Correct sign error

Vectornaut 2024-11-18 22:57:04 +00:00
parent eeccf5fe55
commit ee6276f926

@ -101,7 +101,7 @@ we can express the derivative of $\operatorname{grad}(f)$ as
\[
\begin{align*}
d\operatorname{grad}(f) & = -4 Q\,dA\,\mathcal{P}(\Delta) - 4 Q A\,\mathcal{P}(d\Delta) \\
& = -4 Q\big[dA\,\mathcal{P}(\Delta) + A\,\mathcal{P}(-d\Delta)\big].
& = 4 Q\big[{-dA}\,\mathcal{P}(\Delta) + A\,\mathcal{P}(-d\Delta)\big].
\end{align*}
\]
In the Rust and Julia implementations of the realization routine, we express $d\operatorname{grad}(f)$ as a matrix in the standard basis for $\operatorname{End}(\mathbb{R}^n)$. We apply the cotangent vector $d\operatorname{grad}(f)$ to each standard basis matrix $E_{ij}$ by setting the value of the matrix-valued 1-form $dA$ to $E_{ij}$.