Physics[Vectors][diff]  differentiate expressions taking into account the geometrical relations between curvilinear unit vectors and coordinates of different types

Calling Sequence


diff(A, x1, x2, ...)


Parameters


A



an algebraic expression

x1, x2, ...



names or functions representing the differentiation variables

geometricdifferentiation,



optional, default value is true, to take into account the geometrical relation between curvilinear coordinates and unit vectors even after setting this option to false using Setup.





Description


•

The diff command of Vectors is the same as the diff command of Physics, but for two additional things:

1.

it will take into account the geometrical relation between cartesian, cylindrical and spherical coordinates, as well as the coordinate dependence of curvilinear unit vectors. Example: , so diff(x, rho) returns even when has(x, rho) returns false.


Although taking into account relations between geometrical coordinates, unit vectors and their functional dependency, is natural when working with vectors, you can turn OFF these two differences between Physics[Vectors][diff] and Physics[diff] using the Setup command, entering Setup(geometricaldifferentiation = false). In such a case you can still differentiate taking into account the geometrical relation between curvilinear coordinates unit vectors by passing the optional argument geometricdifferentiation.


The %diff is the inert form of diff, that is: it represents the same mathematical operation while holding the operation unperformed. To activate the operation use value.

•

The result of diff is always expressed in the coordinate system of the differentiation variable. When that is ambiguous (e.g. may be cartesian or cylindrical), the ambiguity is resolved looking at the derivand, whether it is a cartesian or cylindrical vector, and when it is neither, then cartesian coordinates are used. The same approach is used when the differentiation variable is , that could be cylindrical or spherical.

•

The computation of diff(A, q) is performed as follows.


Otherwise, if is a projected vector then

3.

is reprojected in the cartesian orthonormal basis (using ChangeBasis), where unit vectors are constant;

5.

the differentiation is performed using the standard Physics[diff];

6.

the result is reprojected into the original orthonormal basis and returned.


If is a nonprojected vector or a scalar function, the task is restricted to steps 2. and 3. above.

•

For the conventions about the geometrical coordinates and vectors see Identify



Compatibility


•

The Physics[Vectors][diff] command was introduced in Maple 17.



Examples


>


 (1) 
>


 (2) 
The radial cylindrical unit vector
>


 (3) 
>


 (4) 
>


 (5) 
>


 (6) 
Note the difference when you change the order in which derivatives are computed in a 2nd order derivative
>


 (7) 
>


 (8) 
Curvilinear coordinates and related unit vectors can have functional dependency, and so can the differentiation variable. Consider for instance the radial unit vector in cylindrical coordinates as a function of the polar angle which in turn is a function of
>


 (9) 
The derivative with respect to takes into account the geometrical dependency of with respect to , while keeping, in the result, the dependency with respect to of the derivand
>


 (10) 
The derivative of with respect to uses the chain rule taking the result above into account, hence
>


 (11) 
You can turn OFF geometric differentiation using Setup
>


 (12) 
So now, for example, the differentiation above returns
>


 (13) 
and differentiating with respect to x returns 0 instead of eq.(4)
>


 (14) 
You can still use geometric differentiation by passing the optional argument geometricdifferentiation
>


 (15) 
>



