\[\renewcommand{\u}[1]{\boldsymbol{#1}} \renewcommand{\b}[1]{\mathbb} \newcommand{\set}[2]{\left\{#1\,\big|\,#2\right\}} \newcommand{\pr}[1]{\left(#1\right)} \newcommand{\st}[2][]{#2_{\mathscr{s}#1}} \newcommand{\mt}[2][]{#2_{\mathscr{m}#1}} \newcommand{\lsc}[2][\mathscr{l}]{ { }^{#1}\!#2 } \newcommand{\norm}[1]{\lVert #1 \rVert}\]

Spatial velocity field

However, we would like to answer the question what is the velocity of the material particle that has the spatial position vector \(\u{x}\) at time \(\u{\tau}\). In short we would like to construct a field \(\u{v}_{\u{\tau}}(\u{x})\) which would give the velocity of the material partcile \(\u{X}\) that is at \(\u{x}\) at time \(\u{\tau}\). Since the argument of this field is the spatial position vectors \(\u{x}\), this field is called the spatial velocity field. The spatial velocity field \(\u{v}_{\u{\tau}}(\u{x})\) is constructed as follows. Using the inverse deformation mapping \(\u{\varphi}_{\u{\tau}}^{-1}:\u{\kappa}_{\u{\tau}}(\mathcal{B})\to \u{\kappa}_{\rm R}(\mathcal{B})\), we can find the material particle \(\u{X}\) at is positioned at \(\u{x}\) at time \(\u{\tau}\) as \(\u{\varphi}_{\u{\tau}}^{-1}(\u{x})\). The velocity of the material particle \(\u{\varphi}_{\u{\tau}}^{-1}(\u{x})\) at the time instance \(\u{\tau}\) is of course \(\u{V}_{\u{\tau}}\pr{\u{\varphi}_{\u{\tau}}^{-1}(\u{x})}\). Thus, we define \(\begin{equation} \u{v}_{\u{\tau}}\pr{\u{x}}=\u{V}_{\u{\tau}}(\u{\varphi}_{\u{\tau}}^{-1}(\u{x})) \end{equation}\) In component form of the above above equation can be written as, \(\begin{align} \u{v}_{\u{\tau}}\pr{\u{x}}&=\u{V}_{\u{\tau}}(\u{\varphi}_{\u{\tau}}^{-1}(\u{x}))\\ &=x_{i,\tau}(X_i\circ\u{\varphi}_{\u{\tau}}^{-1}(\u{x}),\tau)\u{v}_{i} \end{align}\)