In tensor calculation, I need to do the following thing:

Define a derivative operator Drv[fun,i], such that

Drv[f[i],j]=f[i,j] Drv[f[i,j],k]=f[i,j,k] Drv[f[i]**g[j],k]=f[i,k]**g[j]+f[i]**g[j,k] 

UPDATE

I think this question in fact is quite important to me, and the answer, however is not quite complete, thus I will try to make the question more clearly.

The basic setting is that we want to define a derivative rule for the operator NonCommutativeMultiply, recall what's a derivative, call Drv:

1. To do the derivation, we must know who is/isn't a function with respect to the derivative variable, since Drv acts on them differently. Thus as first step, it should be declare a set of functions that are real functions, let's say f, g, h.
2. The basic rule for a derivative is that:

• Linearity: Drv[c**f]=c**Drv[f] and Drv[f+g]=Drv[f]+Drv[g];
• Distributive: Drv[f**g]=Drv[f]**g+f**Drv[g];

3. for multivariable functions, we just write $Drv[f(x_1,x_2,...,x_n),x_k]$ as $f[k]$, and $Drv[Drv[f(x_1,x_2,...,x_n),x_i],x_j]$ as $f[i,j]$ and so on. For example: $$ Drv[f,i]=f[i]\\ Drv[f[i],j]=f[i,j]\\ Drv[f[i,j],k]=f[i,j,k] $$ the properties of derivative is just reads:

linearity $$ Drv[c**f,i]=c**Drv[f,i]=c**f[i], Drv[f[i]+g,j]=f[i,j]+g[j] $$ for c is a function which is independent on $x_i$. But when $c$ is a real number, $$ Drv[2 f,i]=2f[i] $$ distributive $$ Drv[f**g[i]**h[j,k],l]=f[l]**g[i]**h[j,k]+f**g[i,l]**h[j,k]+f**g[i]**h[j,k,l]. $$

In tensor calculation, I need to do the following thing:

Define a derivative operator Drv[fun,i], such that

Drv[f[i],j]=f[i,j] Drv[f[i,j],k]=f[i,j,k] Drv[f[i]**g[j],k]=f[i,k]**g[j]+f[i]**g[j,k] 

UPDATE

I think this question in fact is quite important to me, and the answer, however is not quite complete, thus I will try to make the question more clearly.

The basic setting is that we want to define a derivative rule for the operator NonCommutativeMultiply, recall what's a derivative, call Drv:

1. To do the derivation, we must know who is/isn't a function with respect to the derivative variable, since Drv acts on them differently. Thus as first step, it should be declare a set of functions that are real functions, let's say f, g, h.
2. The basic rule for a derivative is that:

• Linearity: Drv[c**f]=c**Drv[f] and Drv[f+g]=Drv[f]+Drv[g];
• Distributive: Drv[f**g]=Drv[f]**g+f**Drv[g];

3. for multivariable functions, we just write $Drv[f(x_1,x_2,...,x_n),x_k]$ as $f[k]$, and $Drv[Drv[f(x_1,x_2,...,x_n),x_i],x_j]$ as $f[i,j]$ and so on. For example: $$ Drv[f,i]=f[i]\\ Drv[f[i],j]=f[i,j]\\ Drv[f[i,j],k]=f[i,j,k] $$ the properties of derivative is just reads:

linearity $$ Drv[c**f,i]=c**Drv[f,i]=c**f[i], Drv[f[i]+g,j]=f[i,j]+g[j] $$ for c is a function which is independent on $x_i$. But when $c$ is a real number, $$ Drv[2 f,i]=2f[i] $$ distributive $$ Drv[f**g[i]**h[j,k],l]=f[l]**g[i]**h[j,k]+f**g[i,l]**h[j,k]+f**g[i]**h[j,k,l]. $$

UPDATE FOR 1st Answer

1. could you just make some explanation for your code?
2. I want drv[f[i,j],j] output f[i,j,j] rather than f[i,{j,2}]. I think this is easy to do by flatten, but since I don't understand your code, I can't do it myself.

In tensor calculation, I need to do the following thing:

Define a derivative operator Drv[fun,i], such that

Drv[f[i],j]=f[i,j] Drv[f[i,j],k]=f[i,j,k] Drv[f[i]**g[j],k]=f[i,k]**g[j]+f[i]**g[j,k] 

In tensor calculation, I need to do the following thing:

Define a derivative operator Drv[fun,i], such that

Drv[f[i],j]=f[i,j] Drv[f[i,j],k]=f[i,j,k] Drv[f[i]**g[j],k]=f[i,k]**g[j]+f[i]**g[j,k] 

UPDATE

I think this question in fact is quite important to me, and the answer, however is not quite complete, thus I will try to make the question more clearly.

The basic setting is that we want to define a derivative rule for the operator NonCommutativeMultiply, recall what's a derivative, call Drv:

1. To do the derivation, we must know who is/isn't a function with respect to the derivative variable, since Drv acts on them differently. Thus as first step, it should be declare a set of functions that are real functions, let's say f, g, h.
2. The basic rule for a derivative is that:

• Linearity: Drv[c**f]=c**Drv[f] and Drv[f+g]=Drv[f]+Drv[g];
• Distributive: Drv[f**g]=Drv[f]**g+f**Drv[g];

3. for multivariable functions, we just write $Drv[f(x_1,x_2,...,x_n),x_k]$ as $f[k]$, and $Drv[Drv[f(x_1,x_2,...,x_n),x_i],x_j]$ as $f[i,j]$ and so on. For example: $$ Drv[f,i]=f[i]\\ Drv[f[i],j]=f[i,j]\\ Drv[f[i,j],k]=f[i,j,k] $$ the properties of derivative is just reads:

linearity $$ Drv[c**f,i]=c**Drv[f,i]=c**f[i], Drv[f[i]+g,j]=f[i,j]+g[j] $$ for c is a function which is independent on $x_i$. But when $c$ is a real number, $$ Drv[2 f,i]=2f[i] $$ distributive $$ Drv[f**g[i]**h[j,k],l]=f[l]**g[i]**h[j,k]+f**g[i,l]**h[j,k]+f**g[i]**h[j,k,l]. $$