%0 Journal Article
%T The Partial Jacobian Matrix and the Chain Rule for Functions on Products of Real <i>n</i>-Spaces
%A Amod Agashe
%J Applied Mathematics
%P 420-427
%@ 2152-7393
%D 2025
%I Scientific Research Publishing
%R 10.4236/am.2025.164022
%X The usual chain rule is for compositions of functions from <math xmlns='http://www.w3.org/1998/Math/MathML'>
 <mrow>
  <msup>
   <mstyle mathvariant='bold' mathsize='normal'><mi>R</mi></mstyle>
   <mi>n</mi>
  </msup>
  </mrow>
</math>

 to <math xmlns='http://www.w3.org/1998/Math/MathML'>
 <mrow>
  <msup>
   <mstyle mathvariant='bold' mathsize='normal'><mi>R</mi></mstyle>
   <mi>m</mi>
  </msup>
  </mrow>
</math>

, for some <math xmlns='http://www.w3.org/1998/Math/MathML'>
 <mi>m</mi>
</math>

 and <math xmlns='http://www.w3.org/1998/Math/MathML'>
 <mi>n</mi>
</math>

. We consider functions whose domains and codomains are <i>products</i> of various <math xmlns='http://www.w3.org/1998/Math/MathML'>
 <mrow>
  <msup>
   <mstyle mathvariant='bold' mathsize='normal'><mi>R</mi></mstyle>
   <mi>n</mi>
  </msup>
  </mrow>
</math>

&#8217;s, and introduce the notion of a partial Jacobian matrix, using which we formulate a chain rule in this setting that should be useful for calculations as well as for some theoretical purposes.
%K Chain Rule
%K Partial Derivative
%K Jacobian Matrix
%K Partial Jacobian Matrix
%K Normed Vector Spaces
%K Manifolds
%U http://www.scirp.org/journal/PaperInformation.aspx?PaperID=142431