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Jan 20, 2019 · Notion 2 is available now.

Feb 10, 2019 · Notion Music, the company behind Notion 2, is available as a free standalone app for OS X and Windows, as well as a Music Notation software for iOS and Android.Q:

How can I calculate the Gauss Curvature?

Let $f(x,y)$ be a continuous function defined on $[a,b] \times[c,d] \subset \mathbb{R}^2$. Let $g(u,v) = \frac{\partial f(x,y)}{\partial u} \frac{\partial f(x,y)}{\partial v}$ be a gauss-type curvature on $[a,b] \times [c,d]$. Given that $g(u,v)$ is symmetric, determine the Gauss curvature $K_g(u,v)$ of the surface $f(x,y) = c$.

The book does not give a general formula for the Gauss curvature, but only one for a surface with an upper flat of the form $x^2 + y^2 — a^2 = 1$. How could I generalize this to arbitrary surfaces?

A:

For surfaces with an upper flat I use the formula for a locally defined and positive curvature function:

If $(u,v)$ is a point on the surface (with $u \in (a,b),v \in (c,d)$) with a quadratic differential

$$Q(u,v) := \det \begin{pmatrix} u & v \\ \frac{\partial^2 f}{\partial u^2} & \frac{\partial^2 f}{\partial u\partial v} \\ \frac{\partial^2 f}{\partial v^2} & \frac{\partial^2 f}{\partial v\partial u}\end{pmatrix}$$

then the corresponding Gauss-curvature is given by:

$$K_g(u,v) = \frac{Q(u,v)}{\det \begin{pmatrix} u & v \\ \frac{\partial^2 f}{\partial u^2} & \frac{\partial^2 f}{\partial u\partial v}

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There are dozens of them, and they’re in every town. But, in general, they haven’t been doing so well lately.

The Associated Press reports that of the nearly 800 brick-and-mortar bookstores in the U.S., just 42 are still open. Some are remaining open due to loyal customers, or because they’re home to lots of local attractions. But many are struggling to find customers, and they’re usually the small-town shops that you can’t just drive through.

People still love to read, of course, but they don’t seem to be eager to hit the mall to do so, at least not in significant numbers.

But Amazon is trying to change that. It’s reportedly planning to open a book-rental shop within the next year, in a community where it stores a large number of books.

The proposed location for its first store is in Seattle’s University Village, a business area that also includes a supermarket and a pharmacy.

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Amazon started a bookstore partnership with Borders in 2012, so this probably isn’t too big of a step. And if it succeeds, it’s just one more data point showing that customers prefer the convenience of online shopping to going out to a bookstore.Q:

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