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Correct sign error

Vectornaut 2024-11-18 22:57:04 +00:00
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Gram-matrix-parameterization.md

@ -101,7 +101,7 @@ we can express the derivative of $\operatorname{grad}(f)$ as
\[ \[
\begin{align*} \begin{align*}
d\operatorname{grad}(f) & = -4 Q\,dA\,\mathcal{P}(\Delta) - 4 Q A\,\mathcal{P}(d\Delta) \\ d\operatorname{grad}(f) & = -4 Q\,dA\,\mathcal{P}(\Delta) - 4 Q A\,\mathcal{P}(d\Delta) \\
& = -4 Q\big[dA\,\mathcal{P}(\Delta) + A\,\mathcal{P}(-d\Delta)\big]. & = 4 Q\big[{-dA}\,\mathcal{P}(\Delta) + A\,\mathcal{P}(-d\Delta)\big].
\end{align*} \end{align*}
\] \]
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 $E_{ij}$. 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 $E_{ij}$.