Differentiability at a Point and Jacobian Matrix for Maps Between Euc…

Statement

Let $n,m\in\mathbb{N}$ . Let $U\subseteq \mathbb{R}^n$ be open, let $f=(f_1,\dots,f_m):U\to\mathbb{R}^m$ , and let $a=(a_1,\dots,a_n)\in U$ . Suppose that for every $j\in\{1,\dots,m\}$ and every $i\in\{1,\dots,n\}$ the partial derivative Partial Derivative of a Coordinate Function $\frac{\partial f_j}{\partial x_i}(a)$ exists. The matrix

J_f(a)=\left(\frac{\partial f_j}{\partial x_i}(a)\right)_{1\le j\le m,\ 1\le i\le n}

is called the Jacobian matrix of $f$ at $a$ . We say that $f$ is differentiable at $a$ if for every $\varepsilon>0$ there exists $\delta>0$ with the following property: whenever $h=(h_1,\dots,h_n)\in\mathbb{R}^n$ satisfies

0<\sum_{i=1}^n h_i^2<\delta^2

and $a+h\in U$ , one has

\sum_{j=1}^m \left(f_j(a+h)-f_j(a)-\sum_{i=1}^n \frac{\partial f_j}{\partial x_i}(a) h_i\right)^2 \le \varepsilon^2 \sum_{i=1}^n h_i^2.

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currentDifferentiability at a Point and Jacobian Matrix for Maps Between Euclidean Spacesdef:differentiable-map-at-point-euclidean-2026a

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