What is a Tensor? | Practical Uses & Difference from Vector

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In summary, a tensor is a vector generalized to three dimensions and is commonly used in structural engineering for materials with changing characteristics. It is different from a vector in that it has two dimensions and is written in matrix form. Tensors are commonly used in topics such as inertia, strength, and fluid mechanics.
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yaswanth_040
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Hi all,
What is a tensor? Can any give a practical use of it? How it is different from Vector?

Thank You,
Yaswanth
 
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  • #2
Tensor is a vector generalized to 3Dimensions. Tensors are typically found in structural engineering for instance with anisotropic materials with characteristics which change in direction. Here you can see a typical strain tensor

http://en.wikipedia.org/wiki/Deformation_(mechanics)#Shear_strain

Other topics include inertia, and strength.
 
  • #3
  • #4
Try thinking of a vector but with two dimensions that is written down in matrix form. For example the strain tensor in a solid like Viscousflow mentioned. Or the stress tensor in a fluid. The the components of the stress tensor are

Txx Txy Txz
Tyx Tyy Tyz
Tzx Tzy Tzz

So for an element in the fluid Txx would be the stress on the x face in the x direction. So it is a normal stress or pressure. Txy would be the stress on the x face in the y direction, shear stress caused by viscosity.
 

1. What is a Tensor?

A tensor is a mathematical object that describes the relationship between multiple vectors. It is a generalization of vectors and matrices, and can have multiple dimensions and components.

2. How are Tensors used in practical applications?

Tensors are used in various fields such as physics, engineering, and computer science. They are used to represent physical quantities like forces and velocities, and also in data analysis and machine learning for tasks like image recognition and language processing.

3. What is the difference between a Tensor and a Vector?

A vector is a one-dimensional mathematical object with magnitude and direction, while a tensor can have multiple dimensions and components. Vectors are a special case of tensors, and tensors can have different transformation properties compared to vectors.

4. Can Tensors be visualized?

Yes, tensors can be visualized in certain cases. For example, a two-dimensional tensor can be visualized as a matrix, and a three-dimensional tensor can be visualized as a cube. However, higher-dimensional tensors are more difficult to visualize.

5. How are Tensors related to Einstein's Theory of General Relativity?

In Einstein's theory of general relativity, tensors are used to describe the curvature of spacetime. The Einstein field equations, which describe the relationship between matter and the curvature of spacetime, are written in terms of tensors.

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