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October 7th, 2015, 12:48 PM  #1 
Newbie Joined: Dec 2013 Posts: 28 Thanks: 0  Example of a vector that is not a tensor?
I read somewhere that not all vectors are tensors. Can you give me an example?

October 7th, 2015, 02:49 PM  #2 
Math Team Joined: Jan 2015 From: Alabama Posts: 2,579 Thanks: 668 
That is simply not true. Every vector is a tensor of rank 1. (And every number is a tensor of rank 0.)

October 8th, 2015, 06:35 AM  #3  
Senior Member Joined: Jun 2015 From: England Posts: 644 Thanks: 184  Quote:
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.  
October 8th, 2015, 10:02 AM  #4 
Newbie Joined: Dec 2013 Posts: 28 Thanks: 0  
October 8th, 2015, 01:23 PM  #5 
Senior Member Joined: Jun 2015 From: England Posts: 644 Thanks: 184 
Tensors of any order can be resented by matrices. But Not every matrix or even every square matrix represents a Tensor. 
January 3rd, 2017, 12:55 PM  #6 
Newbie Joined: Jan 2017 From: Canada Posts: 6 Thanks: 1  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? 
January 3rd, 2017, 02:12 PM  #7  
Senior Member Joined: Sep 2015 From: CA Posts: 1,302 Thanks: 666  Quote:
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 $E,~(E_x, E_y, E_z)$  
January 4th, 2017, 03:58 AM  #8 
Math Team Joined: Jul 2011 From: North America, 42nd parallel Posts: 3,347 Thanks: 227 
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 
January 4th, 2017, 08:25 AM  #9  
Senior Member Joined: Sep 2015 From: CA Posts: 1,302 Thanks: 666  Quote:
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.  
January 4th, 2017, 01:52 PM  #10  
Newbie Joined: Jan 2017 From: Canada Posts: 6 Thanks: 1  Quote:
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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