$u_{\mathrm{ij} \cdot \cdot \cdot \mathrm{\ℓ}}^{\mathrm{\α}}\left(\mathrm{\σ}\right) \= \frac{{\partial }^k s^{\mathrm{\α} }\left(x\right)}{\partial x^i \partial x^i\cdot \cdot \cdot \partial x^{\mathrm{\ℓ}}}$ and $1\le i\le j\cdot \cdot \cdot \le \mathrm{\ℓ}\mathit{\le }\mathit{ }$dim$\left(M\right)$.

Then the total derivative of the jet coordinate $u_{\mathrm{ij} \cdot \cdot \cdot \ell }^{\mathrm{\alpha }}$ with respect to the independent variable $x^k$ is ${\mathrm{D}}_k\left(u_{\mathrm{ij} \cdot \cdot \cdot \ell }^{\mathrm{\alpha }}\right) \= u_{\mathrm{ij} \cdot \cdot \cdot \mathrm{\ℓk}}^{\mathrm{\α}}$. If $f \= f$$\left(x^i\, u^{\mathrm{\α}}\, u_{i_{}}^{\mathrm{\α}}\, u_{i_{}j}^{\mathrm{\α}}..\.\right)$ is a function on jet space, then by the chain rule

 ${\mathrm{D}}_k f \= \frac{\partial f }{\partial x^k} \+ u_k^{\mathrm{\α}}\frac{\partial f }{\partial u^{\mathrm{\α}}} \+ u_{\mathrm{ik}}^{\mathrm{\alpha }}\frac{\partial f }{\partial u_i^{\mathrm{\α}}} \+ u_{\mathrm{ijk}}^{\mathrm{\alpha }}\frac{\partial f }{\partial u_{\mathrm{ij}}^{\mathrm{\α}} } \+ \cdot \cdot \cdot$

Similarly, the total derivatives of differential forms ${\mathrm{dx}}^i\,$d$u_{\mathrm{ij} \cdot \cdot \cdot \ell }^{\mathrm{\alpha }}$and the contact form ${\mathrm{\Θ}}_{\mathrm{ij}\cdot \cdot \cdot \mathrm{\ℓ}}^{\mathrm{\α}}$=${\mathrm{du}}_{\mathrm{ij} \cdot \cdot \cdot \ell }^{\mathrm{\alpha }}$- $u_{\mathrm{ij} \cdot \cdot \cdot \mathrm{\ℓm}}^{\mathrm{\alpha }}{\mathrm{dx}}^m$with respect to the independent variable $x^k$are given by

${\mathrm{D}}_k\left({\mathrm{dx}}^i\right) \= 0\,$  ${\mathrm{D}}_k$(d$u_{\mathrm{ij} \cdot \cdot \cdot \ell }^{\mathrm{\alpha }} \) \= d{u}_{\mathrm{ij}\cdot \cdot \cdot \mathrm{\ℓk}}^{\mathrm{\α}}$and  ${\mathrm{D}}_k$$\left({\mathrm{\Θ}}_{\mathrm{ij}\cdot \cdot \cdot \mathrm{\ℓ}}^{\mathrm{\α}}\right) \= {\mathrm{\Θ}}_{\mathrm{ij}\cdot \cdot \cdot \mathrm{\ℓk}}^{\mathrm{\alpha }}$.

If ${\mathrm{\ω}}_1$and ${\mathrm{\ω}}_2$are 2 differential forms on jet space, then ${\mathrm{D}}_k\left({\mathrm{\ω}}_1 \wedge {\mathrm{\ω}}_2 \right) \=$ ${\mathrm{D}}_k\left({\mathrm{\ω}}_1\right)\wedge {\mathrm{\ω}}_2 \+ {\mathrm{\ω}}_1\wedge {\mathrm{D}}_k\left({\mathrm{\ω}}_2\right)$. One can summarize all these formulas by saying that total differentiation with respect to the independent variable $x^k$coincides with Lie differentiation with respect to the total vector field ${\mathrm{D}}_k$. Thus the total derivative with respect to $x^k$ commutes with the exterior derivative, the horizontal exterior derivative, and the vertical exterior derivative, that is,

${\mathrm{D}}_k \circ d \= d\circ {\mathrm{D}}_k \, {\mathrm{D}}_k \circ d{ }_H\= d_H\circ {\mathrm{D}}_k$ and ${\mathrm{D}}_k \circ d{ }_V\= d_V\circ {\mathrm{D}}_k$.

Example 1.

First initialize the jet space for two independent variables $\left(x\, y\right)$and two dependent variables $\left(u\, v\right)$and prolong it to order 3.

Recall that $u\left[1\,2\, 2\right]$represents the mixed 3rd derivative of $u\,$once with respect to $x$and twice with respect to $y\.$The total derivative of $u\left[1\,2\, 2\right]$ with respect to $x$ is $u\left[1\,1\, 2\, 2\right]$which represents the 4th derivative of $u$, twice with respect to $x$and twice with respect to $y$

The total derivative of $u\left[1\,2\,2\right]$ with respect to $y$ is $u\left[1\,2\,2\,2\right]$ which represents the 4th derivative of $u$, once with respect to $x$ and 3 times with respect to $y$.

In place of the independent variables $x$ or $y$the integer 1 or 2 can be used.

Here is a general formula for the total derivative of a function with dependencies on the 2-jet of $u\.$

The total derivative satisfies the usual rules of differentiation.

Multiple total derivatives can also be calculated by using TotalDiff. We differentiate $u\left[2\right]$2 times with respect to $x$ and 3 times with respect to $y$ to get $u\left[1\,1\,2\,2\,2\right]$.

Example 2.

Total differentiation extends to differential forms and contact forms on jet spaces.

Example 3.

The DifferentialGeometry package supports an alternative jet notation. For example, if there are 2 independent variables $\left(x\,y\right)$, then$u\left[1\,2\right]$ would now represent the 3rd mixed partial derivative of $u\,$once with respect to $x$and twice with respect to $y\.$

Revert to the default jet notation.