View source for Leibniz's theorem of calculus



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== Leibniz's theorem of calculus ==
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<div class="definition"><div class="short_definition">A relationship between the derivative of an integral and the integral  of a derivative, that is,  <div class="display-formula"><blockquote>[[File:ams2001glos-Le14.gif|link=|center|ams2001glos-Le14]]</blockquote></div> where ''S''<sub>1</sub> and ''S''<sub>2</sub> are limits of integration, ''s'' is a dummy distance or space [[variable]] such as height  ''z'', ''t'' is time, and ''A'' is some meteorological quantity, such as [[potential temperature]] or [[humidity]],  that is a function of both space and time. If the limits of integration are constant with time, then  the last two terms are zero, and the derivative of the integral equals the integral of the derivative.  However, there are many atmospheric situations, such as a growing [[atmospheric boundary layer]],  where the limits of integration can change with time. Namely, if one wishes to integrate over the  depth of the [[boundary layer]] (between limits ''z'' = 0 to ''z'' = ''z''<sub>''i''</sub>) to find a boundary layer average,  for example, but the top of the boundary layer at height ''z''<sub>''i''</sub>(''t'') is rising with time, then one must  use the full form of Leibniz's theorem to account for this effect.</div><br/> <div class="paragraph"></div><br/> </div>
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