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<center><b><big><big>An Important Reverse Mode Identity</big></big></b></center>
The theorem and the proof below is a restatement
of the results on page 236 of
<a href="bib.xml#Evaluating Derivatives" target="_top"><span style='white-space: nowrap'>Evaluating Derivatives</span></a>
.
<br/>
<br/>
<b><big><a name="Notation" id="Notation">Notation</a></big></b>
<br/>
Given a function
<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow>
<mi mathvariant='italic'>f</mi>
<mo stretchy="false">(</mo>
<mi mathvariant='italic'>u</mi>
<mo stretchy="false">,</mo>
<mi mathvariant='italic'>v</mi>
<mo stretchy="false">)</mo>
</mrow></math>
where
<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow>
<mi mathvariant='italic'>u</mi>
<mo stretchy="false">∈</mo>
<msup><mi mathvariant='italic'>B</mi>
<mi mathvariant='italic'>n</mi>
</msup>
</mrow></math>
we use the notation
<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mrow>
<mfrac><mrow><mo stretchy="false">∂</mo>
<mi mathvariant='italic'>f</mi>
</mrow>
<mrow><mo stretchy="false">∂</mo>
<mi mathvariant='italic'>u</mi>
</mrow>
</mfrac>
<mo stretchy="false">(</mo>
<mi mathvariant='italic'>u</mi>
<mo stretchy="false">,</mo>
<mi mathvariant='italic'>v</mi>
<mo stretchy="false">)</mo>
<mo stretchy="false">=</mo>
<mrow><mo stretchy="true">[</mo><mrow><mfrac><mrow><mo stretchy="false">∂</mo>
<mi mathvariant='italic'>f</mi>
</mrow>
<mrow><mo stretchy="false">∂</mo>
<msub><mi mathvariant='italic'>u</mi>
<mn>1</mn>
</msub>
</mrow>
</mfrac>
<mo stretchy="false">(</mo>
<mi mathvariant='italic'>u</mi>
<mo stretchy="false">,</mo>
<mi mathvariant='italic'>v</mi>
<mo stretchy="false">)</mo>
<mo stretchy="false">,</mo>
<mo stretchy="false">⋯</mo>
<mo stretchy="false">,</mo>
<mfrac><mrow><mo stretchy="false">∂</mo>
<mi mathvariant='italic'>f</mi>
</mrow>
<mrow><mo stretchy="false">∂</mo>
<msub><mi mathvariant='italic'>u</mi>
<mi mathvariant='italic'>n</mi>
</msub>
</mrow>
</mfrac>
<mo stretchy="false">(</mo>
<mi mathvariant='italic'>u</mi>
<mo stretchy="false">,</mo>
<mi mathvariant='italic'>v</mi>
<mo stretchy="false">)</mo>
</mrow><mo stretchy="true">]</mo></mrow>
</mrow></math>
<br/>
<b><big><a name="Reverse Sweep" id="Reverse Sweep">Reverse Sweep</a></big></b>
<br/>
When using <a href="reverse_any.xml" target="_top"><span style='white-space: nowrap'>reverse mode</span></a>
we are given a function
<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow>
<mi mathvariant='italic'>F</mi>
<mo stretchy="false">:</mo>
<msup><mi mathvariant='italic'>B</mi>
<mi mathvariant='italic'>n</mi>
</msup>
<mo stretchy="false">→</mo>
<msup><mi mathvariant='italic'>B</mi>
<mi mathvariant='italic'>m</mi>
</msup>
</mrow></math>
,
a matrix of Taylor coefficients
<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow>
<mi mathvariant='italic'>x</mi>
<mo stretchy="false">∈</mo>
<msup><mi mathvariant='italic'>B</mi>
<mrow><mi mathvariant='italic'>n</mi>
<mo stretchy="false">×</mo>
<mi mathvariant='italic'>p</mi>
</mrow>
</msup>
</mrow></math>
,
and a weight vector
<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow>
<mi mathvariant='italic'>w</mi>
<mo stretchy="false">∈</mo>
<msup><mi mathvariant='italic'>B</mi>
<mi mathvariant='italic'>m</mi>
</msup>
</mrow></math>
.
