JetCalculus[TotalDiff] - take the total derivative of an expression, a differential form or a contact form
f - a Maple expression, a differential form or a bi-form
v - an independent variable, a positive integer or a list of positive integers
The operation of total differentiation is a fundamental one in the study of jet spaces and their application to differential equations and the calculus of variations. Informally, total differentiation of a function on a jet space with respect to an independent variable is the same as ordinary differentiation with respect to that variable if the jet coordinates are treated temporarily as functions of the independent variables.
Let π:E→M be a fiber bundle with base dimension n and fiber dimension m and let πk:JkE →M be the k-th jet bundle. Introduce local coordinates (xi, uα, uiα, uijα, ..., uij ⋅⋅⋅ ℓα, ...) where, as usual, if s:M→E is a section and σ=jksx:M→E is the k-jet of s, then
uij ⋅⋅⋅ ℓασ = ∂k sα x∂xi ∂xi⋅⋅⋅∂xℓ and 1≤i≤j⋅⋅⋅≤ℓ≤ dimM.
Then the total derivative of the jet coordinate uij ⋅⋅⋅ ℓα with respect to the independent variable xk is Dkuij ⋅⋅⋅ ℓα = uij ⋅⋅⋅ ℓkα. If f = fxi, uα, uiα, uijα... is a function on jet space, then by the chain rule
Dk f = ∂f ∂xk + ukα∂f ∂uα + uikα∂f ∂uiα + uijkα∂f ∂uijα + ⋅⋅⋅
Similarly, the total derivatives of differential forms dxi, duij ⋅⋅⋅ ℓα and the contact form Θij⋅⋅⋅ℓα = duij ⋅⋅⋅ ℓα - uij ⋅⋅⋅ ℓmαdxm with respect to the independent variable xk are given by
Dkdxi = 0, Dk(duij ⋅⋅⋅ ℓα ) = duij⋅⋅⋅ℓkα and DkΘij⋅⋅⋅ℓα = Θij⋅⋅⋅ℓkα.
If ω1 and ω2 are 2 differential forms on jet space, then Dkω1 ∧ω2 = Dkω1∧ω2 + ω1∧Dkω2. One can summarize all these formulas by saying that total differentiation with respect to the independent variable xk coincides with Lie differentiation with respect to the total vector field Dk. Thus the total derivative with respect to xk commutes with the exterior derivative, the horizontal exterior derivative, and the vertical exterior derivative, that is,
Dk ∘d = d∘Dk , Dk ∘d H= dH∘Dk and Dk ∘d V= dV∘Dk.
If f is a function or differential form on a jet space and v an independent variable, then TotalDiff(f, v) calculates the total derivative of f with respect to v. If v is a list of r positive integers, then the r-fold iterated total derivative is calculated.
The command TotalDiff is part of the DifferentialGeometry:-JetCalculus package. It can be used in the form TotalDiff(...) only after executing the commands with(DifferentialGeometry) and with(JetCalculus), but can always be used by executing DifferentialGeometry:-JetCalculus:-TotalDiff(...).
First initialize the jet space for two independent variables x, y and two dependent variables u, v and prolong it to order 3.
DGsetup([x, y], [u, v], E, 3):
Recall that u1,2, 2 represents the mixed 3rd derivative of u, once with respect to x and twice with respect to y.The total derivative of u1,2, 2 with respect to x is u1,1, 2, 2which represents the 4th derivative of u, twice with respect to x and twice with respect to y
The total derivative of u1,2,2 with respect to y is u1,2,2,2 which represents the 4th derivative of u, once with respect to x and 3 times with respect to y.
TotalDiff(u[1, 2, 2], x);
TotalDiff(u[1, 2, 2], y);
In place of the independent variables x or y the integer 1 or 2 can be used.
TotalDiff(u[1, 2, 2], 1);
TotalDiff(u[1, 2, 2], 2);
Here is a general formula for the total derivative of a function with dependencies on the 2-jet of u.
vars := x, y, u, u, u, u[1, 1], u[1, 2], u[2, 2]:
F⁡x,y,u,u1,u2,u1,1,u1,2,u2,2⁢will now be displayed as⁢F
The total derivative satisfies the usual rules of differentiation.
TotalDiff(y*u[1, 0]*v[0, 1], y);
f := simplify(TotalDiff(arctan(u[0, 3]/v[2, 0]), x));
Multiple total derivatives can also be calculated by using TotalDiff. We differentiate u2 2 times with respect to x and 3 times with respect to y to get u1,1,2,2,2.
TotalDiff(u, [1, 1, 2, 2, 2]);
TotalDiff(u*v, [1, 1, 1]);
Total differentiation extends to differential forms and contact forms on jet spaces.
TotalDiff(du &w dv, y);
TotalDiff(du, [1, 1, 2]);
TotalDiff(du &w dv, [1, 2]);
The DifferentialGeometry package supports an alternative jet notation. For example, if there are 2 independent variables x,y, then u1,2 would now represent the 3rd mixed partial derivative of u, once with respect to x and twice with respect to y.
DGsetup([x, y], [u, v], J, 3):
TotalDiff(u[1, 2], x);
TotalDiff(u[1, 2], y);
Revert to the default jet notation.
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