Ciro Santilli @cirosantilli 37

One of the defining properties of algebraic structure with two operations such as ring and field:This property shows how the two operations interact.

We define this as the functional equation:It is a bit like cauchy's functional equation but with multiplication instead of addition.

Existence and uniqueness results are fundamental in mathematics because we often define objects by their properties, and then start calling them "the object", which is fantastically convenient.

But calling something "the object" only makes sense if there exists exactly one, and only one, object that satisfies the properties.

One particular context where these come up very explicitly is in solutions to differential equations, e.g. existence and uniqueness of solutions of partial differential equations.

Same remarks as Section "Exam".

Mostly on vintage electronics. Lots of focus on microwave, which he has worked a lot with.

Has been going wild with restoration and reverse engineering of the Apollo moon mission.

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.