How complex can you go with summations?

[SamBridge]

Could I sum a series of points to sum a series of lines to make a plane, then sum a series of planes to make a solid object, then sum solid objects over different 4-D intervals to create a 4-D object? Or in other words, is there a direct way to sum a bunch of 1 dimensional points into a 3-D or 4-D object?

[studiot]

Yes.

 

Incidentally a 'series' is already a sum.

 

A 'sequence' of points is just a list.

Edited April 24, 2014 by studiot

[mathematic]

Could I sum a series of points to sum a series of lines to make a plane, then sum a series of planes to make a solid object, then sum solid objects over different 4-D intervals to create a 4-D object? Or in other words, is there a direct way to sum a bunch of 1 dimensional points into a 3-D or 4-D object?

Is your question about notation? Also the "sums" that you are talking about have to be integrals in order to go up a dimension.

[SamBridge]

 

Incidentally a 'series' is already a sum.

For semantic purposes I have do to identify an object and then identify an action applied to it. I can't just say "could I series of points and then series of lines and then series of planes...?"

 

Is your question about notation? Also the "sums" that you are talking about have to be integrals in order to go up a dimension.

Sort of both, the notation and process of turning a series of only points into a 3-D object.

[studiot]

 

For semantic purposes I have do to identify an object and then identify an action applied to it. I can't just say "could I series of points and then series of lines and then series of planes...?"

 

OK, that is exactly what we do, only we don't call them points, lines or planes.

 

We call them

 

a small element dx

 

and perform the summation on all the dx 's

a small strip dxdy

and perform a double summation on all the dx 's and dy 's

a small section dxdydz

and perform a triple summation on all the dx 's, dy 's and dz 's

 

 

They do not have to be infinite sums, but if they are then the sums go over into integrals, as mathematic says.

I am have tried to avoid this since I don't know your mathematical level, but the notation is

 

A single summation becomes a single integral, which is just called an integral

 

A double summation becomes a double integral, also called an area integral

 

A triple summation becomes a triple integral, also called a volume integral

 

There are some additional complications to this.