
#1
May1007, 09:53 AM

P: 38

Hello Gals,
I know what a scalar is. I know what a vector is. I know what a linear transformation is. But what in the name of sweet aunt petunia is a rank 3 tensor? Love, Plx Mny 



#2
May1007, 10:34 AM

Sci Advisor
P: 4,495

I believe that it is misleading to try to visualize higherrank tensors. Think of vector as collections of 3 or 4 numbers, not as arrows. Then the algebraic generalization to matrices, rank3 tensors, etc, is trivial.




#3
May1007, 12:22 PM

P: 2,955

Pete 



#4
May1007, 12:28 PM

P: 38

Need mental pictureAnyways, I thought a little about this on my lunch break. I came up with "a linear combination of linear combinations". Doesn't seem like a concept worth worrying about. 



#5
May1007, 03:00 PM

P: 2,955

Pete 



#6
May1007, 07:27 PM

Emeritus
Sci Advisor
P: 7,443

One can also think of it as inputting 2 generalized vectors (or a rank 2 tensor), and outputting a vector, or inputting 1 generalized vector, and outputing 2 vectors (or a rank 2 tensor). 



#7
May1107, 12:07 AM

P: 308

I was wondering this same thing. I can't even visualize linear transformations. I keep wondering: does not being able to visualize it make it impossible to do things like GR if you're a "geometric thinker" like me as opposed to a "formula thinker" like... pretty much everyone else?
I'm really bad. My eyes spin in circles whenever I see a sum in sigma notation, and I have to write it out with the ellipsis before I understand what it's saying! 



#8
May1107, 08:22 AM

P: 38

I can definitely visualize a linear transformation. Maps a vector to another vector. Period. 



#9
May1107, 01:55 PM

P: 2,955

Y = aX + B I.e. all linear transformations have this form. Pete 



#10
May1107, 08:46 PM

P: 232

I know what a rank 10 tensor is
I know what a rank 11 tensor is... 



#11
May1207, 12:13 AM

P: 308





#12
May1207, 12:16 AM

P: 308





#13
May1207, 02:59 AM

P: 530

Many tensors can be visualized as ellipsoids.
e.g. the inertia tensor, or the polarizability tensor. 



#14
May1207, 04:17 AM

P: 2,955

Pete 



#15
May1207, 10:29 AM

P: 308

Then, a rank3 tensor is like picturing those three arbitrarily changed/moved vectors in a different coordinate system, and applying a different linear transformation to each one. 



#16
May3009, 12:58 PM

P: 1

m=(((1,2),(2,4)),((2,3),(5,6))) is a rank 3 tensor with dimension 2, a vector of matrices. A tensor is a nested list. An example Eigenmath http://eigenmath.net/ script with a rank 3 tensor is: Maxwell equations in tensor form. See the book Gravitation p. 81.   F + F + F = 0  ab,c bc,a ca,b   ab a  F = 4 pi J  ,b  For this demo, use circular polarized light.  EX = sin(t+z) EY = cos(t+z) EZ = 0 BX = cos(t+z) BY = sin(t+z) BZ = 0 FDD = (( 0, EX, EY, EZ), ( EX, 0, BZ, BY), ( EY, BZ, 0, BX), ( EZ, BY, BX, 0)) See p. 74. Here, DD means "down down" indices. X = (t,x,y,z) Coordinate system FDDD = d(FDD,X) Gradient of F T1 = transpose(transpose(FDDD,2,3),1,2) Transpose bca to abc T2 = transpose(transpose(FDDD,1,2),2,3) Transpose cab to abc check(FDDD + T1 + T2 = 0) guu = ((1,0,0,0),(0,1,0,0),(0,0,1,0),(0,0,0,1)) FDDU = contract(outer(FDDD,guu),3,4) Easier to make FDDU than FUUD. check(contract(FDDU,2,3) = 0) For light J is zero. "OK" The gradient of a rank 2 tensor (matrix) in a coordinate system (vector), is a rank 3 tensor. (GAMUDD) in example below. Another example is the gradient of the metric in general relativity which is the connection. An example of a 4th rank tensor is the Riemann curvature of spacetime. RUDDD in http://eigenmath.net/examples/bondimetric.txt . 



#17
Jun309, 10:01 AM

P: 2

Now some one try to visualize contravariant tensors! I tried to twenty years ago and then decided to do grad school in engineering. At least with fluids you don't get beyond three dimensions!



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