Physlib.Units.FDeriv

Let $M_1$ and $M_2$ be physical vector spaces with dimensions $d_1$ and $d_2$ respectively. Let $f: M_1 \to M_2$ be a differentiable function that is dimensionally correct. For any two systems of unit choices $u_1$ and $u_2$, and for any point $x \in M_1$ and direction $v \in M_1$, the Fréchet derivative of $f$ evaluated at the scaled point $x' = \text{scaleUnit}(u_2, u_1, x)$ satisfies: $$Df(x')(v) = \sigma(u_2, u_1, d_2) \cdot \sigma(u_1, u_2, d_1) \cdot (Df(x)(v))$$ where $\sigma(u_i, u_j, d)$ is the scaling factor for a quantity of dimension $d$ when transitioning from unit system $u_i$ to $u_j$, and $Df(x)(v)$ denotes the derivative of $f$ at $x$ in the direction $v$.

Let $M_1$ and $M_2$ be physical vector spaces. Let $f : M_1 \to M_2$ be a function that is differentiable over $\mathbb{R}$. If $f$ is dimensionally correct, then its Fréchet derivative $Df : M_1 \to (M_1 \to M_2)$ is also dimensionally correct.

Let $M_1$ and $M_2$ be physical vector spaces with dimensions $d_1$ and $d_2$, respectively. Let $f: M_1 \to M_2$ be a differentiable function that is dimensionally correct. For a fixed vector $v_0 \in M_1$, the relation $Df(x)(v_0) = v$ is dimensionally correct, where $x \in M_1$ (with dimension $d_1$) and $v \in M_2$ is a quantity with dimension $d_2 \cdot d_1^{-1}$. Here $Df(x)(v_0)$ denotes the Fréchet derivative of $f$ at $x$ applied to the direction $v_0$.