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<div class="section" id="spivak-notation">
<span id="chapter-spivak-notation"></span><h1>Spivak Notation<a class="headerlink" href="#spivak-notation" title="Permalink to this headline">ΒΆ</a></h1>
<p>To preserve our collective sanities, we will use Spivak’s notation for
derivatives. It is a functional notation that makes reading and
reasoning about expressions involving derivatives simple.</p>
<p>For a univariate function <span class="math">\(f\)</span>, <span class="math">\(f(a)\)</span> denotes its value at
<span class="math">\(a\)</span>. <span class="math">\(Df\)</span> denotes its first derivative, and
<span class="math">\(Df(a)\)</span> is the derivative evaluated at <span class="math">\(a\)</span>, i.e</p>
<div class="math">
\[Df(a) = \left . \frac{d}{dx} f(x) \right |_{x = a}\]</div>
<p><span class="math">\(D^kf\)</span> denotes the <span class="math">\(k^{\text{th}}\)</span> derivative of <span class="math">\(f\)</span>.</p>
<p>For a bi-variate function <span class="math">\(g(x,y)\)</span>. <span class="math">\(D_1g\)</span> and
<span class="math">\(D_2g\)</span> denote the partial derivatives of <span class="math">\(g\)</span> w.r.t the
first and second variable respectively. In the classical notation this
is equivalent to saying:</p>
<div class="math">
\[D_1 g = \frac{\partial}{\partial x}g(x,y) \text{ and } D_2 g = \frac{\partial}{\partial y}g(x,y).\]</div>
<p><span class="math">\(Dg\)</span> denotes the Jacobian of <cite>g</cite>, i.e.,</p>
<div class="math">
\[Dg = \begin{bmatrix} D_1g & D_2g \end{bmatrix}\]</div>
<p>More generally for a multivariate function <span class="math">\(g:\mathbb{R}^n
\longrightarrow \mathbb{R}^m\)</span>, <span class="math">\(Dg\)</span> denotes the <span class="math">\(m\times
n\)</span> Jacobian matrix. <span class="math">\(D_i g\)</span> is the partial derivative of
<span class="math">\(g\)</span> w.r.t the <span class="math">\(i^{\text{th}}\)</span> coordinate and the
<span class="math">\(i^{\text{th}}\)</span> column of <span class="math">\(Dg\)</span>.</p>
<p>Finally, <span class="math">\(D^2_1g\)</span> and <span class="math">\(D_1D_2g\)</span> have the obvious meaning
as higher order partial derivatives.</p>
<p>For more see Michael Spivak’s book <a class="reference external" href="https://www.amazon.com/Calculus-Manifolds-Approach-Classical-Theorems/dp/0805390219">Calculus on Manifolds</a>
or a brief discussion of the <a class="reference external" href="http://www.vendian.org/mncharity/dir3/dxdoc/">merits of this notation</a> by
Mitchell N. Charity.</p>
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