Tensors are complex arrays that have specific and different properties. Not every multifaceted collection is a tensor.
There are two types of one-dimensional tensors: These include vectors and co-vectors. Either vectors or co-vectors can be represented as an accessible array of numbers.
The only difference is that linking those two comes when you have a variety of digits representing the object on one basis and want to find out what numbers complicate the same thing on some different ground.
Transformation signs and rules are slightly dissimilar for vectors and co-vectors. Vectors and co-vectors are usually “columns of numbers” or “lines of numbers,” respectively.
In short, a vector will always be a one-dimensional tensor; if you have a one-dimensional tensor, it will surely be either a vector or co-vector. Two-dimensional tensors are known as matrices.
Four different types of two-dimensional tensors exist, but no specific names exist. In the case of vectors, transformation rules are slightly different when you move from one basis to another, but there are no specific names for these tensors: they are only matrices.
Sooner or later, any two-dimensional array can be called a “matrix,” even if it is not a tensor. Again, for more details about the difference between array and tensor, refer to the earlier discussion.
What to Know About Tensors
Tensors are mathematical objects that can describe substantial properties, the same as scalars and vectors. Tensors are simply an inference of scalars and vectors; a scalar is a 0-rank tensor, and a vector is a 1st-rank tensor.
The rank of a tensor is identified by the number of directions (and hence the array’s dimensionality) necessary to define it. For example, properties that require one approach ( or first rank) can be easily described by a 3×1 column vector.
Furthermore, properties that require two orders (second rank tensors) can be defined by nine numbers, as in a 3×3 matrix general, 3n coefficients can describe the nth rank tensor.
The requirement for second-rank tensors comes when we must think about more than one direction to describe 1 of these physical aspects.
A perfect example is if we need to tell the electrical conductivity of any isotropic crystal. Generally, isotropic conductors are required to obey Ohm’s law, and that is, j=σE. This means that the current density j is parallel to the devoted electric field, E, and that each part of j is linearly proportional to per element of E. (e.g., j1 = σE1).
| Components of Electric Field |
| j1 = σ11E1 + σ12E2 + σ13E3 |
| j2 = σ21E1 + σ22E2 + σ23E3 |
| j3 = σ31E1 + σ32E2 + σ33E3 |
However, the current density induced in an anisotropic material will not necessarily parallel the involved electric field due to the crystal’s different directions of current flow (an excellent example of this is in graphite). This suggests that, in general, each component of the existing density vector can rely on all the parts of the present electric field.
So, in general, electrical conductivity is a 2nd rank tensor and can be fixed by nine independent coefficients, which can be illustrated in a 3×3 matrix.
This means that the current density j is parallel to the dedicated electric field, E, and that every part of j is linearly proportional to per field.
Some Examples of Second Rank Tensors
Some other examples of second rank tensors are:
- Electric susceptibility
- Thermal conductivity
- Stress
They generally relate a vector to another vector or dual-rank tensor to a scalar. Tensors of a high rank are instructed to fully describe properties that tell two second-rank tensors (e.g., Stiffness (4th rank): stress and strain) or a second-rank tensor and a vector (e.g., Piezoelectricity (3rd rank): anxiety and polarization).
To view these and more examples and investigate how changing the components of the tensors affects these properties, go through the Flash program below.
What Is a Vector?
A vector is a 1-dimensional array of numbers, a matrix where m or n equals 1. Like a matrix, it’s possible to perform various mathematical operations on a vector, and it is easy to multiply matrices with vectors and vice versa.
However, a tensor can be considered a generalized matrix that its rank can describe.
The level of a tensor is an integer number of 0 or higher. A scalar can represent a tensor with a rank of 0, a vector can represent a tensor with a rank of one, and a matrix can represent a tensor with a rank of two. There are also tensors of rank three and higher, the latter more difficult to visualize.
In addition to the rank, tensors have specific characteristics related to how they interact with each other mathematical entities. If any entities in an interaction transform the other entity or entities, then the tensor must obey a similar transformation rule.
Difference Between Vectors and Tensors
All vectors are usually tensors. But all tensors can’t be vectors. This means tensors are more widespread than vectors (strictly speaking, though mathematicians assemble tensors through vectors). Tensors are technically described through two different objects:
- Vectors
- One-forms (“dual” vectors)
Vectors are exclusively objects for which you know what counting any two of them (vector addition) indicates to scale-changing it ( also known as scalar multiplication).
One form, likewise, has all the same notions; apart from that, it can operate on vectors and then return scalars. Examples are in order: The most prototypical examples include Euclidean vectors –points of space.
Examples include one-forms would be the magnetic potential “vector” (It’s not a “true” vector) or the gradient operator.
Key Concepts in Tensor Mathematics and Physics
When you add other appropriate assumptions, the most significant property is that one-forms and vectors convert in some manner under a change of coordinates. These are the properties that physicists are most often worried about when consulting about things like the theory of general relativity.
Tensors, by elongation, as mathematical objects, are “multilinear” operators; this is to say, they take in sets of vectors (and one-forms) and return another tensor (as opposed to linear operators, which take in vectors and return vectors). These have varying uses.
Suppose you want to understand the general theory of tensors. In that case, you should realize abstract algebra and incredibly linear algebra), and if you’re going to understand tensor calculus, you should also understand the theory of differentiable manifolds.
Final Thoughts
- Tensors are multidimensional arrays with distinct properties.
- Not every multifaceted collection is a tensor.
- A vector is always a one-dimensional tensor, and a one-dimensional tensor is always either a vector or a co-vector. Matrix is the name given to two-dimensional tensors.
- A vector is a one-dimensional array of numbers, often known as a matrix, where m or n = 1. A vector, like a matrix, can be used to execute various mathematical operations, and it’s simple to multiply matrices with vectors and vice versa.
- On the other hand, a tensor can be conceived of as a generalized matrix described by its rank.
Related Articles
Wizard vs. Warlock (Who is stronger?)
Different Types Of Steaks (T-Bone, Ribeye, Tomahawk, and Filet Mignon)
Differences Between the Cessna 150 and Cessna 152 (Comparison)