Ciro Santilli @cirosantilli 37

Total derivative Updated 2025-07-16

The total derivative of a function assigns for every point of the domain a linear map with same domain, which is the best linear approximation to the function value around this point, i.e. the tangent plane.

E.g. in 1D:

T o t a l d er i v a t i v e = D [f (x_{0})] (x) = f (x_{0}) + \frac{\partial f}{\partial x} (x_{0}) \times x

(1)

and in 2D:

D [f (x_{0}, y_{0})] (x, y) = f (x_{0}, y_{0}) + \frac{\partial f}{\partial x} (x_{0}, y_{0}) \times x + \frac{\partial f}{\partial y} (x_{0}, y_{0}) \times y

(2)

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Euler number Updated 2025-07-16

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Go (game) Updated 2025-07-16

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M,n,k game Updated 2025-07-16

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Newton dot notation Updated 2025-07-16

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Order of a differential equation Updated 2025-07-16

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Order of the highest derivative that appears.

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Partial derivative Updated 2025-07-16

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Biconvex spherical lens Updated 2025-07-16

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Focal length

Each side is a sphere section. They don't have to have the same radius, they are still simple to understand with different radiuses.

The two things you have to have in mind that this does are:

converges parallel light to a point at center at distance $f$ known as the focal length.
This is for example why you can use lenses to burn things with Sun rays, which are basically parallel.
Conversely, if the input is a point light source at the focal length, it gets converted into parallel light.
image formation: it converges all rays coming from a given source point to a single point image. This amplifies the signal, and forms an image at a plane.
The source image can be far away, and the virtual image can be close to the lens. This is exactly what we need for a camera.
For each distance on one side, it only works for another distance on the other side. So when we set the distance between the lens and the detector, this sets the distance of the source object, i.e. the focus. The equation is:
$\frac{1}{f} = \frac{1}{S _{1}} + \frac{1}{S _{2}}$
(1)
where $S_{1}$ and $S_{2}$ are the two distances.

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Bicycle tire sizes Updated 2025-07-16

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The images you have to have in mind are:

Yes, Sheldon he has separate American and British English versions of pages!!!

For example, Kross bicycle (2017) had a Schwalbe tyre with markings:

42-622 (28 x 1.60, 700x40C)

When inflated, the tires were about 3.5cm wide as measured with a ruler.

And the Mavic A319 rim had markings:

622x19C

In this:

ISO (Etrto): 42-622. So:
- 42 is the inner rim width. The actual inflated tire is going to be even wider.
- 622 is the bead seat diameter. The actual inflated tire is going to be even wider.
imperial: 28 x 1.60
French: 700x40C:
- meaning of the "C" asked at: bicycles.stackexchange.com/questions/16190/what-does-the-c-in-bicycle-tire-size-mean
  - www.sheldonbrown.com/tire-sizing.html#french says A is larger than B which is larger than C, and C means 622 mm

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Chain rule Updated 2025-07-16

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Here's an example of the chain rule. Suppose we want to calculate:

\frac{d}{d x} e^{2 x}

(1)

So we have:

f (x) = e^{x} g (x) = 2 x

(2)

and so:

f^{'} (x) = e^{x} g^{'} (x) = 2

(3)

Therefore the final result is:

f^{'} (g (x)) g^{'} (x) = e^{2 x} 2 = 2 e^{2 x}

(4)

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Taxonomic rank Updated 2025-07-16

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Naming taxonomic ranks like genus, domain, etc. is a fucking waste of time, only useful before we developed molecular biology.

All that matters is the tree of clades with examples of species in each clade, and common characteristics shared by the clade.

And with molecular biology, we can build those trees incredibly well for extant species. When extinct species are involved however, things get more complicated.

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Bilinear map Updated 2025-07-16

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Linear map of two variables.

More formally, given 3 vector spaces X, Y, Z over a single field, a bilinear map is a function from:

f : X \times Y \to Z

(1)

that is linear on the first two arguments from X and Y, i.e.:

f (a_{1} x_{1} + a_{2} x_{2}, y) = a_{1} f (x_{1}, y) + a_{2} f (x_{2}, y)

(2)

Note that the definition only makes sense if all three vector spaces are over the same field, because linearity can mix up each of them.

The most important example by far is the dot product from

R^{n} \times R^{n} \to R

, which is more specifically also a symmetric bilinear form.

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Bill Gates Updated 2025-07-16

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The enemy?

You must watch this: Video "Bill Gates vs Steve Jobs by Epic Rap Battles of History (2012)".

It does not matter how many trillions you donate to charity, Bill. If you want to prove your point, make MS Word free and open source and port it to Linux. And then Window implements POSIX-compatible APIs and then deprecate non-POSIX APIs.