Gradient of a vector

[Alexander Lindsay]

I've been using the capabilities in sympy.vector for generating MMS solutions for verifying FEM in MOOSE. I'm curious whether sympy supports calculation of the gradient of a vector field (e.g. with the result being a tensor)? It doesn't appear to me that it does, but I'm hoping I overlooked something. Below is the sample code I'd like to be able to execute and the corresponding backtrace. If the ability to take the gradient of a vector doesn't exist, I'm curious whether it would be difficult to add? If it's not too much anticipated work, I'd be willing to take a stab at it.

from sympy.vector import gradient, CoordSys3D
R = CoordSys3D('R')
u = R.x*R.y * (R.i + R.j)
gradient(u)

---------------------------------------------------------------------------
ValueError                                Traceback (most recent call last)
<ipython-input-49-c095794df196> in <module>
      5 u = R.x*R.y * (R.i + R.j)
      6 
----> 7 gradient(u)

~/miniconda3/envs/moose-boost/lib/python3.7/site-packages/sympy/vector/operators.py in gradient(scalar_field, coord_sys, doit)
    317 
    318         if doit:
--> 319             return (vx * i + vy * j + vz * k).doit()
    320         return vx * i + vy * j + vz * k
    321     else:

~/miniconda3/envs/moose-boost/lib/python3.7/site-packages/sympy/vector/basisdependent.py in doit(self, **hints)
    169         """Calls .doit() on each term in the Dyadic"""
    170         doit_components = [self.components[x].doit(**hints) * x
--> 171                            for x in self.components]
    172         return self._add_func(*doit_components)
    173 

~/miniconda3/envs/moose-boost/lib/python3.7/site-packages/sympy/vector/basisdependent.py in <listcomp>(.0)
    169         """Calls .doit() on each term in the Dyadic"""
    170         doit_components = [self.components[x].doit(**hints) * x
--> 171                            for x in self.components]
    172         return self._add_func(*doit_components)
    173 

~/miniconda3/envs/moose-boost/lib/python3.7/site-packages/sympy/core/decorators.py in __sympifyit_wrapper(a, b)
     89                 if not hasattr(b, '_op_priority'):
     90                     b = sympify(b, strict=True)
---> 91                 return func(a, b)
     92             except SympifyError:
     93                 return retval

~/miniconda3/envs/moose-boost/lib/python3.7/site-packages/sympy/core/decorators.py in binary_op_wrapper(self, other)
    127                     if f is not None:
    128                         return f(self)
--> 129             return func(self, other)
    130         return binary_op_wrapper
    131     return priority_decorator

~/miniconda3/envs/moose-boost/lib/python3.7/site-packages/sympy/vector/basisdependent.py in __mul__(self, other)
     34     @call_highest_priority('__rmul__')
     35     def __mul__(self, other):
---> 36         return self._mul_func(self, other)
     37 
     38     @_sympifyit('other', NotImplemented)

~/miniconda3/envs/moose-boost/lib/python3.7/site-packages/sympy/vector/vector.py in __new__(cls, *args, **options)
    412 
    413     def __new__(cls, *args, **options):
--> 414         obj = BasisDependentMul.__new__(cls, *args, **options)
    415         return obj
    416 

~/miniconda3/envs/moose-boost/lib/python3.7/site-packages/sympy/vector/basisdependent.py in __new__(cls, *args, **options)
    257         # Make sure incompatible types weren't multiplied
    258         if count > 1:
--> 259             raise ValueError("Invalid multiplication")
    260         elif count == 0:
    261             return Mul(*args, **options)

ValueError: Invalid multiplication

[Aaron Meurer]

Hi.

I don't know the answer to your question, as I am not familiar enough
with the vector or tensor modules, but if you don't get an answer here
I would suggest opening an issue about this on the issue tracker.

Aaron Meurer

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[Alan Bromborsky]

You may want to look at the following link -

https://github.com/pygae/galgebra

Also these docs -

https://galgebra.readthedocs.io/en/latest/galgebra_guide.html#Basic-Multilinear-Function-Class-Functions

[S.Y. Lee]

SymPy defines gradient only for scalar valued function.

Do you mean that gradient of vector valued function should return other stuff like jacobian?

[Nicolas Guarin]

You could check a module that I wrote in Continuum Mechanics

https://continuum-mechanics.readthedocs.io/en/latest/

It already has the gradient of a vector for orthogonal coordinates.

[Alexander Lindsay]

On Sat, Apr 25, 2020 at 6:39 AM S.Y. Lee <syle...@gmail.com> wrote:

SymPy defines gradient only for scalar valued function.

Do you mean that gradient of vector valued function should return other stuff like jacobian?

I would like the gradient of a vector valued function to return the Jacobian yes, or the transpose of the Jacobian, I don't really care. I know different people prefer different conventions.

Thanks Alan and Nicolas for sharing those packages; I will look into them. The ideal solution would be to do everything in sympy in order for us to minimize our number of package dependencies, but if sympy can't do this then we very well might use one of the packages you shared.

Aaron, I will go ahead and open a ticket, thanks.

Alex

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[Aaron Meurer]

SymPy does support taking derivatives with respect to matrices, via
diff(). Is this what you are looking for?

Aaron Meurer

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[Alexander Lindsay]

grad(vec) should return a matrix (like the Jacobian https://en.wikipedia.org/wiki/Jacobian_matrix_and_determinant)

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