and there are limiting processes converting sums with finite increments to integrals as increments become arbitrary small.

CB
Sorry, i didn't know what delta x was called :D
Ok, here is my problem, in the following diagram the height (length) of the cylinder:
My calculations: \(\displaystyle Rcos(\theta)-Rcos(\theta+\delta \theta)\)= \(\displaystyle Rcos(\theta)-R(cos(\theta)cos(\delta \theta)+sin(\theta)sin(\delta \theta)\)
\(\displaystyle =R(cos\theta-cos\theta+sin\theta*\delta\theta)\)
\(\displaystyle =Rsin\theta*\delta\theta\) (assumptions \(\displaystyle cos\delta\theta \)becomes 1, \(\displaystyle sin\delta\theta\) becomes \(\displaystyle \delta \theta \)
But the book says that the height is R\delta \theta, what am I doing wrong here?
 

Attachments

CaptainBlack

MHF Hall of Fame
Nov 2005
14,972
5,271
someplace
Sorry, i didn't know what delta x was called :D
Ok, here is my problem, in the following diagram the height (length) of the cylinder:
My calculations: \(\displaystyle Rcos(\theta)-Rcos(\theta+\delta \theta)\)= \(\displaystyle Rcos(\theta)-R(cos(\theta)cos(\delta \theta)+sin(\theta)sin(\delta \theta)\)
\(\displaystyle =R(cos\theta-cos\theta+sin\theta*\delta\theta)\)
\(\displaystyle =Rsin\theta*\delta\theta\) (assumptions \(\displaystyle cos\delta\theta \)becomes 1, \(\displaystyle sin\delta\theta\) becomes \(\displaystyle \delta \theta \)
But the book says that the height is R\delta \theta, what am I doing wrong here?
You are computing different things, you are computing the x extent of the ring while the book is computing the slant height of the ring.

CB
 
  • Like
Reactions: ssadi
Oct 2008
104
0
You are computing different things, you are computing the x extent of the ring while the book is computing the slant height of the ring.

CB
But a cylinder has no slant height, it is a property of cones, and the book instructs me to use elemental cylinders. How do I know that I have to calculate slant height when it tells me to use cylinders, I calculated what I knew to be height of the cylinders!(Nod)
 

CaptainBlack

MHF Hall of Fame
Nov 2005
14,972
5,271
someplace
But a cylinder has no slant height, it is a property of cones, and the book instructs me to use elemental cylinders. How do I know that I have to calculate slant height when it tells me to use cylinders, I calculated what I knew to be height of the cylinders!(Nod)
You are going to be computing the mass of a slice of a spherical shell, this is approximately a part of a cone and does have a slant height (and that is what you need to calculate its volume and hence mass.

CB
 
  • Like
Reactions: ssadi
Oct 2008
104
0
You are going to be computing the mass of a slice of a spherical shell, this is approximately a part of a cone and does have a slant height (and that is what you need to calculate its volume and hence mass.

CB
If I take this as part of a cone, why do I compute its mass using the formula of a cylinder's volume? And if using the formula for cylinder's volume is ok, why is it not ok to use cylinder's height (which I calculated) instead of slant height?
 

CaptainBlack

MHF Hall of Fame
Nov 2005
14,972
5,271
someplace
If I take this as part of a cone, why do I compute its mass using the formula of a cylinder's volume? And if using the formula for cylinder's volume is ok, why is it not ok to use cylinder's height (which I calculated) instead of slant height?
You have not posted the image file showing what you say, so my guess is that you have misconstrued the use of the area formula for a parallelogram for that of a rectangle.

CB
 
  • Like
Reactions: ssadi

HallsofIvy

MHF Helper
Apr 2005
20,249
7,909
If I take this as part of a cone, why do I compute its mass using the formula of a cylinder's volume?
Because, for constant density, weight is equal to "density times volume". And if density is a variable you choose your "pieces" so that the density is at least approximately constant in each piece- sum the pieces and take the limit as their size goes to 0 giving you of the form "density times dVolume".

And if using the formula for cylinder's volume is ok, why is it not ok to use cylinder's height (which I calculated) instead of slant height?
If you are talking about problem 19, a hemi-spherical shell, each band has bases of two different sizes- part of a cone, not a cylinder.
 
  • Like
Reactions: ssadi
Oct 2008
104
0
You have not posted the image file showing what you say, so my guess is that you have misconstrued the use of the area formula for a parallelogram for that of a rectangle.

CB
The diagram is at post #3.