We define the functions
<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow>
<mi mathvariant='italic'>X</mi>
<mo stretchy="false">:</mo>
<mi mathvariant='italic'>B</mi>
<mo stretchy="false">×</mo>
<msup><mi mathvariant='italic'>B</mi>
<mrow><mi mathvariant='italic'>n</mi>
<mo stretchy="false">×</mo>
<mi mathvariant='italic'>p</mi>
</mrow>
</msup>
<mo stretchy="false">→</mo>
<msup><mi mathvariant='italic'>B</mi>
<mi mathvariant='italic'>n</mi>
</msup>
</mrow></math>
,
<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow>
<mi mathvariant='italic'>W</mi>
<mo stretchy="false">:</mo>
<mi mathvariant='italic'>B</mi>
<mo stretchy="false">×</mo>
<msup><mi mathvariant='italic'>B</mi>
<mrow><mi mathvariant='italic'>n</mi>
<mo stretchy="false">×</mo>
<mi mathvariant='italic'>p</mi>
</mrow>
</msup>
<mo stretchy="false">→</mo>
<mi mathvariant='italic'>B</mi>
</mrow></math>
, and
<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow>
<msub><mi mathvariant='italic'>W</mi>
<mi mathvariant='italic'>j</mi>
</msub>
<mo stretchy="false">:</mo>
<msup><mi mathvariant='italic'>B</mi>
<mrow><mi mathvariant='italic'>n</mi>
<mo stretchy="false">×</mo>
<mi mathvariant='italic'>p</mi>
</mrow>
</msup>
<mo stretchy="false">→</mo>
<mi mathvariant='italic'>B</mi>
</mrow></math>
by
<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mrow>
<mtable rowalign="center" ><mtr><mtd columnalign="right" >
<mi mathvariant='italic'>X</mi>
<mo stretchy="false">(</mo>
<mi mathvariant='italic'>t</mi>
<mo stretchy="false">,</mo>
<mi mathvariant='italic'>x</mi>
<mo stretchy="false">)</mo>
</mtd><mtd columnalign="center" >
<mo stretchy="false">=</mo>
</mtd><mtd columnalign="left" >
<msup><mi mathvariant='italic'>x</mi>
<mrow><mo stretchy="false">(</mo>
<mn>0</mn>
<mo stretchy="false">)</mo>
</mrow>
</msup>
<mo stretchy="false">+</mo>
<msup><mi mathvariant='italic'>x</mi>
<mrow><mo stretchy="false">(</mo>
<mn>1</mn>
<mo stretchy="false">)</mo>
</mrow>
</msup>
<mi mathvariant='italic'>t</mi>
<mo stretchy="false">+</mo>
<mo stretchy="false">⋯</mo>
<mo stretchy="false">+</mo>
<msup><mi mathvariant='italic'>x</mi>
<mrow><mo stretchy="false">(</mo>
<mi mathvariant='italic'>p</mi>
<mn>-1</mn>
<mo stretchy="false">)</mo>
</mrow>
</msup>
<msup><mi mathvariant='italic'>t</mi>
<mrow><mi mathvariant='italic'>p</mi>
<mn>-1</mn>
</mrow>
</msup>
</mtd></mtr><mtr><mtd columnalign="right" >
<mi mathvariant='italic'>W</mi>
<mo stretchy="false">(</mo>
<mi mathvariant='italic'>t</mi>
<mo stretchy="false">,</mo>
<mi mathvariant='italic'>x</mi>
<mo stretchy="false">)</mo>
</mtd><mtd columnalign="center" >
<mo stretchy="false">=</mo>
</mtd><mtd columnalign="left" >
<msub><mi mathvariant='italic'>w</mi>
<mn>0</mn>
</msub>
<msub><mi mathvariant='italic'>F</mi>
<mn>0</mn>
</msub>
<mo stretchy="false">[</mo>
<mi mathvariant='italic'>X</mi>
<mo stretchy="false">(</mo>
<mi mathvariant='italic'>t</mi>
<mo stretchy="false">,</mo>
<mi mathvariant='italic'>x</mi>
<mo stretchy="false">)</mo>
<mo stretchy="false">]</mo>
<mo stretchy="false">+</mo>
<mo stretchy="false">⋯</mo>
<mo stretchy="false">+</mo>
<msub><mi mathvariant='italic'>w</mi>
<mrow><mi mathvariant='italic'>m</mi>
<mn>-1</mn>
</mrow>
</msub>
<msub><mi mathvariant='italic'>F</mi>
<mrow><mi mathvariant='italic'>m</mi>
<mn>-1</mn>
</mrow>
</msub>
<mo stretchy="false">[</mo>
<mi mathvariant='italic'>X</mi>
<mo stretchy="false">(</mo>
<mi mathvariant='italic'>t</mi>
<mo stretchy="false">,</mo>
<mi mathvariant='italic'>x</mi>
<mo stretchy="false">)</mo>
<mo stretchy="false">]</mo>
</mtd></mtr><mtr><mtd columnalign="right" >
<msub><mi mathvariant='italic'>W</mi>
<mi mathvariant='italic'>j</mi>
</msub>
<mo stretchy="false">(</mo>
<mi mathvariant='italic'>x</mi>
<mo stretchy="false">)</mo>
</mtd><mtd columnalign="center" >
<mo stretchy="false">=</mo>
</mtd><mtd columnalign="left" >
<mfrac><mrow><mn>1</mn>
</mrow>
<mrow><mi mathvariant='italic'>j</mi>
<mo stretchy="false">!</mo>
</mrow>
</mfrac>
<mfrac><mrow><msup><mo stretchy="false">∂</mo>
<mrow><mi mathvariant='italic'>j</mi>
</mrow>
</msup>
</mrow>
<mrow><mo stretchy="false">∂</mo>
<msup><mrow><mi mathvariant='italic'>t</mi>
</mrow>
<mrow><mi mathvariant='italic'>j</mi>
</mrow>
</msup>
</mrow>
</mfrac>
<mi mathvariant='italic'>W</mi>
<mo stretchy="false">(</mo>
<mn>0</mn>
<mo stretchy="false">,</mo>
<mi mathvariant='italic'>x</mi>
<mo stretchy="false">)</mo>
</mtd></mtr></mtable>
</mrow></math>
where
<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow>
<msup><mi mathvariant='italic'>x</mi>
<mrow><mo stretchy="false">(</mo>
<mi mathvariant='italic'>j</mi>
<mo stretchy="false">)</mo>
</mrow>
</msup>
</mrow></math>
is the <i>j</i>-th column of
<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow>
<mi mathvariant='italic'>x</mi>
<mo stretchy="false">∈</mo>
<msup><mi mathvariant='italic'>B</mi>
<mrow><mi mathvariant='italic'>n</mi>
<mo stretchy="false">×</mo>
<mi mathvariant='italic'>p</mi>
</mrow>
</msup>
</mrow></math>
.
