Fields and their relation to Tensors

In summary, A group of students have been assigned to read a section about Fields from Anadijiban Das' "Tensors". The section defines a field and its axioms, and the students are tasked with finding the definitions of unfamiliar words. One student is looking for a deeper understanding of how fields relate to vector spaces and tensors, as they understand that a vector space presupposes a field. Another student suggests that vectors and other mathematical objects reveal information about the properties of a field, rather than the other way around. The first student asks for further clarification.
  • #1
iamalexalright
164
0
Alright, a little motivation for my question before I ask it... We have been assigned to read a section of our book (Anadijiban Das' "Tensors", section 1.1) and find the definitions of all words we don't know. The section I have been assigned is all about Fields. It gives the definition of a field and the axioms that constitute a field.

Now, I understand, in a very basic way, what a field is. I can prove, using the axioms, whether or not some set constitutes a field. What information I am really looking for is a deeper understanding of a field and specifically how they relate to a vector space or tensors (and I do know that a vector space presupposes some field F).
 
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  • #2
Hrm. My first inclination is to associate them in the exact opposite way. Vectors and stuff tell you about the arithmetic of your field, rather than the field telling you stuff about vectors.
 
  • #3
Alright, I can somewhat see what you are saying (keep in my mind this is really my first time being exposed to these subjects). Could you perhaps expound?
 

Related to Fields and their relation to Tensors

1. What is a tensor?

A tensor is a mathematical object that describes the linear relationship between two sets of vectors, scalars, or other tensors. It is represented by a multi-dimensional array of numbers and can be used to describe physical quantities such as force, velocity, and stress in a particular coordinate system.

2. How are tensors related to fields?

Tensors are closely related to fields, which are mathematical functions that assign a value to every point in space. Tensors can be used to describe the behavior of fields and how they change over space and time. For example, the stress tensor can be used to describe the stress field within a material.

3. What is the difference between a scalar and a tensor field?

A scalar field is a function that assigns a single value to each point in space, while a tensor field assigns a tensor to each point in space. Scalars can be thought of as 0th-order tensors, while vector fields can be thought of as 1st-order tensors.

4. How are tensors used in physics?

Tensors are used extensively in physics to describe the relationships between physical quantities and their properties. They are used in fields such as electromagnetism, fluid dynamics, and general relativity to model and predict the behavior of physical systems.

5. Can tensors be visualized?

While tensors themselves cannot be directly visualized, their effects can be visualized through visualization of the corresponding field. For example, the stress tensor can be visualized through stress contour plots or vector fields, and the electromagnetic field tensor can be visualized through electric and magnetic field lines.

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