Adding infinitesimals

[Jehanne]

Why does adding two or more infinitesimals equal an infinitesimal?  What would happen if one added an infinite number of infinitesimals together?  How about multiplying an infinitesimal against any non-infinitesimal?  How about multiplying an infinitesimal by infinity?  Finally, consider the following:

dx = cos (theta) (dp)

Are 'dx' and 'dp' infinitesimals?  Trying to broaden my understanding of tensor calculus.

[robvalue]

Yes, they are. Such equations represent a relationship between them.

This is why simple addition will never work. To add together infinitesimals, you have to use integration rather than summation. It's a method of continuous adding.

It does seem extremely bizarre to begin with!

When you sum up infinitesimals, you'll always be summing and infinite amount of them, throughout any particular range. For example, between x=0 and X=1, I will cover every value between and there will be a dx for each one.

The easiest way of visualising integration is to look at a graph, such as y = x^2. You use integration to find the area under the graph, between any given point. What you are actually doing is splitting the area up into tiny trapeziums. Think of a very thin one, with the X coordinates almost identical, going up to meet the graph. If you split up the interval into a thousand sections, you'll have a thousand trapeziums, giving a pretty close approximation of the total area. The flat tops of the trapeziums just won't be exactly the same as the curve of the graph.

As you allow the number of sections to approach infinity, the accuracy improves to be exact. You're summing an infinite amount, each with width dx, and the relationship between dx and the values of dA (the change in area dx causes) represent the trapezium approaching zero width.

In general dA = y dx since y is the height of the trapezium at any given point (instead of meeting the graph twice, it only meets it once at an exact point, so there is just one height measurement now)

In this case the relationship is dA = x^2 dx
So A = X^3/3 to be evaluated between any two values of X.

[BrianSoddingBoru4]

two infinitesimals + two infinitesimals = four infinitesimals

Easy peasy.

Boru

[robvalue]

When you put it like that...

[SteelCurtain]

Boru would be an amazing maths teacher, I'd guess.

"SOHCAHTOA. Figure it out.

Dismissed."

[BrianSoddingBoru4]

Boru would be an amazing maths teacher, I'd guess.

"SOHCAHTOA. Figure it out.

Dismissed."

Sohcahtoa was the Indian princess who helped the Wright Brothers discover the Pacific Ocean on their way to find a Western route to Japan. Failing in this, they returned to Dweebmont, Ohio (Styptic Pencil Capital Of The World) where, using some canvas and old bicycle parts, they invented the 3D printer.

I'd be as good a history teacher as a maths teacher, that much seems plain.

Boru

[robvalue]

Please let me know if what I've written isn't clear, and I'll do some diagrams or a video or something

[Alex K]

Why does adding two or more infinitesimals equal an infinitesimal?  What would happen if one added an infinite number of infinitesimals together?  How about multiplying an infinitesimal against any non-infinitesimal?  How about multiplying an infinitesimal by infinity?  Finally, consider the following:

dx = cos (theta) (dp)

Are 'dx' and 'dp' infinitesimals?  Trying to broaden my understanding of tensor calculus.

Can you give some context so we can tell what x and p are in this example? Where did you get this relation from?

[Jehanne]

Why does adding two or more infinitesimals equal an infinitesimal?  What would happen if one added an infinite number of infinitesimals together?  How about multiplying an infinitesimal against any non-infinitesimal?  How about multiplying an infinitesimal by infinity?  Finally, consider the following:

dx = cos (theta) (dp)

Are 'dx' and 'dp' infinitesimals?  Trying to broaden my understanding of tensor calculus.

Can you give some context so we can tell what x and p are in this example? Where did you get this relation from?

It's from a book that I am trying to read, Tensor Calculus for Physics; it's on Amazon.

[Jehanne]

Yes, they are. Such equations represent a relationship between them.

This is why simple addition will never work. To add together infinitesimals, you have to use integration rather than summation. It's a method of continuous adding.

It does seem extremely bizarre to begin with!

When you sum up infinitesimals, you'll always be summing and infinite amount of them, throughout any particular range. For example, between x=0 and X=1, I will cover every value between and there will be a dx for each one.

The easiest way of visualising integration is to look at a graph, such as y = x^2. You use integration to find the area under the graph, between any given point. What you are actually doing is splitting the area up into tiny trapeziums. Think of a very thin one, with the X coordinates almost identical, going up to meet the graph. If you split up the interval into a thousand sections, you'll have a thousand trapeziums, giving a pretty close approximation of the total area. The flat tops of the trapeziums just won't be exactly the same as the curve of the graph.

As you allow the number of sections to approach infinity, the accuracy improves to be exact. You're summing an infinite amount, each with width dx, and the relationship between dx and the values of dA (the change in area dx causes) represent the trapezium approaching zero width.

In general dA = y dx  since y is the height of the trapezium at any given point (instead of meeting the graph twice, it only meets it once at an exact point, so there is just one height measurement now)

In this case the relationship is dA = x^2 dx
So A = X^3/3 to be evaluated between any two values of X.

Thanks; good explanation.