^^ [[Schutz - 2 - Differentiable Manifolds and Tensors|2. Differentiable Manifolds and Tensors]] <<[[Schutz - 2.5 - Curves|2.5 Curves]] | [[Schutz - 2.7 - Vectors and Vector Fields|2.7 Vectors and Vector Fields]]>> # Functions on $M$ > **Function on $M$** > A *function on $M$* is a rule that assigns a real number to each point of $M$, $f:M\rightarrow \mathbb{R}$ > When a region of $M$ is mapped differentiably to a region of $\mathbb{R}^n$, the function becomes a function on $\mathbb{R}^n$, $f:M\rightarrow \mathbb{R}^n$ ![[Pasted image 20210123193628.png]] > **Differentiable function on $M$** > If a function $f:M\rightarrow \mathbb{R}^n$ is *differentiable* on $\mathbb{R}^n$ and maps regions of $M$ differentiably to $\mathbb{R}^n$, then it is said to be a differentiable function on $\mathbb{R}^n$. > More abstractly, the function may be written as $f(P), \, P \in M$. But $P$ has coordinates, so one can express the value of the function by some algebraic expression $f(x^i)$. If this expression is differentiable in all its arguments, then the function is *differentiable*. ^6ded65 The coordinates themselves, are, of course, continuous and infinitely differentiable functions, assuming a $C^\infty$-manifold. From here on out, we will always assume that we can place coordinates $\{x^i, i = 1\ldots n\}$ on the manifold, and that any sufficiently-differentiable set of equations $y^i = y^i(x^j)$, which is locally invertible (i.e. the [[Schutz - 1.2 - Mappings#^76b143|Jacobian]] $|\partial y^i/\partial x_j| \neq 0$) constitutes an acceptable coordinate transformation to new coordinates $\{y^i, i = 1\ldots n\}$. We will avoid referring to the mapping directly.