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Partial derivatives, \(C^k\) functions, chain rule, inverse function theorem, implicit function
theorem.
We begin by reviewing some calculus concepts. A reference for this material is
Hubbard, J. H., & Hubbard, B. B. (2015). Vector calculus, linear algebra, and
differential forms: a unified approach (pp. 818-pages). Matrix Editions.
Partial derivatives Let \(U \subset \mathbb {R}^n\) be an open set, \(f : U \to \mathbb {R}\) a function, \(p \in U\), and \(i \in \{ 1, \ldots , n \}\). The \(i\)-th partial
derivative of \(f\) at \(p\) is defined as
\[ \partial _i f (p) : = \lim _{h \to 0} \frac {f(p + h e_i) - f(p)}{h} , \]
whenever the limit exists, where \(e_i = (0, \ldots , 1, \ldots , 0)\) denotes
the \(i\)-th canonical vector. If this limit exists for all \(p \in U\), it defines a function
\[ \partial _i f : U \to \mathbb {R}. \]
\(C^k\) functions Let \(U \subset \mathbb {R}^n\) be an open set and \(f : U \to \mathbb {R}\) a function. We say \(f\) is \(C^0\) if it is continuous.
Inductively, we say it is \(C^{k+1} \) if for each \(i \in \{ 1, \ldots , n \}\), the function \( \partial _i f : U \to \mathbb {R}\) exists and is \(C^k\). We say that \(f\) is \(C^{\infty }\) if it
is \(C^k\) for all \(k \in \mathbb {N}\). \(C^{\infty }\) functions are called smooth.
Let \(U \subset \mathbb {R}^n\) be an open set. We denote by \(C^k (U)\) the set of \(C^k\) functions \(f: U \to \mathbb {R}\). Same with \(k \) replaced by \(\infty \).
Classical derivative rules Let \(U \subset \mathbb {R}^n\) open and \(f,g \in C^{\infty }(U)\). Then
\begin{gather*} \partial _i (f+ g) = \partial _i f + \partial _i g ,\\ \partial _i (fg) = (\partial _i f) g + f (\partial _i g) . \end{gather*}
If in addition, \(g (x) \neq 0\) for all \(x \in U\), then
For \(m \in \mathbb {N}\), we denote by \(C ^{k} (U ; \mathbb {R}^m)\) the set of functions \(f : U \to \mathbb {R}^m\), where \(f = (f_1, \ldots , f_m ) \) and \(f _j \in C^k (U)\) for each \(j \in \{ 1, \ldots , m \}\). We also denote
Differential of a function Let \(U \subset \mathbb {R}^n\) be an open set, \(f : U \to \mathbb {R}^m \) a function, and \(p \in U\). We say a linear
function \(L : \mathbb {R}^n \to \mathbb {R} ^m \) is a derivative or differential of \(f\) at \(p\) if
In such a case, we say that \(f\) is
differentiable at \(p\).
Smooth implies differentiable Let \(U \subset \mathbb {R}^n\) be an open set. If \(f \in C^{\infty }( U ; \mathbb {R}^m ) \), then \(f\) is differentiable at \(p\) for all
\(p \in U\). Moreover, the differential of \(f\) at \(p\) is unique and given by
Throughout this course, the word “differentiable” will mean “smooth” as in
Definition def:c-k, ignoring the concept of “differentiable” from Definition def:differential-classic. This abuse of
notation is common in the literature of smooth manifolds.
Chain rule Let \(U \subset \mathbb {R}^n\) and \(V \subset \mathbb {R}^m\) be open sets, \(f \in C^{\infty }(U;\mathbb {R}^m)\), \(h \in C^{\infty }(V; \mathbb {R}^{\ell })\), and \(f(U) \subset V\). Then \(h \circ f \in C^{\infty } (U; \mathbb {R}^{\ell })\), and for each \(p \in U\), the differential is
given by
\[ d_p (h \circ f) = ( d_{f(p)} h ) \circ (d_p f ) . \]
Diffeomorphism Let \(U , V \subset \mathbb {R}^n\) be open sets. A bijective smooth function \(f : U \to V\) is called a
diffeomorphism if the inverse function \(f^{-1} : V \to U\) is smooth.
Inverse function theorem Let \(U \subset \mathbb {R}^n\) be open, \(f \in C^{\infty }(U; \mathbb {R}^n)\), and \(p \in U\). Assume the linear function \(d_p f : \mathbb {R}^n \to \mathbb {R}^n\) is
invertible. Then there are open sets \(U_0 \subset \mathbb {R}^n\) and \( V_0 \subset \mathbb {R}^n\) such that \(p \in U_0\), \(f(p )\in V_0\), and \(f : U _ 0 \to V_0\) is a diffeomorphism.
Implicit function theorem Let \(\Omega \subset \mathbb {R}^n \times \mathbb {R}^m = \mathbb {R}^{n+m}\) be open and \(F \in C^{\infty }(\Omega ; \mathbb {R}^m)\). Assume the matrix
is invertible at a
point \((p,q) \in F^{-1} (0) \subset \Omega \subset \mathbb {R}^n \times \mathbb {R}^m \). Then there are open sets \(U \subset \mathbb {R}^n\), \(V \in \mathbb {R}^m\), and \(f \in C^{\infty }(U; \mathbb {R}^m)\) such that \(p \in U\), \(q \in V\), and
\[ F^{-1} (0) \cap ( U \times V) = \{ (x, f(x) ) \in \mathbb {R}^n \times \mathbb {R}^m \, \vert \, x \in U \} . \]
Implicit function theorem again Let \(\Omega \subset \mathbb {R}^{n+m}\) be open and \(F \in C^{\infty }(\Omega ; \mathbb {R}^m)\). Assume the linear map \(d_{p} F : \mathbb {R}^{n+m} \to \mathbb {R}^m\) is
surjective for some \( p \in F^{-1} (0) \subset \Omega \). Then there are open sets \(U \subset \mathbb {R}^n\), \(V \in \mathbb {R}^m\), and \(f \in C^{\infty }(U; \mathbb {R}^m)\) such that after reordering the
coordinates of \(\mathbb {R}^{n+m}\), one has \(p \in U\times V\) and
\[ F^{-1} (0) \cap ( U \times V) = \{ (x, f(x) ) \in \mathbb {R}^n \times \mathbb {R}^m \, \vert \, x \in U \} . \]
For the purposes of this class, the above results were stated for smooth functions, but
they have analogues for functions of less regularity:
Smooth manifolds