^^ [[Schutz - 1 - Some Basic Mathematics|1. Some Basic Mathematics]] << [[Schutz - 1.1 - The Space R^n and its Topology| 1.1 The Space R^n and its Topology]] | [[Schutz - 1.3 - Real Analysis|1.3 Real Analysis]] >> # Mappings > **Definition of a map** > A map $f$ from a space $M$ to a space $N$ is a rule which associates with an element $x$ of $M$ a unique element $y$ of $N$. Notice that a map gives a unique $f(x)$ for every $x$ but not necessarily a unique $x$ for every $f(x)$. > This map can be written as: > $f:M\rightarrow N$ > Or > $f:x\mapsto f(x)$ > See the figure below for an example. ^b0e557 ![[Pasted image 20210122194226.png]] > **Domain, Co-domain, and Image** > Consider a [[Schutz - 1.2 - Mappings#^b0e557|map]] $f:M\rightarrow N$. The *domain* is $M$ and the *co-domain* is $N$. > Consider $S \subset M$, the elements in $N$ mapped from points of $S$ form a set $T \subset N$ called the *image* or the *range* of $S$ under $f$, denoted by $f(S)$. > Conversely, the set $S$ is called the *inverse image* of $T$, denoted by $f^{-1}(T)$. If the map is *many-to-one*, then the inverse image of a single point of $N$ is not a single point of $M$, so there is no *map* $f^{-1}$ from $N$ to $M$ (recall that a map must map each element in its domain to a unique element in its range). In this case, the symbol $f^{-1}(T)$ must be read as a single symbol, $f^{-1}(T)$ is simply a set with that name, not the image of $T$ under a map $f^{-1}$ ^12051e > **Types of Maps** > * A map $f$ is *injective* or (*one-to-one*, or an *injection*) if it maps an element in its [[Schutz - 1.2 - Mappings#^12051e|domain]] to a unique element in its [[Schutz - 1.2 - Mappings#^12051e|range]]. In this case, an inverse map $f^{-1}$ is unique. > * A map $f$ is from $M$ *into* $N$ if it is defined for all points in its domain. > * A map $f$ is called *surjective* or (*onto*, or a *surjection*) if it is *into*, and the image of its domain is isomorphic to its [[Schutz - 1.2 - Mappings#^12051e|co-domain]]. (i.e. it maps all points in its domain to all points in its range, in a many-to-one way.) > * A map $f$ is called a *bijection* if it is both an *injection* and a *surjection*. > See the figure below for a simple diagram. ^6c8b58 ![[Pasted image 20210122200039.png]] > **Continuous Map** > A map $f:M\rightarrow N$ is called continuous at $x$ in $M$ is any [[Schutz - 1.1 - The Space R^n and its Topology#^224639|open set]] of $N$ containing $f(x)$ contains the image of an open set of $M$. > Here, $M$ and $N$ are topological spaces, otherwise, we cannot define open sets and thus cannot define continuity. > More generally, $f$ is continuous on $M$ (or more simply, *continuous*) if it is continuous for all points $x \in M$. ^1aee5f > **Differentiable Functions** > If $f(x_1, x_2, \ldots, x_n)$ is a function defined on some [[Schutz - 1.1 - The Space R^n and its Topology#^224639|open region]] $S$ of $\mathbb{R}^n$, then it is said to be *differentiable of class $C^k$* if all its partial derivatives of order less than or equal to $k$ exist and are [[Schutz - 1.2 - Mappings#^1aee5f|continuous]] functions on $S$. ^319309 > **Differentials of a Map** > If $f$ is an [[Schutz - 1.2 - Mappings#^6c8b58|injection]] of an open set $M$ of $\mathbb{R}^n$ onto another open set $N$ of $\mathbb{R}^n$, it can be expressed concretely as: > $ y_i = f_i(x_1, x_2, \ldots, x_n) \quad \text{or} \,\, \pmb{y} = \pmb{f}(\pmb{x})$ > where $\{x_i, i = 1, \ldots, n\}$ define a point $\pmb{x}$ of $M$ and $\{y_i, i = 1, \ldots n\}$ likewise define a points $\pmb{y}$ of $N$. If the functions are all $C^k$-[[Schutz - 1.2 - Mappings#^319309|differentiable]], then the map is said to be $C^k$-differentiable. > **The Jacobian Matrix and the Jacobian** > The *Jacobian Matrix* of a $C^1$ map is the matrix of partial derivatives $\partial f_i/\partial x_j$. The determinant of the matrix is called the *Jacobian* and is often denoted by: > $J = \partial(f_1, \ldots f_n)/\partial(x_1, \ldots x_n)$ ^76b143 Consider the mapping from the function $g$ to $g_*$ as follows: $g_*(f_1(x_1 , \ldots x_n), \ldots f_n(x_1, \ldots x_n)) = g(x_1, \ldots x_n)$ Then $ \int_M g(x_1, \ldots x_n) dx_1 \ldots dx_n = \int_N g_*(y_1, \ldots y_n) J dy_1 \ldots dy_n $ Since $g$ and $g^*$ have the same value at appropriate points, it is often said that the volume element $dx_1 \ldots dx_n$ has changed to $J dy_1 \ldots dy_n$, a useful point of view if we view the mapping $f$ as a coordinate change.