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Mathematics for Physics

169 Functions

A mapping that assigns to each element of one set a value in another set that is used to express relations between variables.

definition (Alternating Function) A property of a \(k\)-linear function from the \(k\)-fold product of a vector space to the real numbers, \(T: V^k \rightarrow \mathbb{R}\), where the following conditions apply:

  • \(T(v_{\sigma(1)}, \dots, v_{\sigma(k)}) = (\text{sgn } \sigma) T(v_1, \dots, v_k)\) for every permutation \(\sigma \in S_k\).
  • \(T(v_1, \dots, v_i, \dots, v_j, \dots, v_k) = -T(v_1, \dots, v_j, \dots, v_i, \dots, v_k)\) for any interchange of two arguments.
  • \(T(v_1, \dots, v_k) = 0\) whenever two of the vectors \(v_1, \dots, v_k\) are equal.

where

  • \(V\) is a vector space.
  • \(V^k\) is the \(k\)-fold Cartesian product \(V \times \dots \times V\).
  • \(S_k\) is the permutation group of \(k\) objects.
  • \(\text{sgn } \sigma\) is the sign of the permutation \(\sigma\), which is \(+1\) if the permutation is even and \(-1\) if it is odd.
  • \(v_1, \dots, v_k\) are vectors in \(V\).
  • \(k\) is a positive integer representing the degree of the function.

definition (Alternating k-linear function) A multilinear function from the \(k\)-fold product of a vector space to the real numbers, \(f: V^k \rightarrow \mathbb{R}\), where the following conditions apply:

  • \(f(v_{\sigma(1)}, \dots, v_{\sigma(k)}) = (\text{sgn } \sigma) f(v_1, \dots, v_k)\) for all \(\sigma \in S_k\).
  • \(f(v_1, \dots, v_i, \dots, v_j, \dots, v_k) = -f(v_1, \dots, v_j, \dots, v_i, \dots, v_k)\) for any interchange of two arguments.
  • \(f(v_1, \dots, v_k) = 0\) whenever two of the vectors \(v_1, \dots, v_k\) are equal.

where

  • \(V\) is a vector space.
  • \(V^k\) is the \(k\)-fold Cartesian product \(V \times \dots \times V\).
  • \(k\) is a positive integer representing the degree of the function.
  • \(S_k\) is the permutation group of \(k\) objects.
  • \(\text{sgn } \sigma\) is the sign of the permutation \(\sigma\).
  • \(v_1, \dots, v_k \in V\) are vectors.
  • \(A^k(V)\) is the vector space of all alternating \(k\)-linear functions on \(V\).

Note:

  • \(A^k(V)\) is also written \(\Lambda^k(V^*)\).
  • k-covector, multicovector of degree k, and alternating k-tensor are synonyms for an alternating \(k\)-linear function.

169.1 Elementary Example

169.1.1 Simple

A function maps each domain element to exactly one codomain value.

\[ f : A \rightarrow B \]

\[ A = \{ 1,\ 2,\ 3 \} \]

\[ B = \{ 2,\ 4,\ 9 \} \]

\[ f(1) = 2,\quad f(2) = 4,\quad f(3) = 9 \]

where

  • \(A\) is the domain.
  • \(B\) is the codomain.
  • \(f\) is the function.