Intuition on second order partial derivatives

Question

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.

Answer 1

I may have misunderstood the question, but I'll attempt to give a geometric interpretation of what you have written:

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.

Is that what you were looking for?

Answer 2

Well, not a direct answer, but in an attempt to answer 'why this would be better?' (in one of your comments above), I'll prefer to put it in real/physical entities for intuition. A very simple example is (instantaneous) acceleration which is a second order derivative of distance covered by say a vehicle wrt time. Acceleration practically gives you useful information which could be used by its driver to take decisions (such as overtaking).

Apologies for being barely mathematical, but I feel if you really want good intuitive idea, real world examples help a lot.