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- https://math.libretexts.org/Bookshelves/Calculus/Supplemental_Modules_(Calculus)/Vector_Calculus/3%3A_Multiple_Integrals/3.9%3A_Substitutions_in_Multiple_IntegralsThis section discusses the translation of a graph from the xy Cartesian plane to the uv Cartesian plane and defines the Jacobian. The Jacobian measures how much the volume at a certain point chang...This section discusses the translation of a graph from the xy Cartesian plane to the uv Cartesian plane and defines the Jacobian. The Jacobian measures how much the volume at a certain point changes when being transformed from one coordinate system to another.
- https://math.libretexts.org/Bookshelves/Linear_Algebra/Interactive_Linear_Algebra_(Margalit_and_Rabinoff)/04%3A_DeterminantsThis page discusses matrix equations, focusing on solving Ax=b, determining eigenvalues and eigenvectors, and finding approximate solutions. The current chapter emphasizes determinants, covering t...This page discusses matrix equations, focusing on solving Ax=b, determining eigenvalues and eigenvectors, and finding approximate solutions. The current chapter emphasizes determinants, covering their definition, properties, and computation methods. It includes cofactor expansions as a recursive calculation method and explores the geometric interpretation of determinants in relation to volumes in multivariable calculus.
- https://math.libretexts.org/Bookshelves/Linear_Algebra/Linear_Algebra_with_Applications_(Nicholson)/03%3A_Determinants_and_Diagonalization/3.01%3A_The_Cofactor_ExpansionIf a multiple of one row of A is added to a different row (or if a multiple of a column is added to a different column), the determinant of the resulting matrix is detA. \[\begin{aligned} \d...If a multiple of one row of A is added to a different row (or if a multiple of a column is added to a different column), the determinant of the resulting matrix is detA. detB=(aq1+uap1)cq1(A)+(aq2+uap2)cq2(A)+⋯+(aqn+uapn)cqn(A)=[aq1cq1(A)+aq2cq2(A)+⋯+aqncqn(A)]+u[ap1cq1(A)+ap2cq2(A)+⋯+apncqn(A)]=detA+udetC