The theorem below implies that
<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mrow>
<mfrac><mrow><mo stretchy="false">∂</mo>
<msub><mi mathvariant='italic'>W</mi>
<mi mathvariant='italic'>j</mi>
</msub>
</mrow>
<mrow><mo stretchy="false">∂</mo>
<msup><mi mathvariant='italic'>x</mi>
<mrow><mo stretchy="false">(</mo>
<mi mathvariant='italic'>i</mi>
<mo stretchy="false">)</mo>
</mrow>
</msup>
</mrow>
</mfrac>
<mo stretchy="false">(</mo>
<mi mathvariant='italic'>x</mi>
<mo stretchy="false">)</mo>
<mo stretchy="false">=</mo>
<mfrac><mrow><mo stretchy="false">∂</mo>
<msub><mi mathvariant='italic'>W</mi>
<mrow><mi mathvariant='italic'>j</mi>
<mo stretchy="false">-</mo>
<mi mathvariant='italic'>i</mi>
</mrow>
</msub>
</mrow>
<mrow><mo stretchy="false">∂</mo>
<msup><mi mathvariant='italic'>x</mi>
<mrow><mo stretchy="false">(</mo>
<mn>0</mn>
<mo stretchy="false">)</mo>
</mrow>
</msup>
</mrow>
</mfrac>
<mo stretchy="false">(</mo>
<mi mathvariant='italic'>x</mi>
<mo stretchy="false">)</mo>
</mrow></math>
A <a href="reverse_any.xml" target="_top"><span style='white-space: nowrap'>general reverse sweep</span></a>
calculates the values
<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mrow>
<mfrac><mrow><mo stretchy="false">∂</mo>
<msub><mi mathvariant='italic'>W</mi>
<mrow><mi mathvariant='italic'>p</mi>
<mn>-1</mn>
</mrow>
</msub>
</mrow>
<mrow><mo stretchy="false">∂</mo>
<msup><mi mathvariant='italic'>x</mi>
<mrow><mo stretchy="false">(</mo>
<mi mathvariant='italic'>i</mi>
<mo stretchy="false">)</mo>
</mrow>
</msup>
</mrow>
</mfrac>
<mo stretchy="false">(</mo>
<mi mathvariant='italic'>x</mi>
<mo stretchy="false">)</mo>
<mspace width='1cm'/>
<mo stretchy="false">(</mo>
<mi mathvariant='italic'>i</mi>
<mo stretchy="false">=</mo>
<mn>0</mn>
<mo stretchy="false">,</mo>
<mo stretchy="false">…</mo>
<mo stretchy="false">,</mo>
<mi mathvariant='italic'>p</mi>
<mn>-1</mn>
<mo stretchy="false">)</mo>
</mrow></math>
But the return values for a reverse sweep are specified
in terms of the more useful values
<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mrow>
<mfrac><mrow><mo stretchy="false">∂</mo>
<msub><mi mathvariant='italic'>W</mi>
<mi mathvariant='italic'>j</mi>
</msub>
</mrow>
<mrow><mo stretchy="false">∂</mo>
<msup><mi mathvariant='italic'>x</mi>
<mrow><mo stretchy="false">(</mo>
<mn>0</mn>
<mo stretchy="false">)</mo>
</mrow>
</msup>
</mrow>
</mfrac>
<mo stretchy="false">(</mo>
<mi mathvariant='italic'>x</mi>
<mo stretchy="false">)</mo>
<mspace width='1cm'/>
<mo stretchy="false">(</mo>
<mi mathvariant='italic'>j</mi>
<mo stretchy="false">=</mo>
<mn>0</mn>
<mo stretchy="false">,</mo>
<mo stretchy="false">…</mo>
<mo stretchy="false">,</mo>
<mi mathvariant='italic'>p</mi>
<mn>-1</mn>
<mo stretchy="false">)</mo>
</mrow></math>
<br/>
<b><big><a name="Theorem" id="Theorem">Theorem</a></big></b>
<br/>
Suppose that
<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow>
<mi mathvariant='italic'>F</mi>
<mo stretchy="false">:</mo>
<msup><mi mathvariant='italic'>B</mi>
<mi mathvariant='italic'>n</mi>
</msup>
<mo stretchy="false">→</mo>
<msup><mi mathvariant='italic'>B</mi>
<mi mathvariant='italic'>m</mi>
</msup>
</mrow></math>
is a
<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow>
<mi mathvariant='italic'>p</mi>
</mrow></math>
times
continuously differentiable function.
Define the functions
<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow>
<mi mathvariant='italic'>Z</mi>
<mo stretchy="false">:</mo>
<mi mathvariant='italic'>B</mi>
<mo stretchy="false">×</mo>
<msup><mi mathvariant='italic'>B</mi>
<mrow><mi mathvariant='italic'>n</mi>
<mo stretchy="false">×</mo>
<mi mathvariant='italic'>p</mi>
</mrow>
</msup>
<mo stretchy="false">→</mo>
<msup><mi mathvariant='italic'>B</mi>
<mi mathvariant='italic'>n</mi>
</msup>
</mrow></math>
,
<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow>
<mi mathvariant='italic'>Y</mi>
<mo stretchy="false">:</mo>
<mi mathvariant='italic'>B</mi>
