Euclidean spatial calculus: gradient, divergence and material derivative

We use one-based component labels: for $x\in {\mathbb{R}}^3$, $x_i:=x(i-1)$ for $1\le i\le 3$, and $e_i$ denotes the imported standard unit vector at index $i-1$. Thus ${\mathrm{\partial }}_i$ here is the imported partial derivative ${\mathrm{\partial }}_{i-1}$, and an entry labelled $(i,j)$ of a $3\times 3$ matrix is its imported entry $(i-1,j-1)$. All component formulas below use this relabelling; the underlying maps, matrices, inner product and trace are unchanged.

Throughout, ${\mathbb{R}}^3$ carries its standard ordered orthonormal basis $e_1,e_2,e_3$, its Euclidean inner product $⟨\cdot ,\cdot {⟩}_2$ (The Euclidean inner product $⟨x,y⟩={\sum }_{k<n}{x}_k{y}_k$ on ${\mathbb{R}}^n$).

Let $O\subseteq {\mathbb{R}}^3$ be open, let $a\in O$ and let $v:O\to {\mathbb{R}}^3$ be of class $C^1$ ($C^k$ Euclidean maps and diffeomorphisms), so that the total derivative $Dv(a):{\mathbb{R}}^3\to {\mathbb{R}}^3$ is a linear map (The total (Fréchet) derivative $Df(a)$ as the linear first-order approximation with $o(\mathrm{\parallel }h{\mathrm{\parallel }}_2)$ remainder, A linear map $L:{\mathbb{R}}^m\to {\mathbb{R}}^n$ in Euclidean coordinates) whose matrix in the standard bases is $({\mathrm{\partial }}_j{v}_i(a){)}_{1\le i,j\le 3}$ (The Jacobian matrix of partial derivatives and the gradient in the scalar-valued case, A total derivative computes every directional derivative, and its matrix is the Jacobian). The divergence of $v$ at $a$ is

$\mathrm{div}v(a):=\mathrm{tr}(Dv(a))={\sum }_{i=1}^3{\mathrm{\partial }}_i{v}_i(a),$

the trace of the total derivative (The trace $\mathrm{tr}(A)$ as the sum of the diagonal entries), read in the standard basis of $M_3(\mathbb{R})$ (The vector space $M_{m\times n}(F):=F^{m\times n}$ of $m$ by $n$ matrices over a field, with entrywise operations).

For a $C^1$ scalar field $f:O\to \mathbb{R}$ and $a\in O$, the gradient $\mathrm{\nabla }f(a)\in {\mathbb{R}}^3$ is the vector of partial derivatives of The Jacobian matrix of partial derivatives and the gradient in the scalar-valued case; it is the unique vector with $Df(a)h=⟨\mathrm{\nabla }f(a),h{⟩}_2$ for every $h\in {\mathbb{R}}^3$ (Directional derivatives and partial derivatives of a map $U\subseteq {\mathbb{R}}^m\to {\mathbb{R}}^n$, For a differentiable scalar field, $D_{v}f(a)=⟨\mathrm{\nabla }f(a),v⟩$ and the unit direction of steepest ascent is the normalized gradient). We write $⟨\mathrm{\nabla }f(a),v(a){⟩}_2=:\mathrm{\nabla }f(a)\cdot v(a)$.

Let $I\subseteq \mathbb{R}$ be a nonempty open interval and let $u:I\times O\to {\mathbb{R}}^3$ and $f:I\times O\to \mathbb{R}$ be $C^1$. For $(t,x)\in I\times O$ write $D_{x}f(t,x):{\mathbb{R}}^3\to \mathbb{R}$ for the total derivative of $y\mapsto f(t,y)$ at fixed $t$, and ${\mathrm{\nabla }}_{x}f(t,x)$ for its gradient. The derivative $Df(t,x)$ of the full map on $I\times O$ acts on $\mathbb{R}\times {\mathbb{R}}^3$; it is distinct from $D_{x}f$. Define the partial time derivative ${\mathrm{\partial }}_{t}f(t,x):=\frac{d}{ds}{\mid }_{s=t}f(s,x)$ and the material derivative of $f$ along $u$

$D_{t}f(t,x):={\mathrm{\partial }}_{t}f(t,x)+D_{x}f(t,x)(u(t,x))={\mathrm{\partial }}_{t}f(t,x)+{\mathrm{\nabla }}_{x}f(t,x)\cdot u(t,x)\mathrm{.}$

Spatial derivatives $Du$, $Dw$, $\mathrm{\nabla }f$ and $\mathrm{div}u$ of time-dependent fields always mean $D_{x}u$, $D_{x}w$, ${\mathrm{\nabla }}_{x}f$ and spatial divergence at fixed time; a stationary field $u(x)$ is included by setting $u(t,x)=u(x)$. The same definitions apply on any open spacetime domain, by restriction to a product neighborhood of each point.

For a $C^1$ vector field $w:I\times O\to {\mathbb{R}}^3$ the material derivative $D_{t}w$ is defined componentwise, and $\mathrm{div}w(t,x)$ means the divergence of $x\mapsto w(t,x)$ at $x$ at each fixed $t$.

• All objects above are defined in the single fixed ordered orthonormal basis $(e_1,e_2,e_3)$; gradients and divergences are not attached to any physical frame, unit system or observer. Coordinates are pure numbers here. A physical reading (length units, time units, a chosen inertial chart) is additional structure supplied by a consuming model, never by this definition.

• The divergence is the trace of the total derivative, not a sum of partial derivatives read in an arbitrary coordinate system; the two agree in the standard orthonormal coordinates used here, and both expressions are recorded so that no later item may silently use a different convention. The gradient likewise depends on the chosen Euclidean inner product.

• The chain rule (The chain rule for total derivatives: $D(g\circ f)(a)=Dg(f(a))\circ Df(a)$) gives, for a $C^1$ curve $\gamma :J\to O$ with ${\gamma }^{\mathrm{\prime }}(t)=u(t,\gamma (t))$, the identity $\frac{d}{dt}f(t,\gamma (t))=D_{t}f(t,\gamma (t))$ whenever $f$ is $C^1$; this is the sense in which $D_t$ differentiates along the flow of $u$ and is used by the transport theorem of this collection.

• Integrals and derivatives of fields are taken where they are defined: the material derivative requires $f$ to be $C^1$ on a neighbourhood of the point in question, and no extension of $u$ or $f$ is implicit. Boundary behaviour is never assumed from this definition alone.