---- > [!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 $g(\operatorname{grad}f, X)=\overbrace{ df(X) }^{ Xf }$ > 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. ```tikz \usepackage{tikz-cd} \usepackage{amsmath} \usepackage{amsfonts} \usepackage[mathscr]{euscript} \usepackage{tikzit} \begin{document} % TiKZ style file generated by TikZiT. You may edit this file manually, % but some things (e.g. comments) may be overwritten. To be readable in % TikZiT, the only non-comment lines must be of the form: % \tikzstyle{NAME}=[PROPERTY LIST] % Node styles \tikzstyle{red dot}=[fill={rgb,255: red,191; green,0; blue,64}, draw=black, shape=circle] \tikzstyle{green dot}=[fill={rgb,255: red,0; green,128; blue,128}, draw=black, shape=circle] \tikzstyle{rectangle}=[fill=white, draw=black, shape=rectangle] % Edge styles \tikzstyle{new edge style 0}=[-] %% paste tikzpicture here % https://tikzcd.yichuanshen.de/#N4Igdg9gJgpgziAXAbVABwnAlgFyxMJZABgBpiBdUkANwEMAbAVxiRAGEA9AHW6zABmOAJ4AKALIBKEAF9S6TLnyEUZAIxVajFmy69+QsVNnyQGbHgJE1pDdXrNWiELwDidALYe6ogCrG5BQtla3JNBx1nXgB5DxgAczpONQlpGU0YKHj4IlABACcIDyQbEBwIJDItRzZeAGMCeJM8wuLEKvKkAGZqACMYMChuqoinECHAkAKipAAmak7EHurIl24BBjocZqnWuYWKpcnptvmyw+XN-oYABUVLFRB8rHiAC237bTH6xp2T7oOlT6AyGSxGX1q3FgDBwdAAvFBOAAqP57RClRZnfqDYbUBj8MZQOhwV6ZECfGpRbgQNAwfJbCD5MCeGDAeL0qAyOG8Zm9TaomaIM6LUrY0FdKr4sCE4mkoYU1YAWl4NLpDKZLOAUCwNC5yu4vM2AAJ6lAINt0jIgA \begin{tikzcd}[scale=2.0] C^\infty(M) \arrow[r, "d", bend left] & \Omega^1(M) \arrow[l, "\delta=d^*", bend left] \\ C^\infty(M) \arrow[u, "\cong"] \arrow[r, "\operatorname{grad}=\nabla", dashed, bend left] & \Gamma(TM) \arrow[u, "\flat"] \arrow[u] \arrow[u, "\cong"'] \arrow[l, "-\operatorname{div}=-\nabla \cdot", dashed, bend left] \end{tikzcd} %% end paste \end{document} ``` > [!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 > ```