<mo stretchy="false">×</mo>
<msup><mi mathvariant='italic'>B</mi>
<mrow><mi mathvariant='italic'>n</mi>
<mo stretchy="false">×</mo>
<mi mathvariant='italic'>p</mi>
</mrow>
</msup>
<mo stretchy="false">→</mo>
<msup><mi mathvariant='italic'>B</mi>
<mi mathvariant='italic'>m</mi>
</msup>
</mrow></math>
,
and
<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow>
<msup><mi mathvariant='italic'>y</mi>
<mrow><mo stretchy="false">(</mo>
<mi mathvariant='italic'>j</mi>
<mo stretchy="false">)</mo>
</mrow>
</msup>
<mo stretchy="false">:</mo>
<msup><mi mathvariant='italic'>B</mi>
<mrow><mi mathvariant='italic'>n</mi>
<mo stretchy="false">×</mo>
<mi mathvariant='italic'>p</mi>
</mrow>
</msup>
<mo stretchy="false">→</mo>
<msup><mi mathvariant='italic'>B</mi>
<mi mathvariant='italic'>m</mi>
</msup>
</mrow></math>
by
<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mrow>
<mtable rowalign="center" ><mtr><mtd columnalign="right" >
<mi mathvariant='italic'>Z</mi>
<mo stretchy="false">(</mo>
<mi mathvariant='italic'>t</mi>
<mo stretchy="false">,</mo>
<mi mathvariant='italic'>x</mi>
<mo stretchy="false">)</mo>
</mtd><mtd columnalign="center" >
<mo stretchy="false">=</mo>
</mtd><mtd columnalign="left" >
<msup><mi mathvariant='italic'>x</mi>
<mrow><mo stretchy="false">(</mo>
<mn>0</mn>
<mo stretchy="false">)</mo>
</mrow>
</msup>
<mo stretchy="false">+</mo>
<msup><mi mathvariant='italic'>x</mi>
<mrow><mo stretchy="false">(</mo>
<mn>1</mn>
<mo stretchy="false">)</mo>
</mrow>
</msup>
<mi mathvariant='italic'>t</mi>
<mo stretchy="false">+</mo>
<mo stretchy="false">⋯</mo>
<mo stretchy="false">+</mo>
<msup><mi mathvariant='italic'>x</mi>
<mrow><mo stretchy="false">(</mo>
<mi mathvariant='italic'>p</mi>
<mn>-1</mn>
<mo stretchy="false">)</mo>
</mrow>
</msup>
<msup><mi mathvariant='italic'>t</mi>
<mrow><mi mathvariant='italic'>p</mi>
<mn>-1</mn>
</mrow>
</msup>
</mtd></mtr><mtr><mtd columnalign="right" >
<mi mathvariant='italic'>Y</mi>
<mo stretchy="false">(</mo>
<mi mathvariant='italic'>t</mi>
<mo stretchy="false">,</mo>
<mi mathvariant='italic'>x</mi>
<mo stretchy="false">)</mo>
</mtd><mtd columnalign="center" >
<mo stretchy="false">=</mo>
</mtd><mtd columnalign="left" >
<mi mathvariant='italic'>F</mi>
<mo stretchy="false">[</mo>
<mi mathvariant='italic'>Z</mi>
<mo stretchy="false">(</mo>
<mi mathvariant='italic'>t</mi>
<mo stretchy="false">,</mo>
<mi mathvariant='italic'>x</mi>
<mo stretchy="false">)</mo>
<mo stretchy="false">]</mo>
</mtd></mtr><mtr><mtd columnalign="right" >
<msup><mi mathvariant='italic'>y</mi>
<mrow><mo stretchy="false">(</mo>
<mi mathvariant='italic'>j</mi>
<mo stretchy="false">)</mo>
</mrow>
</msup>
<mo stretchy="false">(</mo>
<mi mathvariant='italic'>x</mi>
<mo stretchy="false">)</mo>
</mtd><mtd columnalign="center" >
<mo stretchy="false">=</mo>
</mtd><mtd columnalign="left" >
<mfrac><mrow><mn>1</mn>
</mrow>
<mrow><mi mathvariant='italic'>j</mi>
<mo stretchy="false">!</mo>
</mrow>
</mfrac>
<mfrac><mrow><msup><mo stretchy="false">∂</mo>
<mrow><mi mathvariant='italic'>j</mi>
</mrow>
</msup>
</mrow>
<mrow><mo stretchy="false">∂</mo>
<msup><mrow><mi mathvariant='italic'>t</mi>
</mrow>
<mrow><mi mathvariant='italic'>j</mi>
</mrow>
</msup>
</mrow>
</mfrac>
<mi mathvariant='italic'>Y</mi>
<mo stretchy="false">(</mo>
<mn>0</mn>
<mo stretchy="false">,</mo>
<mi mathvariant='italic'>x</mi>
<mo stretchy="false">)</mo>
</mtd></mtr></mtable>
</mrow></math>
where
<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow>
<msup><mi mathvariant='italic'>x</mi>
<mrow><mo stretchy="false">(</mo>
<mi mathvariant='italic'>j</mi>
<mo stretchy="false">)</mo>
</mrow>
</msup>
</mrow></math>
denotes the <i>j</i>-th column of
<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow>
<mi mathvariant='italic'>x</mi>
<mo stretchy="false">∈</mo>
<msup><mi mathvariant='italic'>B</mi>
<mrow><mi mathvariant='italic'>n</mi>
<mo stretchy="false">×</mo>
<mi mathvariant='italic'>p</mi>
</mrow>
</msup>
</mrow></math>
.
