# Tagged Questions

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### Properties and notation of third-order (and higher) partial-derivatives

This question has been bothering me for quite a while and I still haven't found a satisfying answer anywhere on the internet or in any of my books (which may not be that advanced, mind you...). Since ...
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### Characterization of the derivative as a tensor field

I was thinking about the derivative, and I wanted to make sure I’m thinking about it the right way. Suppose we have a $C^{\infty}$ function $f: {V}\to \mathbb{R}$, where $V$ is a finite-dimensional ...
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### What exactly do the terms of $(f \circ g)'''$ mean?

Say, $g: X\to Y$ and $f: Y\to Z$ are smooth. One can find $(f \circ g)'''$ by using the Faà di Bruno's formula: $$(f \circ g)''' =(f'''\circ g)(g')^3 + 3(f''\circ g)g'g'' + (f'\circ g)g'''$$ But my ...
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### Eigenvalues of a second derivative

I have a function f(r) that describes a Gaussian random field. A second derivative can be formed $\nabla_i \nabla_j f(r)$. I am looking at a paper that claims that in finding the extremum, the ...
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### Differentiation of a vector (in index notation) with respect to its square

I have a formula which contains the derivative $\partial_{z^2}$(with respect to the square of a vector $z^\mu$). However, as the functions inside the formula might not directly depend on $z^2$ but on ...
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### Is $H^{\alpha}_{ij,k}=H^{\alpha}_{ik,j}$ for submanifolds in $R^n$?

Consider $\Sigma^n\subset {\bf R}^{n+p}$ a submanifold and $\{e_i, e_\alpha\}$ an orthonormal frame where the Greek indexes stand for the normal vectors. Then define \$H^{\alpha}_{ij,k}=e_k\langle ...