可逆的导数，可逆的映射
线性化支配局部行为。若 C^1 映射 f: R^n -> R^n 在 a 处导数可逆——即雅可比矩阵行列式不为零——则 f 在 a 附近看起来就像它可逆的线性部分，因此它自身在附近也应可逆。这就是反函数定理。
Inverse function theorem.
Let f: R^n -> R^n be C^1 near a, and suppose Df(a) is invertible,
i.e. det Df(a) != 0. Then there are open sets U about a and V about
f(a) such that f : U -> V is a bijection, the inverse g : V -> U is
C^1, and by the chain rule applied to g(f(x)) = x:
Dg( f(a) ) = ( Df(a) )^{-1}.
Proof idea: solving f(x) = y is finding a fixed point of
T(x) = x + Df(a)^{-1} ( y - f(x) ),
which is a [[contraction-mapping|contraction]] near a; the
[[banach-fixed-point-theorem|Banach fixed-point theorem]] gives a
unique solution, and it varies smoothly with y.隐式地解方程
水平集 F(x, y) = 0 通常把 y 定义为 x 的函数，即使你无法用公式解出。隐函数定理使之精确：只要关于你想解出的变量的偏导数可逆，就存在光滑的局部解，并可通过对方程求导算出其导数。
Implicit function theorem (scalar case). Let F(x, y) be C^1 with F(a, b) = 0 and dF/dy (a, b) != 0. Then near a there is a unique C^1 function y = h(x) with h(a) = b and F(x, h(x)) = 0. Differentiate that identity in x: dF/dx + (dF/dy) * h'(x) = 0 => h'(x) = - (dF/dx) / (dF/dy). Example: the unit circle F(x, y) = x^2 + y^2 - 1 = 0 at (a, b)=(0, 1). dF/dy = 2y = 2 != 0, so y = sqrt(1 - x^2) exists locally, and h'(x) = - (2x) / (2y) = - x / y. At (0,1): slope 0, as expected. At (1, 0) instead dF/dy = 0 — and indeed near (1,0) the circle is NOT a graph y = h(x); the hypothesis fails exactly where it must.
一枚硬币的两面
这两个定理在“彼此可迅速互推”的意义上是等价的。要由反函数定理导出隐函数定理，对映射 (x, y) -> (x, F(x, y)) 应用前者；当且仅当 dF/dy 可逆时它的雅可比矩阵可逆，而其逆便给出解 y = h(x)。两者都基于同一引擎：可逆的线性全导数在局部支配着非线性映射。