It follows that
for all
<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow>
<mi mathvariant='italic'>i</mi>
<mo stretchy="false">,</mo>
<mi mathvariant='italic'>j</mi>
</mrow></math>
such that
<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow>
<mi mathvariant='italic'>i</mi>
<mo stretchy="false">≤</mo>
<mi mathvariant='italic'>j</mi>
<mo stretchy="false"><</mo>
<mi mathvariant='italic'>p</mi>
</mrow></math>
,
<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mrow>
<mtable rowalign="center" ><mtr><mtd columnalign="right" >
<mfrac><mrow><mo stretchy="false">∂</mo>
<msup><mi mathvariant='italic'>y</mi>
<mrow><mo stretchy="false">(</mo>
<mi mathvariant='italic'>j</mi>
<mo stretchy="false">)</mo>
</mrow>
</msup>
</mrow>
<mrow><mo stretchy="false">∂</mo>
<msup><mi mathvariant='italic'>x</mi>
<mrow><mo stretchy="false">(</mo>
<mi mathvariant='italic'>i</mi>
<mo stretchy="false">)</mo>
</mrow>
</msup>
</mrow>
</mfrac>
<mo stretchy="false">(</mo>
<mi mathvariant='italic'>x</mi>
<mo stretchy="false">)</mo>
</mtd><mtd columnalign="center" >
<mo stretchy="false">=</mo>
</mtd><mtd columnalign="left" >
<mfrac><mrow><mo stretchy="false">∂</mo>
<msup><mi mathvariant='italic'>y</mi>
<mrow><mo stretchy="false">(</mo>
<mi mathvariant='italic'>j</mi>
<mo stretchy="false">-</mo>
<mi mathvariant='italic'>i</mi>
<mo stretchy="false">)</mo>
</mrow>
</msup>
</mrow>
<mrow><mo stretchy="false">∂</mo>
<msup><mi mathvariant='italic'>x</mi>
<mrow><mo stretchy="false">(</mo>
<mn>0</mn>
<mo stretchy="false">)</mo>
</mrow>
</msup>
</mrow>
</mfrac>
<mo stretchy="false">(</mo>
<mi mathvariant='italic'>x</mi>
<mo stretchy="false">)</mo>
</mtd></mtr></mtable>
</mrow></math>
<br/>
<b><big><a name="Proof" id="Proof">Proof</a></big></b>
<br/>
If follows from the definitions that
<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mrow>
<mtable rowalign="center" ><mtr><mtd columnalign="right" >
<mfrac><mrow><mo stretchy="false">∂</mo>
<msup><mi mathvariant='italic'>y</mi>
<mrow><mo stretchy="false">(</mo>
<mi mathvariant='italic'>j</mi>
<mo stretchy="false">)</mo>
</mrow>
</msup>
</mrow>
<mrow><mo stretchy="false">∂</mo>
<msup><mi mathvariant='italic'>x</mi>
<mrow><mo stretchy="false">(</mo>
<mi mathvariant='italic'>i</mi>
<mo stretchy="false">)</mo>
</mrow>
</msup>
</mrow>
</mfrac>
<mo stretchy="false">(</mo>
<mi mathvariant='italic'>x</mi>
<mo stretchy="false">)</mo>
</mtd><mtd columnalign="center" >
<mo stretchy="false">=</mo>
</mtd><mtd columnalign="left" >
<mfrac><mrow><mn>1</mn>
</mrow>
<mrow><mi mathvariant='italic'>j</mi>
<mo stretchy="false">!</mo>
</mrow>
</mfrac>
<mfrac><mrow><mo stretchy="false">∂</mo>
</mrow>
<mrow><mo stretchy="false">∂</mo>
<msup><mi mathvariant='italic'>x</mi>
<mrow><mo stretchy="false">(</mo>
<mi mathvariant='italic'>i</mi>
<mo stretchy="false">)</mo>
</mrow>
</msup>
</mrow>
</mfrac>
<msub><mrow><mo stretchy="true">[</mo><mrow><mfrac><mrow><msup><mo stretchy="false">∂</mo>
<mrow><mi mathvariant='italic'>j</mi>
</mrow>
</msup>
</mrow>
<mrow><mo stretchy="false">∂</mo>
<msup><mrow><mi mathvariant='italic'>t</mi>
</mrow>
<mrow><mi mathvariant='italic'>j</mi>
</mrow>
</msup>
</mrow>
</mfrac>
<mo stretchy="false">(</mo>
<mi mathvariant='italic'>F</mi>
<mo stretchy="false">∘</mo>
<mi mathvariant='italic'>Z</mi>
<mo stretchy="false">)</mo>
<mo stretchy="false">(</mo>
<mi mathvariant='italic'>t</mi>
<mo stretchy="false">,</mo>
<mi mathvariant='italic'>x</mi>
<mo stretchy="false">)</mo>
</mrow><mo stretchy="true">]</mo></mrow>
<mrow><mi mathvariant='italic'>t</mi>
<mo stretchy="false">=</mo>
<mn>0</mn>
</mrow>
</msub>
</mtd></mtr><mtr><mtd columnalign="right" >
</mtd><mtd columnalign="center" >
<mo stretchy="false">=</mo>
</mtd><mtd columnalign="left" >
<mfrac><mrow><mn>1</mn>
</mrow>
<mrow><mi mathvariant='italic'>j</mi>
<mo stretchy="false">!</mo>
</mrow>
</mfrac>
<msub><mrow><mo stretchy="true">[</mo><mrow><mfrac><mrow><msup><mo stretchy="false">∂</mo>
<mrow><mi mathvariant='italic'>j</mi>
</mrow>
</msup>
</mrow>
<mrow><mo stretchy="false">∂</mo>
<msup><mrow><mi mathvariant='italic'>t</mi>
</mrow>
<mrow><mi mathvariant='italic'>j</mi>
</mrow>
</msup>
</mrow>
</mfrac>
<mfrac><mrow><mo stretchy="false">∂</mo>
</mrow>
<mrow><mo stretchy="false">∂</mo>
<msup><mi mathvariant='italic'>x</mi>
<mrow><mo stretchy="false">(</mo>
<mi mathvariant='italic'>i</mi>
