---- > [!definition] Definition. ([[gradient]]) > Let $(M, g)$ be a [[Riemannian manifold]] and $f \in C^{\infty}(M)$ a function. The [[musical isomorphism induced by a nondegenerate bilinear form|musical isomorphism(s)]] give a unique [[vector field]] $\text{grad } f \in \Gamma(TM)$ satisfying the identity $g(\operatorname{grad}f, X)=df(X)$ > for all $X \in \Gamma(TM)$, called the **gradient** of $f$. That is, $\operatorname{grad}f=(df)^{\sharp}.$ > Note that in [[coordinate chart|local coordinates]], with $(g^{ij})$ denoting the [[nondegenerate bilinear form|metric inverse]], one has $\operatorname{grad}f=g^{ij}\frac{ \partial f }{ \partial x^{j} } \frac{ \partial }{ \partial x^{i} } .$ > Note that when $g^{ij}=\delta^{ij}$ (e.g. [[dot product]] on $\mathbb{R}^{n}$) this expression becomes $\text{grad }f=\frac{ \partial f }{ \partial x^{i} } \frac{ \partial }{ \partial x^{i} }$, an expression familiar from multivariable calculus. > [!specialization] > For a $C^1$ [[scalar field]] $f: \Omega \subset \mathbb{R}^m \to \mathbb{R}$, its **gradient at $x \in \rr ^{m}$** is the $m \times 1$ [[vector]] defined as $(\text{grad }f)(x) = \begin{bmatrix} D_1f(x) \\ \vdots \\ D_mf (x) \end{bmatrix} = Df(x)^\top.$ ^specialization ---- #### ---- #### 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 > ```