---- Let $\Omega \subset \mathbb{R}^{m}$ be [[open set|open in]] $\mathbb{R}^{m}$. Let $f: \Omega \to \mathbb{R} ^{n}$ such that all [[partial derivative]]s of $f$ exist at $p \in \Omega$. > [!definition] Definition. ([[Jacobian]]) > Define the **Jacobian** $J$ of $f$ at $p$ by the formula $\begin{bmatrix} \frac{ \partial f_{1} }{ \partial x_{1} } & \dots & \frac{ \partial f_{1} }{ \partial x_{m} } \\ \vdots & \ddots & \vdots \\ \frac{ \partial f_{n} }{ \partial x_{1} } & \dots & \frac{ \partial f_{n} }{ \partial x_{m} } \end{bmatrix} = \begin{bmatrix} D_{1}f_{1}(\vec p) & \dots & D_{m}f_{1}(\vec p) \\ \vdots & \ddots & \vdots \\ D_{1}f_{n}(\vec p) & \dots & D_{m}f_{n}(\vec p) \end{bmatrix}.$ > [!warning] Warning. $J$ might exist when $Df(\vec p)$ does not! [[existence of derivative guarantees existence of all directional derivatives|The converse is false.]] > ---- #### ---- #### References > [!backlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM [[]] > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ``` > [!frontlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM outgoing([[]]) > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ```