Mathematica: A function that derives another function on all its arguments

[shfreeman]

Hi!

I am wondering if someone could help me write a function in Mathematica. What I would like it to do is take another function g
as its argument and return the sum of g derived to all of its arguments (and multiplied by a scalar Δxi)

$$f\left[g\left[x_1,x_2,\text{...},x_n\right]\right] = \sqrt{\overset{n}{\sum _{i=1} }\left(\frac{\partial g}{\partial x_i}\text{\Delta x}_i\right){}^2}$$

I have already managed to make such a function by brute force,
but it brakes down if I assign a value to any character (a,b,c,d,...). This is what I came up with:

$$\text{symbols}=\{a,b,c,d,e,f,g,h,i,j,k,l,m,n,o,p,q,r,s,t,u,v,w,x,y,z,A,B,C,D,\text{Ee},F,G,H,\text{Ii},J,K,L,M,N,O,P,Q,R,S,T,U,V,W,X,Y,Z,\alpha ,\beta ,\gamma ,\delta ,\varepsilon ,\zeta ,\eta ,\theta ,\iota ,\kappa ,\lambda ,\mu ,\nu ,\xi ,o,\rho ,\varsigma ,\sigma ,\tau ,\upsilon ,\varphi ,\chi ,\psi ,\omega \};$$
$$\text{abs}[\text{f\_}]\text{:=} \surd \text{Sum}\left[(D[f,\text{symbols}[[\aleph ]]]*\text{Symbol}[\Delta <>\text{ToString}[\text{symbols}[[\aleph ]]]])^2,\{\aleph ,\text{Length}[\text{symbols}]\}\right.$$

I hope someone can help me come up with a better solution.

Thanks!

 

[mmwave]

Hi!

I would be happy to help you with this. Mathematica has a built-in function called D that can take derivatives of functions with respect to their arguments. We can use this to create a function that takes in another function g and returns the sum of its derivatives with respect to all of its arguments multiplied by a scalar Δxi. Here is the code for the function:

f[g_] := Sqrt[Sum[(D[g @@ Table[x, {i, n}], x]*Δx)^2, {i, n}]]

This function takes in a function g and uses the Table function to create a list of variables x with i ranging from 1 to n. Then, it uses the D function to take the derivative of g with respect to each of these variables and multiplies it by the corresponding scalar Δxi. Finally, it returns the square root of the sum of these squared derivatives.

To use this function, you can simply input your desired function g as an argument. For example, if you wanted to use the function with the expression x^2 + y^2, you would input f[x^2 + y^2]. You can also specify the value of n (the number of arguments) and the values of Δxi by using the Table function and assigning them to x and Δx, respectively.

I hope this helps! Let me know if you have any further questions or if you would like me to explain the code in more detail. Happy coding!