<mo stretchy="false">)</mo>
</mrow>
</msup>
</mrow>
</mfrac>
<mo stretchy="false">(</mo>
<mi mathvariant='italic'>F</mi>
<mo stretchy="false">∘</mo>
<mi mathvariant='italic'>Z</mi>
<mo stretchy="false">)</mo>
<mo stretchy="false">(</mo>
<mi mathvariant='italic'>t</mi>
<mo stretchy="false">,</mo>
<mi mathvariant='italic'>x</mi>
<mo stretchy="false">)</mo>
</mrow><mo stretchy="true">]</mo></mrow>
<mrow><mi mathvariant='italic'>t</mi>
<mo stretchy="false">=</mo>
<mn>0</mn>
</mrow>
</msub>
</mtd></mtr><mtr><mtd columnalign="right" >
</mtd><mtd columnalign="center" >
<mo stretchy="false">=</mo>
</mtd><mtd columnalign="left" >
<mfrac><mrow><mn>1</mn>
</mrow>
<mrow><mi mathvariant='italic'>j</mi>
<mo stretchy="false">!</mo>
</mrow>
</mfrac>
<msub><mrow><mo stretchy="true">{</mo><mrow><mfrac><mrow><msup><mo stretchy="false">∂</mo>
<mrow><mi mathvariant='italic'>j</mi>
</mrow>
</msup>
</mrow>
<mrow><mo stretchy="false">∂</mo>
<msup><mrow><mi mathvariant='italic'>t</mi>
</mrow>
<mrow><mi mathvariant='italic'>j</mi>
</mrow>
</msup>
</mrow>
</mfrac>
<mrow><mo stretchy="true">[</mo><mrow><msup><mi mathvariant='italic'>t</mi>
<mi mathvariant='italic'>i</mi>
</msup>
<mo stretchy="false">(</mo>
<msup><mi mathvariant='italic'>F</mi>
<mrow><mo stretchy="false">(</mo>
<mn>1</mn>
<mo stretchy="false">)</mo>
</mrow>
</msup>
<mo stretchy="false">∘</mo>
<mi mathvariant='italic'>Z</mi>
<mo stretchy="false">)</mo>
<mo stretchy="false">(</mo>
<mi mathvariant='italic'>t</mi>
<mo stretchy="false">,</mo>
<mi mathvariant='italic'>x</mi>
<mo stretchy="false">)</mo>
</mrow><mo stretchy="true">]</mo></mrow>
</mrow><mo stretchy="true">}</mo></mrow>
<mrow><mi mathvariant='italic'>t</mi>
<mo stretchy="false">=</mo>
<mn>0</mn>
</mrow>
</msub>
</mtd></mtr></mtable>
</mrow></math>
For
<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow>
<mi mathvariant='italic'>k</mi>
<mo stretchy="false">></mo>
<mi mathvariant='italic'>i</mi>
</mrow></math>
, the <i>k</i>-th
partial of
<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow>
<msup><mi mathvariant='italic'>t</mi>
<mi mathvariant='italic'>i</mi>
</msup>
</mrow></math>
with respect to
<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow>
<mi mathvariant='italic'>t</mi>
</mrow></math>
is zero.
Thus, the partial with respect to
<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow>
<mi mathvariant='italic'>t</mi>
</mrow></math>
is given by
<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mrow>
<mtable rowalign="center" ><mtr><mtd columnalign="right" >
<mfrac><mrow><msup><mo stretchy="false">∂</mo>
<mrow><mi mathvariant='italic'>j</mi>
</mrow>
</msup>
</mrow>
<mrow><mo stretchy="false">∂</mo>
<msup><mrow><mi mathvariant='italic'>t</mi>
</mrow>
<mrow><mi mathvariant='italic'>j</mi>
</mrow>
</msup>
</mrow>
</mfrac>
<mrow><mo stretchy="true">[</mo><mrow><msup><mi mathvariant='italic'>t</mi>
<mi mathvariant='italic'>i</mi>
</msup>
<mo stretchy="false">(</mo>
<msup><mi mathvariant='italic'>F</mi>
<mrow><mo stretchy="false">(</mo>
<mn>1</mn>
<mo stretchy="false">)</mo>
</mrow>
</msup>
<mo stretchy="false">∘</mo>
<mi mathvariant='italic'>Z</mi>
<mo stretchy="false">)</mo>
<mo stretchy="false">(</mo>
<mi mathvariant='italic'>t</mi>
<mo stretchy="false">,</mo>
<mi mathvariant='italic'>x</mi>
<mo stretchy="false">)</mo>
</mrow><mo stretchy="true">]</mo></mrow>
</mtd><mtd columnalign="center" >
<mo stretchy="false">=</mo>
</mtd><mtd columnalign="left" >
<munderover><mo displaystyle='true' largeop='true'>∑</mo>
<mrow><mi mathvariant='italic'>k</mi>
<mo stretchy="false">=</mo>
<mn>0</mn>
</mrow>
<mi mathvariant='italic'>i</mi>
</munderover>
<mrow><mo stretchy="true">(</mo><mrow><mtable rowalign="center" ><mtr><mtd columnalign="center" >
<mi mathvariant='italic'>j</mi>
</mtd></mtr><mtr><mtd columnalign="center" >
<mi mathvariant='italic'>k</mi>
</mtd></mtr></mtable>
</mrow><mo stretchy="true">)</mo></mrow>
<mfrac><mrow><mi mathvariant='italic'>i</mi>
<mo stretchy="false">!</mo>
</mrow>
<mrow><mo stretchy="false">(</mo>
<mi mathvariant='italic'>i</mi>
<mo stretchy="false">-</mo>
<mi mathvariant='italic'>k</mi>
<mo stretchy="false">)</mo>
<mo stretchy="false">!</mo>
</mrow>
</mfrac>
<msup><mi mathvariant='italic'>t</mi>
<mrow><mi mathvariant='italic'>i</mi>
<mo stretchy="false">-</mo>
<mi mathvariant='italic'>k</mi>
