Let \((M,g)\) be a smooth Riemannian manifold and \(f\) a real-valued smooth function on \(M\). Then the gradient of \(f\) is defined by
\begin{equation*} \grad f = \hat{g}^{-1}(df), \end{equation*}where \(\hat{g}\) denotes the tangent-cotangent isomorphism and \(df\) the differentiation of \(f\).
Remarks
- Using musical isomorphisms notation, we can also write \begin{equation*} \grad f = (df)^{\sharp}. \end{equation*}
- The vector field \(\grad f\) is the unique vector field which satisfies \begin{equation*} \langle \grad f, \cdot\rangle_g=df. \end{equation*}