All these concepts belong to the toolbox of physicists. I read them quite often on our forum and their usage is sometimes a bit confused. Physicists learn how to apply them, but occasionally I get the impression, that the concepts behind are forgotten. So what are they? Especially when it comes to the adjoint representation […]
Let me start with a counter-question. What is a number? Before you laugh, there is more to this question as one might think. A number can be something we use to count, or more advanced an element of a field like the real numbers. Students might answer that a number is a scalar. This is […]
Important Theorems – biased, of course Implicit Function Theorem [1] Jacobi Matrix (Chain Rule). Let ## (x_0,y_0 ) ## be a point in $$U_1 \times U_2 = \{x \in \mathbb{R}^k\,\vert \,||x-x_0||< \varepsilon_1 \} \times \{y \in \mathbb{R}^m\,\vert \,||y-y_0||< \varepsilon_2\}$$ and ## f: U_1 \times U_2 \rightarrow \mathbb{R}^m ## a function with ##f(x_0,y_0)=0## which is totally […]
Lie Derivatives A Lie derivative is in general the differentiation of a tensor field along a vector field. This allows several applications, since a tensor field includes a variety of instances, e.g. vectors, functions or differential forms. In case of vector fields we additionally get a Lie algebra structure. This is, although formulated in […]
Some Topology Whereas the terminology of vector fields, trajectories and flows almost by itself suggests its origins and physical relevance, the general treatment of vector fields, however, require some abstractions. The following might appear to be purely mathematical constructions, and I will restrict myself to a minimum, but they actually occur in modern physics: […]
Generalizations Beyond ##\mathbb{R}## and ##\mathbb{C}## As mentioned in the section of complex functions (The Pantheon of Derivatives – Part I), the main parts of defining a differentiation process are a norm and a direction. So to extend the differentiation concepts on normed vector spaces seems to be the obvious thing to do. Fréchet Derivative […]
Differentiation in a Nutshell I want to gather the various concepts at one place, to reveal the similarities between them, as they are often hidden by the serial nature of a curriculum. There are many terms and special cases, which deal with the process of differentiation. The basic idea, however, is the same in […]
First of all: What is a representation? It is the description of a mathematical object like a Lie group or a Lie algebra by its actions on another space 1). We further want this action to preserve the given structure, because its structure is exactly what we’re interested in. And this other space here should […]