</mrow>
</msup>
<mspace width='.3em'/>
<mfrac><mrow><msup><mo stretchy="false">∂</mo>
<mrow><mi mathvariant='italic'>j</mi>
<mo stretchy="false">-</mo>
<mi mathvariant='italic'>k</mi>
</mrow>
</msup>
</mrow>
<mrow><mo stretchy="false">∂</mo>
<msup><mrow><mi mathvariant='italic'>t</mi>
</mrow>
<mrow><mi mathvariant='italic'>j</mi>
<mo stretchy="false">-</mo>
<mi mathvariant='italic'>k</mi>
</mrow>
</msup>
</mrow>
</mfrac>
<mo stretchy="false">(</mo>
<msup><mi mathvariant='italic'>F</mi>
<mrow><mo stretchy="false">(</mo>
<mn>1</mn>
<mo stretchy="false">)</mo>
</mrow>
</msup>
<mo stretchy="false">∘</mo>
<mi mathvariant='italic'>Z</mi>
<mo stretchy="false">)</mo>
<mo stretchy="false">(</mo>
<mi mathvariant='italic'>t</mi>
<mo stretchy="false">,</mo>
<mi mathvariant='italic'>x</mi>
<mo stretchy="false">)</mo>
</mtd></mtr><mtr><mtd columnalign="right" >
<msub><mrow><mo stretchy="true">{</mo><mrow><mfrac><mrow><msup><mo stretchy="false">∂</mo>
<mrow><mi mathvariant='italic'>j</mi>
</mrow>
</msup>
</mrow>
<mrow><mo stretchy="false">∂</mo>
<msup><mrow><mi mathvariant='italic'>t</mi>
</mrow>
<mrow><mi mathvariant='italic'>j</mi>
</mrow>
</msup>
</mrow>
</mfrac>
<mrow><mo stretchy="true">[</mo><mrow><msup><mi mathvariant='italic'>t</mi>
<mi mathvariant='italic'>i</mi>
</msup>
<mo stretchy="false">(</mo>
<msup><mi mathvariant='italic'>F</mi>
<mrow><mo stretchy="false">(</mo>
<mn>1</mn>
<mo stretchy="false">)</mo>
</mrow>
</msup>
<mo stretchy="false">∘</mo>
<mi mathvariant='italic'>Z</mi>
<mo stretchy="false">)</mo>
<mo stretchy="false">(</mo>
<mi mathvariant='italic'>t</mi>
<mo stretchy="false">,</mo>
<mi mathvariant='italic'>x</mi>
<mo stretchy="false">)</mo>
</mrow><mo stretchy="true">]</mo></mrow>
</mrow><mo stretchy="true">}</mo></mrow>
<mrow><mi mathvariant='italic'>t</mi>
<mo stretchy="false">=</mo>
<mn>0</mn>
</mrow>
</msub>
</mtd><mtd columnalign="center" >
<mo stretchy="false">=</mo>
</mtd><mtd columnalign="left" >
<mrow><mo stretchy="true">(</mo><mrow><mtable rowalign="center" ><mtr><mtd columnalign="center" >
<mi mathvariant='italic'>j</mi>
</mtd></mtr><mtr><mtd columnalign="center" >
<mi mathvariant='italic'>i</mi>
</mtd></mtr></mtable>
</mrow><mo stretchy="true">)</mo></mrow>
<mi mathvariant='italic'>i</mi>
<mo stretchy="false">!</mo>
<mfrac><mrow><msup><mo stretchy="false">∂</mo>
<mrow><mi mathvariant='italic'>j</mi>
<mo stretchy="false">-</mo>
<mi mathvariant='italic'>i</mi>
</mrow>
</msup>
</mrow>
<mrow><mo stretchy="false">∂</mo>
<msup><mrow><mi mathvariant='italic'>t</mi>
</mrow>
<mrow><mi mathvariant='italic'>j</mi>
<mo stretchy="false">-</mo>
<mi mathvariant='italic'>i</mi>
</mrow>
</msup>
</mrow>
</mfrac>
<mo stretchy="false">(</mo>
<msup><mi mathvariant='italic'>F</mi>
<mrow><mo stretchy="false">(</mo>
<mn>1</mn>
<mo stretchy="false">)</mo>
</mrow>
</msup>
<mo stretchy="false">∘</mo>
<mi mathvariant='italic'>Z</mi>
<mo stretchy="false">)</mo>
<mo stretchy="false">(</mo>
<mi mathvariant='italic'>t</mi>
<mo stretchy="false">,</mo>
<mi mathvariant='italic'>x</mi>
<mo stretchy="false">)</mo>
</mtd></mtr><mtr><mtd columnalign="right" >
</mtd><mtd columnalign="center" >
<mo stretchy="false">=</mo>
</mtd><mtd columnalign="left" >
<mfrac><mrow><mi mathvariant='italic'>j</mi>
<mo stretchy="false">!</mo>
</mrow>
<mrow><mo stretchy="false">(</mo>
<mi mathvariant='italic'>j</mi>
<mo stretchy="false">-</mo>
<mi mathvariant='italic'>i</mi>
<mo stretchy="false">)</mo>
<mo stretchy="false">!</mo>
</mrow>
</mfrac>
<mfrac><mrow><msup><mo stretchy="false">∂</mo>
<mrow><mi mathvariant='italic'>j</mi>
<mo stretchy="false">-</mo>
<mi mathvariant='italic'>i</mi>
</mrow>
</msup>
</mrow>
<mrow><mo stretchy="false">∂</mo>
<msup><mrow><mi mathvariant='italic'>t</mi>
</mrow>
<mrow><mi mathvariant='italic'>j</mi>
<mo stretchy="false">-</mo>
<mi mathvariant='italic'>i</mi>
</mrow>
</msup>
</mrow>
</mfrac>
<mo stretchy="false">(</mo>
<msup><mi mathvariant='italic'>F</mi>
<mrow><mo stretchy="false">(</mo>
<mn>1</mn>
<mo stretchy="false">)</mo>
</mrow>
</msup>
<mo stretchy="false">∘</mo>
<mi mathvariant='italic'>Z</mi>
<mo stretchy="false">)</mo>
<mo stretchy="false">(</mo>
<mi mathvariant='italic'>t</mi>
<mo stretchy="false">,</mo>
<mi mathvariant='italic'>x</mi>
<mo stretchy="false">)</mo>
</mtd></mtr><mtr><mtd columnalign="right" >
<mfrac><mrow><mo stretchy="false">∂</mo>
