Subset of space

J

Jignesh77

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Dec 8, 2020
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Definition — In vector calculus and physics, a vector field is an assignment of a vector to each point in a subset of space.
I work as a pharmacy technician but I love learning maths. I studied maths till grade 12.
What does "field and space" mean in simple words in the context of vector space?
What does subset of space mean in the above definition?
I am learning for fun. It has been more than 25 years since I left school. I did Masters in Organic Chemistry.

Thank you in advance!
 
Jignesh77 said:
Definition — In vector calculus and physics, a vector field is an assignment of a vector to each point in a subset of space.
I work as a pharmacy technician but I love learning maths. I studied maths till grade 12.
What does "field and space" mean in simple words in the context of vector space?
What does subset of space mean in the above definition?
I am learning for fun. It has been more than 25 years since I left school. I did Masters in Organic Chemistry.

Thank you in advance!
Click to expand...
Use Google with keywords vector field and space,

Please tell us what you find and if something particular in those responses that you don't understand.

Read the Wikipedia references - carefully.
 
Thanks. I know the meaning of space as we use in geometry but I got confused by the phrase "subspace of space".
I still remember vectors from physics but don't understand the term "field". What is the difference between a space in geometry and a space used here?
"In mathematics, a space is a set (sometimes called a universe) with some added structure.". Now, I am more confused by the "added structure".
What does structure mean? I am so sorry. All these words seem to have special meaning.
 
Jignesh77 said:
Thanks. I know the meaning of space as we use in geometry but I got confused by the phrase "subspace of space".
I still remember vectors from physics but don't understand the term "field". What is the difference between a space in geometry and a space used here?
"In mathematics, a space is a set (sometimes called a universe) with some added structure.". Now, I am more confused by the "added structure".
What does structure mean? I am so sorry. All these words seem to have special meaning.
Click to expand...
That added structure defines addition, multiplication, division, etc.

In vector space, addition and product (multiplication) are defined - but you will see that

addition in vector space is totally different from addition in number space . And in vector space you will have two types of products (scalar product and cross-product) - moreover there is NO such thing as vector division.​
 
Jignesh77 said:
The region in which objects exist.

The small ball takes up less space than the big ball. l.

Space Definition (Illustrated Mathematics Dictionary)

Illustrated definition of Space: The region in which objects exist. The small ball takes up less space than the big...
www.mathsisfun.com
Click to expand...
That is the definition of "physical space". What about "mathematical space"? Do a Google search for "mathematical space".

Did you do a Google search of "vector field and space," - i.e. "mathematical space"?
 
In mathematics, a space is a set with some added structure. While modern mathematics uses many types of spaces, such as Euclidean spaces, linear spaces, topological spaces, Hilbert spaces, or probability spaces, it does not define the notion of "space" itself. Wikipedia
I understand that set is a collection of distinct objects but I don't understand the above.
Thank you.
 
Jignesh77 said:
In mathematics, a space is a set with some added structure. While modern mathematics uses many types of spaces, such as Euclidean spaces, linear spaces, topological spaces, Hilbert spaces, or probability spaces, it does not define the notion of "space" itself. Wikipedia
I understand that set is a collection of distinct objects but I don't understand the above.
Thank you.
Click to expand...
Did you read any of the reference paper attached to the "Google"search page? e.g. https://www.sciencedirect.com/topics/mathematics/mathematical-space

or did you watch any of the videos attached to it:
 
Jignesh77 said:
In mathematics, a space is a set with some added structure. While modern mathematics uses many types of spaces, such as Euclidean spaces, linear spaces, topological spaces, Hilbert spaces, or probability spaces, it does not define the notion of "space" itself. Wikipedia
I understand that set is a collection of distinct objects but I don't understand the above.
Click to expand...
What is there to understand? Each of the listed spaces is simply a set of objects with a set of axioms that creative a particular structure of the given space. If one really wants to understand the jump in and swim. Try not to drown.
The best way not to drown is to prepare oneself with a firm foundation in sets & logic.
 
Jignesh77 said:
In mathematics, a space is a set with some added structure. While modern mathematics uses many types of spaces, such as Euclidean spaces, linear spaces, topological spaces, Hilbert spaces, or probability spaces, it does not define the notion of "space" itself. Wikipedia
I understand that set is a collection of distinct objects but I don't understand the above.
Thank you.
Click to expand...
Perhaps a problem that you are having is trying to visualize an actual, physical, space. Mathematical spaces do not necessarily have any "real" counterpart. For example, the vector space of quaternions over the field of complex numbers does not represent anything like a space you would look out your window to see. But as the 3D vector space as it is normally formulated actually is a physical space I suppose that's where the name came from.

A space is simply a set of objects with a few added features. Many of them don't even admit anything like a distance structure. The word "space" is just a label and its definition changes between different branches of Mathematics. In Algebra the first usage of the word "space" appears in the term vector space, but again it does not generally denote anything like what you would see out your window.

-Dan
 
Jignesh77 said:
Definition — In vector calculus and physics, a vector field is an assignment of a vector to each point in a subset of space.
I work as a pharmacy technician but I love learning maths. I studied maths till grade 12.
What does "field and space" mean in simple words in the context of vector space?
What does subset of space mean in the above definition?
I am learning for fun. It has been more than 25 years since I left school. I did Masters in Organic Chemistry.

Thank you in advance!
Click to expand...

In vector calculus and physics, a vector field assigns a vector to every point in a specified area of space. "Space" refers to our three-dimensional environment, and a "subset of space" is a smaller, defined portion within it. It's like mapping out quantities (represented as vectors) across a specific region. As a pharmacy technician, you can think of it as assigning different medications to specific shelves in your pharmacy, with each shelf representing a subset of the pharmacy's space. Learning about vector fields can enhance your understanding of mathematical concepts, even if it's been years since you studied them formally.
 
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