Intuition on second order partial derivatives

Inspired by smooth submanifolds of $\mathbb{R}^n$, I am looking for a good geometric way to think of second order partial derivatives of a locally smooth function $f:\mathbb{R}^n \rightarrow \mathbb{R}$.

To clarify, I do not want to think of the first order partial derivatives as stand-alone functions $\mathbb{R}^n \rightarrow \mathbb{R}$, of which we then again take first-order derivatives, but I would like to see the connection to the original function.

Thanks.

• 2nd order derivatives are (modulo a correction factor) an indication on how (into which direction and to which extend) the graph of the function curves. – Thomas Apr 13 '14 at 12:14
• @Thomas, I know this fact (and the theory behind it as learnt in calculus), however I am having difficulty imagining visually why this would be true (or better - obvious). – Pandora Apr 13 '14 at 12:20
• $R^2$ surfaces embedded in 3-space $r t - s^2 =0$ makes the surface flat. Can that at first be be intuitively understood? – Narasimham Jun 26 '15 at 15:55

We will take some topological space. This space gives us a relationship between points and neighbourhoods (which will come in handy based on the definition of a smooth manifold). A manifold is this space which looks and acts like Euclidean space on $n-$dimensional Euclidean space ($\mathbb{R}^n$). This happens locally on the manifold (hence the relation with neighbourhoods). By definition, we can do calculus locally.
Take a function which takes vectors from our $n-$dimensional space, and gives us a number. The manifold possesses the smoothness property, so we know that the first and second derivatives exist. One way you can think of the derivatives is to consider the graph of the function like a section on the surface. For the first derivative, as we perform the linear transformation from $\mathbb{R}^n$ to $\mathbb{R}$, a translation takes place across the surface. The graph of the linear transformation that gives the best approximation via a "line-like" function to the graph of our original function is the derivative. The second derivative can now be thought of as taking the graph and hollowing/bending it to test its concavity/curvature.