1.14.8: Calculus
- Page ID
- 352512
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)Consider a variable u defined by the independent variables \(x\) and \(y\).
\[\text { We write } u=u[x, y] \nonumber \]
Equation (b) is the general exact differential of equation (a).
\[\mathrm{du}=\left(\frac{\partial \mathrm{u}}{\partial \mathrm{x}}\right)_{\mathrm{y}} \, \mathrm{dx}+\left(\frac{\partial \mathrm{u}}{\partial \mathrm{y}}\right)_{\mathrm{x}} \, \mathrm{dy} \nonumber \]
In other words the change in u is related to the differential dependence of \(\mathrm{u}\) on \(x\) at constant \(y\) and the differential dependence of \(\mathrm{u}\) on \(y\) at constant \(x\). For the case where u does not change,
\[\left(\frac{\partial u}{\partial x}\right)_{y}=-\left(\frac{\partial u}{\partial y}\right)_{x} \,\left(\frac{\partial y}{\partial x}\right)_{u}=0 \text { and }\left(\frac{\partial y}{\partial x}\right)_{u}=-\left(\frac{\partial u}{\partial x}\right)_{y} \,\left(\frac{\partial y}{\partial u}\right)_{x} \nonumber \]
A variable \(z\) is defined by the independent variables \(x\) and \(y\).
\[z=z[x, y] \nonumber \]
Equation (e) is the general differential of equation (d). d z=\left(\frac{\partial z}{\partial x}\right)_{y} \, d x+\left(\frac{\partial z}{\partial y}\right)_{x} \, d y \nonumber \]
We direct attention to the dependence of \(z\) on \(x\) along a pathway for which \(\mathrm{u}\) is constant.
\[\text { Then }\left(\frac{\partial z}{\partial x}\right)_{u}=\left(\frac{\partial z}{\partial x}\right)_{y}+\left(\frac{\partial z}{\partial y}\right)_{x} \,\left(\frac{\partial y}{\partial x}\right)_{u} \nonumber \]
The latter equation contains the differential dependence of \(y\) on \(x\) at constant \(\mathrm{u}\). The latter dependence can be reformulated using equation (c). Therefore
\[\left(\frac{\partial z}{\partial x}\right)_{u}=\left(\frac{\partial z}{\partial x}\right)_{y}-\left(\frac{\partial u}{\partial x}\right)_{y} \,\left(\frac{\partial y}{\partial u}\right)_{x} \,\left(\frac{\partial z}{\partial y}\right)_{x} \nonumber \]
The key point to emerge from this exercise centres is the way in which the condition on the partial differential \((\partial z / \partial x)\) can be changed from ‘at constant \(y\)’ to ‘at constant \(\mathrm{u}\)’.
Another important operation concerns a variable \(\mathrm{q}\).
\[\text { Thus, }\left(\frac{\partial x}{\partial y}\right)_{z}=\left(\frac{\partial x}{\partial q}\right)_{z} \,\left(\frac{\partial q}{\partial y}\right)_{z} \nonumber \]
For composite functions such as \(z=z[\mathrm{u}, \mathrm{v}]\), where \(z=z[x, y]\), and \(\mathrm{u}=\mathrm{u}[\mathrm{x}, \mathrm{y}]\), further important equations are found [1].
\[\text { Thus }\left(\frac{\partial z}{\partial u}\right)_{v}=\left(\frac{\partial z}{\partial x}\right)_{y} \,\left(\frac{\partial x}{\partial u}\right)_{v}+\left(\frac{\partial z}{\partial y}\right)_{x} \,\left(\frac{\partial y}{\partial u}\right)_{v} \nonumber \]
Equation (i) is an example of the well-known chain rule, a similar equation holding for \((\partial z / \partial v)_{u}\). This rule allows the total change of independent variables from \(z=z[\mathrm{u}, \mathrm{v}]\) to \(z=z[x, y]\).
\[\text { Also }\left(\frac{\partial z}{\partial x}\right)_{y}=\left(\frac{\partial z}{\partial x}\right)_{y, v}+\left(\frac{\partial z}{\partial v}\right)_{y, x} \,\left(\frac{\partial v}{\partial x}\right)_{y} \nonumber \]
The latter equation is useful for introducing an extra constraint on a given differential.
Footnote
[1] H. B. Callen, Thermodynamics and an Introduction to Thermostatics, Wiley, New York, 2dn. Edn.,1985, Appendix A.