Question: Help with Computing Jacobian and Derivatives of Time Dependent Vectors / Functions


I'm a Maple newbie and I am struggling to figure out how to compute the following:

1) Jacobian of a time dependent function:

For example I have a vector Phi = (x1(t),x2(t))
and I wish to compute the Jacobian of this vector with respect to x1(t) and x2(t) and I want to maintain the explicit reference to these entries as time dependent components.

What I can do:

If I remove the reference to the time dependency I can compute what I require, however, in a follow up computation, I need to compute a time derivative of the resultant Jacobian wrt to time explicitly.

(i.e. with the following commands I can compute the Jacobian of a time independent problem

Phi := <x1,x2>;

Now what I want to do is:

Phi := <x1(t),x2(t)>;

But I receive the following error:

"Error, invalid input: VectorCalculus:-Jacobian expects its 2nd argument, v, to be of type list(name), but received [x1(t), x2(t)]"

Can anyone help me figure this out?

2) Differentiate a time dependent function with respect to a derivative

I come across this problem while trying to solve dynamics problems using Lagrangian mechanics.

In this case I try and compute the derivative of a function wrt a time dependent derivative coordinate

(i.e.  Let's say I have a function L(t) = theta_dot(t) + 10, where theta_dot = d(theta)/dt
I want to compute d(L(t))/d(theta_dot) which equals 1 )

Not sure where to start with this problem without a change of variables.

I'm using Maple v9.5

Any help is greatly appreciated!  :)



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