Ciro Santilli @cirosantilli  40

Only certain battery voltages exist, because this voltage depends intrinsically on the battery's chemical composition.

learn.sparkfun.com/tutorials/battery-technologies/all (CC BY-SA) has a very good summary list, reordered from lowest to highest voltage:

Bibliography:

TODO what is the point of them? Why not just sum over every index that appears twice, regardless of where it is, as mentioned at: www.maths.cam.ac.uk/postgrad/part-iii/files/misc/index-notation.pdf.

Vectors with the index on top such as $x^i$ are the "regular vectors", they are called covariant vectors.

Those in indices on bottom are called contravariant vectors.

It is possible to change between them by Raising and lowering indices.

The values are different only when the metric signature matrix is different from the identity matrix.

Then a specific metric is involved, sometimes we want to automatically add it to products.

E.g., in a context considering the common Minkowski inner product matrix where the $\eta$ 4x4 matrix and $\mu$ is a vector in ${\mathbb{R}}^4$which leads to the change of sign of some terms.

The Einstein summation convention works will with partial derivatives and it is widely used in particle physics.

In particular, the divergence and the Laplacian can be succinctly expressed in this notation:

In order to express partial derivatives, we must use what Ciro Santilli calls the "partial index partial derivative notation", which refers to variables with indices such as $x_0$, $x_1$, $x_2$, ${\partial }_0$, ${\partial }_1$ and ${\partial }_2$ instead of the usual letters $x$, $y$ and $z$.

See: eigenvalues and eigenvectors.

Every invertible matrix $M$ can be written as:where:Note therefore that this decomposition is unique up to swapping the order of eigenvectors. We could fix a canonical form by sorting eigenvectors from smallest to largest in the case of a real number.

Intuitively, Note that this is just the change of basis formula, and so:

See: eigenvalues and eigenvectors.

See also: e-learning website.

Whenever Ciro Santilli walks in front of a school and sees the tall gates it makes him sad. Maybe 8 year olds need gates. But do we need to protect 15 year olds like that? Students should be going out to see the world, both good and evil not hiding from it! We should instead be guiding them to the world. But instead, we are locking them up in brainwashing centers.

Video "The Purpose of Education by Noam Chomsky (2012)" puts it well, education can be either be:He has spoken about that infinitely, e.g. from when he was thin: www.youtube.com/watch?v=JVqMAlgAnlo

Bibliography: