Why in physics the elementary work is written as $delta W$ instead of $dW$?












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Why an elementary work is written $delta W$ instead of $dW$? For example, it's often written $$delta W=Fcdot dr$$ if $dr$ is the elementary displacement. Why don't we write as usual $dW=Fcdot dr$ ? I saw an answer here but it doesn't really answer to the question (at my opinion). By the way, since at the end $W_{AB}=int_A^B delta W$, I really don't understand this $delta W$. Is there mathematically a sense ?






derivatives differential-forms







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  • $begingroup$
    Some thermodynamics books write this as a d with a small slash through the top (I can't do this easily in MathJax).
    $endgroup$
    – Ted Shifrin
    Dec 21 '18 at 21:34










  • $begingroup$
    @Ted, one could try moving a bar horizontally: $newcommand{dbar}{dhspace{-0.35em}bar{}}dbar W$
    $endgroup$
    – rych
    Dec 25 '18 at 13:19












  • $begingroup$
    @rych: Thanks. Too high and too wide, but I know I could design it if I really wanted to make the effort. :)
    $endgroup$
    – Ted Shifrin
    Dec 25 '18 at 21:20
















1












$begingroup$


Why an elementary work is written $delta W$ instead of $dW$? For example, it's often written $$delta W=Fcdot dr$$ if $dr$ is the elementary displacement. Why don't we write as usual $dW=Fcdot dr$ ? I saw an answer here but it doesn't really answer to the question (at my opinion). By the way, since at the end $W_{AB}=int_A^B delta W$, I really don't understand this $delta W$. Is there mathematically a sense ?






derivatives differential-forms







share|cite|improve this question









$endgroup$












  • $begingroup$
    Some thermodynamics books write this as a d with a small slash through the top (I can't do this easily in MathJax).
    $endgroup$
    – Ted Shifrin
    Dec 21 '18 at 21:34










  • $begingroup$
    @Ted, one could try moving a bar horizontally: $newcommand{dbar}{dhspace{-0.35em}bar{}}dbar W$
    $endgroup$
    – rych
    Dec 25 '18 at 13:19












  • $begingroup$
    @rych: Thanks. Too high and too wide, but I know I could design it if I really wanted to make the effort. :)
    $endgroup$
    – Ted Shifrin
    Dec 25 '18 at 21:20














1












1








1


1



$begingroup$


Why an elementary work is written $delta W$ instead of $dW$? For example, it's often written $$delta W=Fcdot dr$$ if $dr$ is the elementary displacement. Why don't we write as usual $dW=Fcdot dr$ ? I saw an answer here but it doesn't really answer to the question (at my opinion). By the way, since at the end $W_{AB}=int_A^B delta W$, I really don't understand this $delta W$. Is there mathematically a sense ?






derivatives differential-forms







share|cite|improve this question









$endgroup$




Why an elementary work is written $delta W$ instead of $dW$? For example, it's often written $$delta W=Fcdot dr$$ if $dr$ is the elementary displacement. Why don't we write as usual $dW=Fcdot dr$ ? I saw an answer here but it doesn't really answer to the question (at my opinion). By the way, since at the end $W_{AB}=int_A^B delta W$, I really don't understand this $delta W$. Is there mathematically a sense ?






derivatives differential-forms




derivatives differential-forms






share|cite|improve this question













share|cite|improve this question











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asked Dec 21 '18 at 10:01









user623855user623855

1407




1407












  • $begingroup$
    Some thermodynamics books write this as a d with a small slash through the top (I can't do this easily in MathJax).
    $endgroup$
    – Ted Shifrin
    Dec 21 '18 at 21:34










  • $begingroup$
    @Ted, one could try moving a bar horizontally: $newcommand{dbar}{dhspace{-0.35em}bar{}}dbar W$
    $endgroup$
    – rych
    Dec 25 '18 at 13:19












  • $begingroup$
    @rych: Thanks. Too high and too wide, but I know I could design it if I really wanted to make the effort. :)
    $endgroup$
    – Ted Shifrin
    Dec 25 '18 at 21:20


















  • $begingroup$
    Some thermodynamics books write this as a d with a small slash through the top (I can't do this easily in MathJax).
    $endgroup$
    – Ted Shifrin
    Dec 21 '18 at 21:34










  • $begingroup$
    @Ted, one could try moving a bar horizontally: $newcommand{dbar}{dhspace{-0.35em}bar{}}dbar W$
    $endgroup$
    – rych
    Dec 25 '18 at 13:19












  • $begingroup$
    @rych: Thanks. Too high and too wide, but I know I could design it if I really wanted to make the effort. :)
    $endgroup$
    – Ted Shifrin
    Dec 25 '18 at 21:20
















$begingroup$
Some thermodynamics books write this as a d with a small slash through the top (I can't do this easily in MathJax).
$endgroup$
– Ted Shifrin
Dec 21 '18 at 21:34




$begingroup$
Some thermodynamics books write this as a d with a small slash through the top (I can't do this easily in MathJax).
$endgroup$
– Ted Shifrin
Dec 21 '18 at 21:34












$begingroup$
@Ted, one could try moving a bar horizontally: $newcommand{dbar}{dhspace{-0.35em}bar{}}dbar W$
$endgroup$
– rych
Dec 25 '18 at 13:19






$begingroup$
@Ted, one could try moving a bar horizontally: $newcommand{dbar}{dhspace{-0.35em}bar{}}dbar W$
$endgroup$
– rych
Dec 25 '18 at 13:19














$begingroup$
@rych: Thanks. Too high and too wide, but I know I could design it if I really wanted to make the effort. :)
$endgroup$
– Ted Shifrin
Dec 25 '18 at 21:20




$begingroup$
@rych: Thanks. Too high and too wide, but I know I could design it if I really wanted to make the effort. :)
$endgroup$
– Ted Shifrin
Dec 25 '18 at 21:20










2 Answers
2






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$begingroup$

The notation $delta W$ rather than $mathrm{d} W$ aims to underline that work (like heat) is an improper differential, i.e. its integral depends crucially on the integration path taken.




