Skip to main content
Engineering LibreTexts

15.2: D.2 The ε-δ relation

  • Page ID
  • \( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \) \( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} \)\(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\) \(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\)

    As was stated without proof in section 3.3.7, the alternating tensor is related to the 2nd-order identity tensor by

    \[\varepsilon_{i j k} \varepsilon_{k l m}=\delta_{i l} \delta_{j m}-\delta_{i m} \delta_{j l}.\label{eqn:1} \]

    The easiest way to convince yourself of this is to try a few tests. First, set \(i\) = \(j\) and verify that the right-hand side is zero, as it should be. Then try interchanging \(i\) and \(j\) and check that the right-hand side changes sign, as it should. The same tests work with \(l\) and \(m\). To remember Equation \(\ref{eqn:1}\), note that the first \(\delta\) on the right-hand side has subscripts \(i\) and \(l\); these are the first free indices of the two \(\varepsilon\)’s on the left-hand side. After this, the remaining pairs of indices fall into place naturally.

    Figure \(\PageIndex{1}\): Relation of the triple product to the volume of the parallelepiped bounded by three vectors \(\vec{u}\), \(\vec{v}\) and \(\vec{w}\).

    15.2: D.2 The ε-δ relation is shared under a not declared license and was authored, remixed, and/or curated by LibreTexts.

    • Was this article helpful?