<msup><mi mathvariant='italic'>y</mi>
<mrow><mo stretchy="false">(</mo>
<mi mathvariant='italic'>j</mi>
<mo stretchy="false">)</mo>
</mrow>
</msup>
</mrow>
<mrow><mo stretchy="false">∂</mo>
<msup><mi mathvariant='italic'>x</mi>
<mrow><mo stretchy="false">(</mo>
<mi mathvariant='italic'>i</mi>
<mo stretchy="false">)</mo>
</mrow>
</msup>
</mrow>
</mfrac>
<mo stretchy="false">(</mo>
<mi mathvariant='italic'>x</mi>
<mo stretchy="false">)</mo>
</mtd><mtd columnalign="center" >
<mo stretchy="false">=</mo>
</mtd><mtd columnalign="left" >
<mfrac><mrow><mn>1</mn>
</mrow>
<mrow><mo stretchy="false">(</mo>
<mi mathvariant='italic'>j</mi>
<mo stretchy="false">-</mo>
<mi mathvariant='italic'>i</mi>
<mo stretchy="false">)</mo>
<mo stretchy="false">!</mo>
</mrow>
</mfrac>
<mfrac><mrow><msup><mo stretchy="false">∂</mo>
<mrow><mi mathvariant='italic'>j</mi>
<mo stretchy="false">-</mo>
<mi mathvariant='italic'>i</mi>
</mrow>
</msup>
</mrow>
<mrow><mo stretchy="false">∂</mo>
<msup><mrow><mi mathvariant='italic'>t</mi>
</mrow>
<mrow><mi mathvariant='italic'>j</mi>
<mo stretchy="false">-</mo>
<mi mathvariant='italic'>i</mi>
</mrow>
</msup>
</mrow>
</mfrac>
<mo stretchy="false">(</mo>
<msup><mi mathvariant='italic'>F</mi>
<mrow><mo stretchy="false">(</mo>
<mn>1</mn>
<mo stretchy="false">)</mo>
</mrow>
</msup>
<mo stretchy="false">∘</mo>
<mi mathvariant='italic'>Z</mi>
<mo stretchy="false">)</mo>
<mo stretchy="false">(</mo>
<mi mathvariant='italic'>t</mi>
<mo stretchy="false">,</mo>
<mi mathvariant='italic'>x</mi>
<mo stretchy="false">)</mo>
</mtd></mtr></mtable>
</mrow></math>
Applying this formula to the case where
<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow>
<mi mathvariant='italic'>j</mi>
</mrow></math>
is replaced by
<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow>
<mi mathvariant='italic'>j</mi>
<mo stretchy="false">-</mo>
<mi mathvariant='italic'>i</mi>
</mrow></math>
and
<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow>
<mi mathvariant='italic'>i</mi>
</mrow></math>
is replaced by zero,
we obtain
<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mrow>
<mfrac><mrow><mo stretchy="false">∂</mo>
<msup><mi mathvariant='italic'>y</mi>
<mrow><mo stretchy="false">(</mo>
<mi mathvariant='italic'>j</mi>
<mo stretchy="false">-</mo>
<mi mathvariant='italic'>i</mi>
<mo stretchy="false">)</mo>
</mrow>
</msup>
</mrow>
<mrow><mo stretchy="false">∂</mo>
<msup><mi mathvariant='italic'>x</mi>
<mrow><mo stretchy="false">(</mo>
<mn>0</mn>
<mo stretchy="false">)</mo>
</mrow>
</msup>
</mrow>
</mfrac>
<mo stretchy="false">(</mo>
<mi mathvariant='italic'>x</mi>
<mo stretchy="false">)</mo>
<mo stretchy="false">=</mo>
<mfrac><mrow><mn>1</mn>
</mrow>
<mrow><mo stretchy="false">(</mo>
<mi mathvariant='italic'>j</mi>
<mo stretchy="false">-</mo>
<mi mathvariant='italic'>i</mi>
<mo stretchy="false">)</mo>
<mo stretchy="false">!</mo>
</mrow>
</mfrac>
<mfrac><mrow><msup><mo stretchy="false">∂</mo>
<mrow><mi mathvariant='italic'>j</mi>
<mo stretchy="false">-</mo>
<mi mathvariant='italic'>i</mi>
</mrow>
</msup>
</mrow>
<mrow><mo stretchy="false">∂</mo>
<msup><mrow><mi mathvariant='italic'>t</mi>
</mrow>
<mrow><mi mathvariant='italic'>j</mi>
<mo stretchy="false">-</mo>
<mi mathvariant='italic'>i</mi>
</mrow>
</msup>
</mrow>
</mfrac>
<mo stretchy="false">(</mo>
<msup><mi mathvariant='italic'>F</mi>
<mrow><mo stretchy="false">(</mo>
<mn>1</mn>
<mo stretchy="false">)</mo>
</mrow>
</msup>
<mo stretchy="false">∘</mo>
<mi mathvariant='italic'>Z</mi>
<mo stretchy="false">)</mo>
<mo stretchy="false">(</mo>
<mi mathvariant='italic'>t</mi>
<mo stretchy="false">,</mo>
<mi mathvariant='italic'>x</mi>
<mo stretchy="false">)</mo>
<mo stretchy="false">=</mo>
<mfrac><mrow><mo stretchy="false">∂</mo>
<msup><mi mathvariant='italic'>y</mi>
<mrow><mo stretchy="false">(</mo>
<mi mathvariant='italic'>j</mi>
<mo stretchy="false">)</mo>
</mrow>
</msup>
</mrow>
<mrow><mo stretchy="false">∂</mo>
<msup><mi mathvariant='italic'>x</mi>
<mrow><mo stretchy="false">(</mo>
<mi mathvariant='italic'>i</mi>
<mo stretchy="false">)</mo>
</mrow>
</msup>
</mrow>
</mfrac>
<mo stretchy="false">(</mo>
<mi mathvariant='italic'>x</mi>
<mo stretchy="false">)</mo>
</mrow></math>
which completes the proof
<hr/>Input File: omh/reverse_identity.omh
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