MaplePrimes - answers and comments on Question, Tensor vs Vector operations

using tensors

jakubi — Tue, 16 Feb 2010 10:43:29 Z

The package VectorCalculus does not handle tensors, just vectors. On the other hand, an option for explicit tensor calculations is the new package DifferentialGeometry. Note however that it is designed for more general calculations than tensors in Euclidean space and cartesian coordinates. So, it may be a bit too much for these particular calculations, but for the record you could do something like this.

Your 3x3 matrix:

sigma := Matrix(3, 3, {(1, 1) = A*x^2, (1, 2) = 2*A*x*y, 
(1, 3) = 0, (2, 1) = 0, (2, 2) = A*y^2, (2, 3) = 0, (3, 1) = 0, 
(3, 2) = 0, (3, 3) = 1}):

Define a 3D manifold M for the Euclidean space:

with(DifferentialGeometry):
with(Tools):with(Tensor):
DGsetup([x,y,z], M):

Convert the matrix sigma to a tensor object, corresponding to sigma_{ij}:

sigma1:=convert(sigma, DGtensor, [["con_bas", "con_bas"], []]);
               2                                2
  sigma1 := A x  D_x D_x + 2 A x y D_x D_y + A y  D_y D_y + 
D_z D_z

For differentiating it, the connection must be defined. And it is zero in this flat (not curved) Euclidean space:

C := Connection(0 &mult (dx &t D_x &t dx) );
                           C := 0 dx D_x dx

Then sigma_{ij,k} is calculated by applying the covariant derivative with this connection:

ds:=CovariantDerivative(sigma1, C);
  ds := 2 A x D_x D_x dx + 2 A y D_x D_y dx + 2 A x D_x D_y dy

         + 2 A y D_y D_y dy

And then, the indices j and k are contracted to form sigma_{i,j,j}:

ds2:=ContractIndices(ds, [[2, 3]]);
                     ds2 := 4 A x D_x + 2 A y D_y

Finally, this divergence displayed as an Array:

convert(ds2,DGArray);
                          [4 A x, 2 A y, 0]

OUCH, my level of MAPLE skills would have

tsunamiBTP — Mon, 08 Mar 2010 01:40:18 Z

OUCH, my level of MAPLE experience would have precluded me from figuring that out. Is MAPLE working on tools for such operations or is Physics & Mechanics far too specialized for them to bother?

Tensor Divergence

josephap83 — Wed, 26 Feb 2014 12:46:38 Z

@jakubi Dear friend,

I think this is a good answer. I'm trying to calculate the stress tensor divergence (momentum equation) in curvilinear coordinates, however the results that I get in Maple, using your procedure, are a little different, comparing with some literature. I can`t understand why. I `m calculating the connection using Christoffel from Differential Geometry package.

This is the tensor

This my results:

This is the result from literature:

I really appreciate your help

Easier using Physics

ecterrab — Fri, 28 Feb 2014 04:51:36 Z

@josephap83 and @tsunamiBTP

Although you could do these computations using DifferentialGeometry, I believe in cases like this one it is simpler using the tensor capabilities of the Physics package.

Independent of that, there is an issue with your post about the Divergence of a Tensor. You show a tensor 'tau' defined as an equation with a matrix on the right-hand side, but you do not show the indices of tau: is this the all contravariant tau, as in or the all covariant or just the "physical" components frequently denotated that are not covariant nor contravariant? For the relationship between convariant, contravariant and "physical" components, see for instance A Primer on Tensor Calculus, page 12, formulas (32) and (33).

Here I will assume that you are describing

Set the dimension to 3 and work with the coordiantes you indicated

(1)

To set the metric, the simplest way is to indicate the line element

(2)

(3)

Visual check on the metric

(4)

Define now the stress tensor

(5)

And that is all. You do not need to define anything else, basis, Christoffel symbols, nothing.

Just compute the (covariant) divergence using standard tensorial notation.

Moreover: to avoid repetitive display of (r,theta,z) everywhere, use PDEtools:-declare

(6)

(7)

The components of this Divergence are

(8)

If what you need is this expression in terms of the "physical components" , from the formulas (32) and (33) of the reference mentioned in the first paragraph, you have:

(9)

(10)

(11)

(12)

And this is the result you show, that you were expecting.

_________________________________________________________________________

Details

Note the compact display of derivatives indexed, and the functionality of the components of omited (this is what PDEtools:-declare does). But is only a display trick.

The actual Maple objects are behind this display. If you want to see them in standard Maple notation, use show

(13)

Also, in (12) you see and also . If you want the symmetry of the stress tensor - that you indicated when you defined the tensor - to be taken into account, use Physics:-Simplify (not simplify)

(14)

If you prefer for the symmetry properties to be taken into account automatically, among other ways you can directly define the tensor as a matrix with the appropriate symmetry, for example:

(15)

(16)

Define now a tensor Tau with this equation (see Physics:-Define , the paragraph about defining using tensorial equations)

(17)

You can now compute taking the symmetry into account automatically

(18)

(19)

By the way you may want to compute with this object itself, as a tensor of 1 index. You can do that the same way, defining a tensor whose components are the components of this divergence, as in

(20)

(21)

Now you have direct access to the covariant OR contravariant components using indices. For example, the 2nd covariant and contravariant components are different

(22)

(23)

(24)

NOTE: the computations above are performed using the newest Physics library, available for download at Maplesoft's Physics Research & Development webpage

Download DivergenceOfATensor.mw

Edgardo S. Cheb-Terrab
Physics, Differential Equations and Mathematical Functions, Maplesoft

Try an orthonormal frame

Torre — Thu, 27 Feb 2014 18:35:27 Z

It appears that the "result from literature" is using the orthonormal basis associated to cylindrical coordinates while your calculation is using the coordinate basis associated to cylindrical coordinates.

To get the desired result using DifferentialGeometry, you need to initialize the orthonormal basis, define the metric and Christoffel symbols relative to that basis, and then take the divergence using CovariantDerivative and contracting indices using the metric.

Does this help? If you would like, I can try to include some commands, or maybe a worksheet.

curvilinear coordinates

jakubi — Fri, 28 Feb 2014 03:38:04 Z

@josephap83

Good that Charles has addressed already your question. The administrators of this site have removed all the subscriptions to threads previous to the reform of early June 2010. So, I have not received any email notification about your comment...

Certainly, the standard way to make these vector/tensor calculations in Euclidean flat space is using an orthonormal basis, and this is what formulas in the literature show. For cartesian coordinates, as in the example of April 2010, the coordinate basis is also orthonormal, so there was no need to go into this distinction. But for curvilinear coordinates, they are different, so care has to be taken...

Note also that, since 2010, some further developments in the area of tensor calculus have occured in Maple.