  • In physical terms, the work done on a system will not only depend on the initial and final condition of the system, but on the transformation chosen to drive the system from its initial to its final condition.


  • In more mathematical terms, an inexact differential cannot be expressed as the the gradient of another function, making an integral of that differential path-dependent.



I found the wikipedia article on the topic quite helpful; any good textbook on thermodynamics will discuss this, for example Schwabl's Statistical Mechanics.






share|cite|improve this answer









$endgroup$





















    1












    $begingroup$

    In physical literature the symbol $d W$ is used to indicate an exact differential form, and $delta W$ is used because, in general, the infinitesimal work $delta W=F cdot dvec r$ is not an exact differential form. It is an exact form if the force $F$ is conservative, so, in this case we write $delta W=dW$.






    share|cite|improve this answer











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      2 Answers
      2






      active

      oldest

      votes








      2 Answers
      2






      active

      oldest

      votes









      active

      oldest

      votes






      active

      oldest

      votes









      4












      $begingroup$

      The notation $delta W$ rather than $mathrm{d} W$ aims to underline that work (like heat) is an improper differential, i.e. its integral depends crucially on the integration path taken.




      • In physical terms, the work done on a system will not only depend on the initial and final condition of the system, but on the transformation chosen to drive the system from its initial to its final condition.


      • In more mathematical terms, an inexact differential cannot be expressed as the the gradient of another function, making an integral of that differential path-dependent.



      I found the wikipedia article on the topic quite helpful; any good textbook on thermodynamics will discuss this, for example Schwabl's Statistical Mechanics.






      share|cite|improve this answer









      $endgroup$


















        4












        $begingroup$

        The notation $delta W$ rather than $mathrm{d} W$ aims to underline that work (like heat) is an improper differential, i.e. its integral depends crucially on the integration path taken.




        • In physical terms, the work done on a system will not only depend on the initial and final condition of the system, but on the transformation chosen to drive the system from its initial to its final condition.


        • In more mathematical terms, an inexact differential cannot be expressed as the the gradient of another function, making an integral of that differential path-dependent.



        I found the wikipedia article on the topic quite helpful; any good textbook on thermodynamics will discuss this, for example Schwabl's Statistical Mechanics.






        share|cite|improve this answer









        $endgroup$
















          4












          4








          4





          $begingroup$

          The notation $delta W$ rather than $mathrm{d} W$ aims to underline that work (like heat) is an improper differential, i.e. its integral depends crucially on the integration path taken.




          • In physical terms, the work done on a system will not only depend on the initial and final condition of the system, but on the transformation chosen to drive the system from its initial to its final condition.


          • In more mathematical terms, an inexact differential cannot be expressed as the the gradient of another function, making an integral of that differential path-dependent.



          I found the wikipedia article on the topic quite helpful; any good textbook on thermodynamics will discuss this, for example Schwabl's Statistical Mechanics.






          share|cite|improve this answer









          $endgroup$



          The notation $delta W$ rather than $mathrm{d} W$ aims to underline that work (like heat) is an improper differential, i.e. its integral depends crucially on the integration path taken.




          • In physical terms, the work done on a system will not only depend on the initial and final condition of the system, but on the transformation chosen to drive the system from its initial to its final condition.


          • In more mathematical terms, an inexact differential cannot be expressed as the the gradient of another function, making an integral of that differential path-dependent.



          I found the wikipedia article on the topic quite helpful; any good textbook on thermodynamics will discuss this, for example Schwabl's Statistical Mechanics.







          share|cite|improve this answer












          share|cite|improve this answer



          share|cite|improve this answer










          answered Dec 21 '18 at 10:14









          workandheatworkandheat

          587




          587























              1












              $begingroup$

              In physical literature the symbol $d W$ is used to indicate an exact differential form, and $delta W$ is used because, in general, the infinitesimal work $delta W=F cdot dvec r$ is not an exact differential form. It is an exact form if the force $F$ is conservative, so, in this case we write $delta W=dW$.






              share|cite|improve this answer











              $endgroup$


















                1












                $begingroup$

                In physical literature the symbol $d W$ is used to indicate an exact differential form, and $delta W$ is used because, in general, the infinitesimal work $delta W=F cdot dvec r$ is not an exact differential form. It is an exact form if the force $F$ is conservative, so, in this case we write $delta W=dW$.






                share|cite|improve this answer











                $endgroup$
















                  1












                  1








                  1





                  $begingroup$

                  In physical literature the symbol $d W$ is used to indicate an exact differential form, and $delta W$ is used because, in general, the infinitesimal work $delta W=F cdot dvec r$ is not an exact differential form. It is an exact form if the force $F$ is conservative, so, in this case we write $delta W=dW$.






                  share|cite|improve this answer











                  $endgroup$



                  In physical literature the symbol $d W$ is used to indicate an exact differential form, and $delta W$ is used because, in general, the infinitesimal work $delta W=F cdot dvec r$ is not an exact differential form. It is an exact form if the force $F$ is conservative, so, in this case we write $delta W=dW$.







                  share|cite|improve this answer














                  share|cite|improve this answer



                  share|cite|improve this answer








                  edited Dec 21 '18 at 11:44

























                  answered Dec 21 '18 at 10:14









                  Emilio NovatiEmilio Novati

                  52.2k43574




                  52.2k43574






























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