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October 7th, 2015, 12:48 PM   #1
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Example of a vector that is not a tensor?

I read somewhere that not all vectors are tensors. Can you give me an example?
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October 7th, 2015, 02:49 PM   #2
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That is simply not true. Every vector is a tensor of rank 1. (And every number is a tensor of rank 0.)
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October 8th, 2015, 06:35 AM   #3
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Quote:
I read somewhere that not all vectors are tensors. Can you give me an example?
This is perfectly true, but, in fact all tensors are vectors.

But then you have to be using the mathematical definition of the word vector as a member of a vector space.

Vectors in a vector space are (mathematical) objects that obey the vector space axioms, in particular they have a binary combination law that obeys the the associated Field.

So from this point of view definite integrals form a vector space where the vectors are definite integrals.

Fourier series form another example, where the vectors have the form asinx or bcosx.
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October 8th, 2015, 10:02 AM   #4
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This is perfectly true, but, in fact all tensors are vectors.
Aren't second-order tensors supposed to be matrices?
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October 8th, 2015, 01:23 PM   #5
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Tensors of any order can be resented by matrices.

But

Not every matrix or even every square matrix represents a Tensor.
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January 3rd, 2017, 12:55 PM   #6
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Tensors are invariant.

I'm just learning tensors in detail myself but the secret here, I believe, is in the definition of a tensor ... "tensors are independent of a particular choice of coordinate system" or ... "tensors are invariant when the frame of reference is changed". So you have to know what a scalar or a vector are referring to before you know whether it is a tensor or not.
Assume you have two frames of reference: a stationary frame A and a moving frame B. If someone in each frame measures the temperature (T) of a point P the temperature will be the same. If, however, they measure the frequency (f) of the light coming from a point P the frequency will be different due to relativity (red or blue shifted depending on whether the B frame is moving away from or towards the point P). So T is invariant and therefore a scalar tensor (rank 0) whereas f varies and is not a tensor. So, the scalar 71 could be a tensor or not a tensor depending on what it is measuring.
Similarly with a vector it must be invariant to be a rank 1 tensor. So a vector showing velocity would vary depending on the observers and isn't a tensor. I can't off hand think of a vector that is a tensor ... maybe the force on an object observed by each observer.
Can anyone else with a deeper understanding of tensors let me know if I'm correct here?
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January 3rd, 2017, 02:12 PM   #7
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Originally Posted by photogo01 View Post
I'm just learning tensors in detail myself but the secret here, I believe, is in the definition of a tensor ... "tensors are independent of a particular choice of coordinate system" or ... "tensors are invariant when the frame of reference is changed". So you have to know what a scalar or a vector are referring to before you know whether it is a tensor or not.
Assume you have two frames of reference: a stationary frame A and a moving frame B. If someone in each frame measures the temperature (T) of a point P the temperature will be the same. If, however, they measure the frequency (f) of the light coming from a point P the frequency will be different due to relativity (red or blue shifted depending on whether the B frame is moving away from or towards the point P). So T is invariant and therefore a scalar tensor (rank 0) whereas f varies and is not a tensor. So, the scalar 71 could be a tensor or not a tensor depending on what it is measuring.
Similarly with a vector it must be invariant to be a rank 1 tensor. So a vector showing velocity would vary depending on the observers and isn't a tensor. I can't off hand think of a vector that is a tensor ... maybe the force on an object observed by each observer.
Can anyone else with a deeper understanding of tensors let me know if I'm correct here?
This is basically correct. The difference between vector and tensor is that any collection of numbers partitioned in the proper way is a scalar, vector, matrix, etc.

Tensors on the other hand have to obey coordinate transformation rules and remain the same object independent of the choice of coordinate system.

A vector that isn't a tensor (your mom's age, the height of your house, how many kids you have)

A vector that is a tensor, given some electric field $E,~(E_x, E_y, E_z)$
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January 4th, 2017, 03:58 AM   #8
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A vector must have magnitude and direction. Your mothers age has magnitude but what is the direction?

I suppose you can put a 'little arrow' above the numerical magnitude and turn it into a vector ... lol

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January 4th, 2017, 08:25 AM   #9
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A vector must have magnitude and direction. Your mothers age has magnitude but what is the direction?

I suppose you can put a 'little arrow' above the numerical magnitude and turn it into a vector ... lol

your mother's age is just a component of that vector.

The direction of that vector is the usual it's just that the coordinate system is a bit odd, which was sort of the point.
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January 4th, 2017, 01:52 PM   #10
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A vector that isn't a tensor (your mom's age, the height of your house, how many kids you have)
Each one of these examples is a scalar; if you put them together as (78, 7, 2) it may form a matrix but not a vector.

You may want to read "An Introduction to Tensors for Students of Physics and Engineering, Joseph C. Kolecki, Glenn Research Center, Cleveland, Ohio"

It is a well thought out introduction. He provides the following examples of vectors that are or are not tensors.

"Now, let V be the position vector extending from the origin of a coordinate space K to a particular point P, and V* be the position vector extending from the origin of another coordinate space K* to that same point. Assume that the origins of K and K* do not coincide; then V ≠ V*. The position vector is very definitely coordinate dependent and is not a tensor because it does not satisfy the condition of coordinate independence.
But suppose that V1 and V2 were position vectors of points P1 and P2 in K, and that V1* and V2* were position vectors to the same points P1 and P2 in K*. The vector extending from P1 to P2 must be the same vector in both systems. This vector is V2 – V1 in K and V2* – V1* in K*. Thus we have

V2 –V1 =V2*–V1*,

i.e., while the position vector itself is not a tensor, the difference between any two position vectors is a tensor of rank 1! Similarly, for any position vectors V and V*, dV = dV*; i.e., the differential of the position vector is a tensor of rank 1."
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