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Positive Integers and the Mathematical Induction 2. Permutations 1. Permutations 2. Combinations 1. Combinations 2. Properties of the Function nk 3. Newton's Multinomial Formula 1. Newton's Binomial Formula 2. Multiplication of Permutations 1. Definition 2. Factorization of a Permutation into Cycles 3. Transpositions III.

Fields 1. Number Fields 2. General Fields 3. Isomorphism of Fields 4. Field of Real Numbers 5. Properties of Complex Numbers 1. Geometric Interpretation of Complex Numbers 2. The Modulus and Conjugate Complex Numbers 3. Roots of Complex Numbers 1. Square Roots of Complex Numbers 2. Roots of Higher Degree of Complex Numbers 3. Primitive Roots of Unity 4. Definition of a Determinant 1.

Introduction 2.

Inversions 3. Applications of Inversions to the Theory of Permutations 4. Definition of a Matrix 5.

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Laplace Expansion 1. Minors 2. Examples 1. Simple Examples 2. Example of a Cyclic Determinant 3. Vandermonde Determinant 4. Cauchy's Theorem and its Generalizations 1. Cauchy's Theorem 2. Cyclic Determinant 3. Generalization of Cauchy's Theorem V. Vector Spaces 1. Linear Independence 3.

## Higher-Dimensional Algebra VI: Lie 2-Algebras

Linear Subspaces 4. Rank of a Matrix 1. Simplest Properties 2. Investigation of the Rank of a Matrix by means of Minors 3. Linear Equations 1. General Systems of Linear Equations 2. Axiomatic Definition of the Determinant VI. Operations on Polynomials 1. Polynomials 2.

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Differentiation of Polynomials 3. The Arithmetic of the Ring K[x] 1. The Arithmetic of Integers 2. Division of Polynomials 3. Roots of a Polynomial 1. Multiple Roots 2. Divisibility of a Polynomial by Linear Factors 3. Interpolation Formulae 1. Lagrange's Formula 2. Newton's Interpolation Formula 3.

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Some Notions on Finite Differences 4. Rational Functions 1. Definitions 2. Partial Fractions VII. The Fundamental Theorem of Algebra 1. The Fundamental Theorem of Algebra 3. Polynomials of the Ring R[x] 1. Fundamental Theorems 2. Sturm's Theorem 3. Quadratic Equations in the Domain of Complex Numbers 1. Simplification of Equations by Substitution 2. Cubic Equations 1. Cardan's Formulae 2. By using our site, you acknowledge that you have read and understand our Cookie Policy , Privacy Policy , and our Terms of Service. Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields.

It only takes a minute to sign up. Is there a way to easily extend this formula to higher dimensions, analogous to the way that the volume of a sphere can be extended to higher dimensions? Sign up to join this community. The best answers are voted up and rise to the top. Home Questions Tags Users Unanswered. How to extend the parallelepiped volume formula to higher dimensions? Ask Question. Asked 3 years, 8 months ago.

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