tensors and vectors

[richardings]

whats the difference between a tensor and a vector?

are they even related?

[Arnaud Antoine ANDRIEU]

whats the difference between a tensor and a vector?

are they even related?

Hello, wholesale to describe it to quickly, the vector draw the path of the tensor. Cordially.

[timo]

whats the difference between a tensor and a vector?

are they even related?

The two things are quite closely related, but I am not sure if there is "the" definition of either. Already the very common term "vector" has different default meanings for a mathematician (element of a vector space), a physicist working with relativity (object that has a certain behavior under coordinate transformations) and an engineer or computer scientist (combination of values (x,y,z)) that are related but not identical.

Generally, in the framework that I know the term tensor from, a vector could be considered be a special case of a tensor (a rank-1 tensor). In fact, something along the lines of "a tensor is a generalization of a vector" may be the explanation you may hear most often. A definition of "tensor" that I like is that it is a function that maps a number of vectors on a value [and that is linear in each of its arguments].

[ajb]

A definition of "tensor" that I like is that it is a function that maps a number of vectors on a value [and that is linear in each of its arguments].

 

You can think of tensors and vectors in this way, as linear mappings. A little more useful in physics and differential geometry is to think of such objects as sections of particular natural bundles over a (smooth) manifold. This is scary mathematics stuff now, but the core thing is that coordinate transformations on the base manifold induce bundle automorphisms. Or in other words, (components of) tensors and tensor-like objects can be defined (locally) by their transformation properties under changes of local coordinates. This is probably the closest to the informal meaning that physicists use.

[elfmotat]

In n-dimensional space, a scalar field (rank (0,0) tensor field) has n⁰=1 components (i.e. it's just a number). It has the same value no matter what coordinate system you use. An example of a scalar field is the temperature at every point in 3-D space. What you have is a function T(xⁱ). The xⁱ can be any type of coordinates (Cartesian, cylindrical, spherical, some crazy coordinate system). It doesn't matter what you call a point, it has whatever temperature it has. So if yⁱ is the same point in some other type of coordinates, what you find is that T(xⁱ)=T(yⁱ)

 

 

A vector field (rank (1,0) tensor field) has n¹ components. For example, in 3-D space it takes three numbers to make up a vector. The thing that characterizes vectors is that their components transform in such a way that the magnitude of a vector (the scalar product of a vector with itself) is the same in any type of coordinates. This makes sense because, for example, a car traveling at 20 mph is still traveling at 20 mph no matter what type of coordinates you label the road with, what direction you call North, etc.

 

 

Then there are tensors of higher rank. A (2,0) tensor has n² components. So a rank (2,0) tensor in 3-D space has 9 components. The components of higher rank tensors transform in such a way that scalar contractions of tensors with other tensors have the same value in any coordinate system. It is this property that makes tensors interesting and useful. Physical quantities need to have to the property of coordinate invariance because they do not depend on how you label points in space, so all physical quantities must be represented by tensors.