Hyperfläche/3 Variablen/Matrix zu Hauptteilen/Beispiel

${\begin{pmatrix}&1&A&B&C&A^{2}&AB&AC&B^{2}&BC&C^{2}\\1&0&0&0&0&0&0&0&0&0&0\\A&{\frac {\partial }{\partial X}}&0&0&0&0&0&0&0&0&0\\B&{\frac {\partial }{\partial Y}}&0&0&0&0&0&0&0&0&0\\C&{\frac {\partial }{\partial Z}}&0&0&0&0&0&0&0&0&0\\A^{2}&{\frac {1}{2}}{\frac {\partial ^{2}}{\partial ^{2}X}}&{\frac {\partial }{\partial X}}&0&0&0&0&0&0&0&0\\AB&{\frac {\partial ^{2}}{\partial ^{2}XY}}&{\frac {\partial }{\partial Y}}&{\frac {\partial }{\partial X}}&0&0&0&0&0&0&0\\AC&{\frac {\partial ^{2}}{\partial ^{2}XZ}}&{\frac {\partial }{\partial Z}}&0&{\frac {\partial }{\partial X}}&0&0&0&0&0&0\\B^{2}&{\frac {1}{2}}{\frac {\partial ^{2}}{\partial ^{2}Y}}&0&{\frac {\partial }{\partial Y}}&0&0&0&0&0&0&0\\BC&{\frac {\partial ^{2}}{\partial ^{2}YZ}}&0&{\frac {\partial }{\partial Z}}&{\frac {\partial }{\partial Y}}&0&0&0&0&0&0\\C^{2}&{\frac {1}{2}}{\frac {\partial ^{2}}{\partial ^{2}Z}}&0&0&{\frac {\partial }{\partial Z}}&0&0&0&0&0&0\\A^{3}&{\frac {1}{6}}{\frac {\partial ^{3}}{\partial ^{3}XXX}}&{\frac {1}{2}}{\frac {\partial ^{2}}{\partial ^{2}X}}&0&0&{\frac {\partial }{\partial X}}&0&0&0&0&0\\A^{2}B&{\frac {1}{2}}{\frac {\partial ^{3}}{\partial ^{3}XXY}}&{\frac {\partial ^{2}}{\partial ^{2}XY}}&{\frac {1}{2}}{\frac {\partial ^{2}}{\partial ^{2}X}}&0&{\frac {\partial }{\partial Y}}&{\frac {\partial }{\partial X}}&0&0&0&0\\A^{2}C&{\frac {\partial ^{3}}{\partial ^{3}XXZ}}&{\frac {\partial ^{2}}{\partial ^{2}XZ}}&0&{\frac {1}{2}}{\frac {\partial ^{2}}{\partial ^{2}X}}&{\frac {\partial }{\partial Z}}&0&{\frac {\partial }{\partial X}}&0&0&0\\AB^{2}&{\frac {1}{2}}{\frac {\partial ^{3}}{\partial ^{3}XYY}}&{\frac {1}{2}}{\frac {\partial ^{2}}{\partial ^{2}Y}}&{\frac {\partial ^{2}}{\partial ^{2}XY}}&0&0&{\frac {\partial }{\partial Y}}&0&{\frac {\partial }{\partial X}}&0&0\\ABC&{\frac {\partial ^{3}}{\partial ^{3}XYZ}}&{\frac {\partial ^{2}}{\partial ^{2}YZ}}&{\frac {\partial ^{2}}{\partial ^{2}XZ}}&{\frac {\partial ^{2}}{\partial ^{2}XY}}&0&{\frac {\partial }{\partial Z}}&{\frac {\partial }{\partial Y}}&0&{\frac {\partial }{\partial X}}&0\\AC^{2}&{\frac {1}{2}}{\frac {\partial ^{3}}{\partial ^{3}XZZ}}&{\frac {1}{2}}{\frac {\partial ^{2}}{\partial ^{2}Z}}&0&{\frac {\partial ^{2}}{\partial ^{2}XZ}}&0&0&{\frac {\partial }{\partial Z}}&0&0&{\frac {\partial }{\partial X}}\\B^{3}&{\frac {1}{6}}{\frac {\partial ^{3}}{\partial ^{3}YYY}}&0&{\frac {1}{2}}{\frac {\partial ^{2}}{\partial ^{2}Y}}&0&0&0&0&{\frac {\partial }{\partial Y}}&0&0\\B^{2}C&{\frac {1}{2}}{\frac {\partial ^{3}}{\partial ^{3}YYZ}}&0&{\frac {\partial ^{2}}{\partial ^{2}YZ}}&{\frac {1}{2}}{\frac {\partial ^{2}}{\partial ^{2}Y}}&0&0&0&{\frac {\partial }{\partial Z}}&{\frac {\partial }{\partial Y}}&0\\BC^{2}&{\frac {1}{2}}{\frac {\partial ^{3}}{\partial ^{3}YZZ}}&0&{\frac {1}{2}}{\frac {\partial ^{2}}{\partial ^{2}Z}}&{\frac {\partial ^{2}}{\partial ^{2}YZ}}&0&0&0&0&{\frac {\partial }{\partial Z}}&{\frac {\partial }{\partial Y}}\\C^{3}&{\frac {1}{6}}{\frac {\partial ^{3}}{\partial ^{3}Z}}&0&0&{\frac {1}{2}}{\frac {\partial ^{2}}{\partial ^{2}Z}}&0&0&0&0&0&{\frac {\partial }{\partial Z}}\end{pmatrix}}.$