Hello community, firstly I wish you a Merry Christmas since it's the 2nd day of Christmas celebration.

Then I've got a question:

When I want to differentiate a function with more than 1 variable, then, for the total differential, I need to do the following:

Let f(x1, x2, ....., xn) be a function of n variables. Then

Df/Dx1 + Df/Dx2 + ... + Df/Dxn = df/d(x1,...,xn)

(while D stands for the partial derivative symbol).

Why is it NOT

sqrt [(Df/Dx1)² + ... + (Df/Dxn)²] ???

Since, when I consider it in a geometrical sense, for example in 2 dimensions,

then I've got the change in x-direction and at the same time in y-direction.

Now these changes would be, as far as I know, added vectorically, i.e.

sqrt ((dx)² + (dy)²) , given a Euclidean distance between two points.

Could somebody explain to me where my mstake is, please? And further, one could also choose still other metrics or norms, which would yield yet other formulas.

Thanks a lot