What is a tensor and why is it useful?

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In summary, a tensor is a generalization of a vector in three dimensional space, with a scalar having one component, a vector having 3 components, and a second order tensor having 9 components. The components of a tensor change homogeneously when changing coordinate systems, meaning they can be represented as a sum of products with the components of the tensor in the old coordinates. This is important because it allows for equations representing natural laws to be independent of coordinate systems. If a tensor equation is true in one coordinate system, it is true in all. Additionally, there have been many discussions on tensors on this site, with one long post by mathwonk providing a detailed explanation.
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What is a tensor and why is it useful?
 
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A "Tensor" is a generalization of "vector". In three dimensional space, a "scalar" (zero order tensor) has one component (number), a vector (first order tensor) has 3, a second order tensor has 9, etc. But they can't be just any numbers. The basic concept is that if you change coordinate systems, the components of the tensor change "homogeneously"- basically that means each component in the new coordinates is a sum of products of numbers with the components of the tensor in the old coordinates.
The point of that is: If a tensor has all its components 0 in one coordinate system, then (since we are multiplying all those numbers by the 0 components) it has all components 0 in any coordinate system.

Why is that important? Because coordinate systems are not "natural"- we make them up ourselves- so any equations representing "natural laws" should be independent of the coordinates system. If I can write a "natural law" as A= B where A and B are both tensors, that is the same as A-B= 0. And if that is true in one coordinates system, then it is true in all. That is: if a tensor equation is true in any coordinate system, then it is true in all coordinate systems.
 
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young person, there have been approximately 100,000 words written on this question in the last 9 months here. is it possible to search the site on this topic?
 
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1. What is a tensor?

A tensor is a mathematical object that represents a physical quantity in a multi-dimensional space. It contains information about the magnitude and direction of the quantity and can be visualized as a multi-dimensional array of numbers.

2. How is a tensor different from a scalar or a vector?

Unlike a scalar, which is a single value representing a quantity, and a vector, which is a one-dimensional array of values, a tensor can have multiple dimensions and can represent complex physical quantities such as stress, strain, or electric fields.

3. What is the significance of tensors in physics and engineering?

Tensors are used to describe and model physical phenomena in fields such as mechanics, fluid dynamics, electromagnetism, and general relativity. They allow for a more accurate and comprehensive understanding of complex systems and their behavior.

4. How are tensors useful in machine learning and artificial intelligence?

In machine learning and artificial intelligence, tensors are used to represent and manipulate multi-dimensional data, such as images, audio, and text. They are also used in deep learning algorithms, which use layers of tensors to extract features and make predictions.

5. Are there different types of tensors?

Yes, there are several types of tensors, including scalars (0-dimensional tensors), vectors (1-dimensional tensors), matrices (2-dimensional tensors), and higher-order tensors (3 or more dimensions). There are also special types of tensors, such as symmetric tensors, anti-symmetric tensors, and tensors with specific transformation properties, that are used in different fields of study.

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