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Section 14.1 The chain rule in higher dimensions
Objectives
Track how a multivariable function changes along curves or surfaces using the Chain Rule.
Rewrite total differentials in terms of partial derivatives to relate dependent quantities.
Analyze implicit relationships and contexts where variables are not independent.
Suppose we move along a parametric curve
\((x(t),y(t))\text{.}\) How does the value of
\(f(x,y)\) change with
\(t\text{?}\)
We can now write the total differential that we saw in Part I in terms of partial derivatives:
\begin{equation*}
\dd f=\pdv{f}{x}\dd{x}+\pdv{f}{y}\dd{y}
\end{equation*}
Dividing by \(\dd t\text{,}\) we now get the Chain Rule in higher dimensions:
\begin{equation*}
\dv{f}{t}=\pdv{f}{x}\cdot\dv{x}{t}+\pdv{f}{y}\cdot\dv{y}{t}
\end{equation*}
If
\(x\) and
\(y\) are both functions of
\(s\) and
\(t\text{,}\) then we have
\begin{align*}
\pdv{f}{s} \amp = \pdv{f}{x}\cdot\pdv{x}{s}+\pdv{f}{y}\cdot\pdv{y}{s}\\
\pdv{f}{t} \amp = \pdv{f}{x}\cdot\pdv{x}{t}+\pdv{f}{y}\cdot\pdv{y}{t}
\end{align*}
New way to do implicit differentiation! Rewrite implicit curve as a level curve \(F(x,y)=0\text{;}\) then
\begin{equation*}
\dv{y}{x} = -\frac{\partial F/\partial x}{\partial F/\partial y}
\end{equation*}
Sometimes we need to be more explicit about which variables we’re holding constant. Use Hughes-Hallett for reference here, they have a great explanation. Let \(U(P,V,T)\) be the internal energy of some amount of gas. According to the Ideal Gas Law, we have
\begin{equation*}
PV=nRT
\end{equation*}
where \(n\) is the (constant) amount of gas and \(R\) is the ideal gas constant. In particular, \(P\text{,}\) \(V\text{,}\) and \(T\) are NOT independent.
Suppose we want to find \(\DS\qty(\pdv{U}{T})_{P}\text{,}\) that is, how \(U\) depends on \(T\) while \(P\) is being held constant. This means we’re implicitly thinking of \(U\) as a function of \(T\) and \(P\text{,}\) with \(V\) as an intermediate variable. This makes our total differential
\begin{equation*}
\dd U=\qty(\pdv{U}{T})_{P}\dd{T}+\qty(\pdv{U}{P})_{T}\dd{P}\text{.}
\end{equation*}
We could also have thought of \(U\) as depending on \(V\) and \(T\text{,}\) with \(P\) as an intermediate variable:
\begin{equation*}
\dd U=\qty(\pdv{U}{T})_{V}\dd{T}+\qty(\pdv{U}{V})_{T}\dd{V}
\end{equation*}
But since \(V\) is a function of \(P\) and \(T\) we have:
\begin{equation*}
\dd V=\qty(\pdv{V}{T})_{P}\dd{T}+\qty(\pdv{V}{P})_{T}\dd{P}
\end{equation*}
Substituting this gives us:
\begin{align*}
\dd{U} \amp = \qty(\pdv{U}{T})_{V}\dd{T}+\qty(\pdv{U}{V})_{T}\qty(\pdv{V}{T})_{P}\dd{T}+\qty(\pdv{U}{V})_{T}\qty(\pdv{V}{P})_{T}\dd{P})\\
\amp = \qty(\qty(\pdv{U}{T})_{V}+\qty(\pdv{U}{V})_{T}\qty(\pdv{V}{T})_{P})\dd{T}+\qty(\pdv{U}{V})_{T}\qty(\pdv{V}{P})_{T}\dd{P}
\end{align*}
We now have two expressions for \(\dd{U}\text{.}\) Setting the \(\dd T\) parts equal we get
\begin{equation*}
\qty(\pdv{U}{T})_{P}=\qty(\pdv{U}{T})_{V}+\qty(\pdv{U}{V})_{T}\qty(\pdv{V}{T})_{P}\text{,}
\end{equation*}
and setting the \(\dd P\) parts equal we get
\begin{equation*}
\qty(\pdv{U}{P})_{T}=\qty(\pdv{U}{V})_{T}\qty(\pdv{V}{P})_{T}\text{.}
\end{equation*}
May decide NOT to do it the way described above ... instead we can just extend the tree diagram so that the only ‘leaves’ are the independent variables. Explain in terms of the ambiguity of the partial derivative notation and what we’re actually looking for.
Should we use the
\(D\) notation to clean things up more?
