Calculus Small Changes, Big Ideas

Section 12.4 Higher order partial derivatives

• Second-order partial derivatives of \(z=f(x,y)\text{:}\)

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  \begin{align*} \pdv[2]{z}{x} \amp = f_{xx}(x,y)\\ \pdv[2]{z}{y} \amp = f_{yy}(x,y)\\ \pdv{z}{y}{x} \amp = f_{xy}(x,y)\\ \pdv{z}{x}{y} \amp = f_{yx}(x,y) \end{align*}
  Yes, the mixed partial notation is confusing. In differential notation, the operators are applied inside-out, while in function notation, the operators are applied left to right.

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• Theorem 9. Clairaut’s Theorem.

  If \(f\) is defined on a disk containing \((a,b)\) and \(f_{xy}\) and \(f_{yx}\) are continuous in that disk, then

  \begin{equation*} f_{xy}(a,b)=f_{yx}(a,b)\text{.} \end{equation*}

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• Some partial differential equations:

  • Wave equation:

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    \begin{align*} \pdv[2]{u}{t} \amp = a^{2}\pdv[2]{u}{x} \\ \pdv[2]{u}{t} \amp = a^{2}\qty(\pdv[2]{u}{x}+\pdv[2]{u}{y}) \end{align*}
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  • Heat equation:

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    \begin{align*} \pdv{u}{t} \amp = a^{2}\pdv[2]{u}{x} \\ \pdv{u}{t} \amp = a^{2}\qty(\pdv[2]{u}{x}+\pdv[2]{u}{y}) \end{align*}
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  • Laplace’s equation:

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    \begin{align*} \pdv[2]{u}{x} + \pdv[2]{u}{y} \amp = 0 \\ \pdv[2]{u}{x} + \pdv[2]{u}{y} + \pdv[2]{u}{z} \amp = 0 \end{align*}
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• Begin by recalling the single-variable Taylor expansion:

  \begin{equation*} f(x) \approx f(a) + f'(a)(x-a) + \frac{1}{2} f''(a)(x-a)^{2}\text{.} \end{equation*}

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  The tangent line is just the first-order approximation, and adding higher-order terms improves accuracy.

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• Connect to the previous section: The tangent plane

  \begin{equation*} z \approx z_{0} + \pdv{f}{x}(x-x_{0}) + \pdv{f}{y}(y-y_{0}) \end{equation*}

  is the first-order Taylor polynomial for \(f(x,y)\text{.}\)

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• Introduce second-order terms using higher partial derivatives:

  \begin{equation*} f(x,y) \approx f(x_{0},y_{0}) + \pdv{f}{x}(x-x_{0}) + \pdv{f}{y}(y-y_{0}) \end{equation*}

  \begin{equation*} \quad + \frac{1}{2} \pdv[2]{f}{x}(x-x_{0})^{2} + \pdv[2]{f}{x}{y}(x-x_{0})(y-y_{0}) + \frac{1}{2} \pdv[2]{f}{y}(y-y_{0})^{2}. \end{equation*}

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• Interpret each term:

  • First-order terms describe tilt.

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  • Pure second derivatives describe curvature in coordinate directions.

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  • Mixed partials describe interaction between variables.

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• Emphasize symmetry of mixed partials when conditions allow (Clairaut’s theorem).

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• Provide a worked example:

  • Compute first and second partial derivatives at a point.

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  • Write the quadratic Taylor approximation.

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  • Use it to estimate a nearby value.

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  • Compare to the actual value.

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• Geometric meaning:

  • First-order: tangent plane.

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  • Second-order: quadratic surface approximating curvature.

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  • The quadratic form determines local shape.

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• General multivariable Taylor expansion:

  \begin{equation*} f(x,y) = \sum_{k=0}^{n}\text{(all $k$th-order partial terms in $(x-x_{0}),(y-y_{0})$)}+ R_{n},\text{,} \end{equation*}

  where each term involves partial derivatives of total order \(k\) evaluated at \((x_{0},y_{0})\text{.}\)

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  Each degree \(k\) term consists of all products

  \begin{equation*} (x-x_{0})^{i} (y-y_{0})^{j} \end{equation*}

  with \(i+j=k\text{,}\) multiplied by the corresponding partial derivative

  \begin{equation*} \frac{\partial^{k}f}{\partial x^{i}\partial y^{j}}(x_{0},y_{0}). \end{equation*}

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  The approximation improves as higher-order terms are included.

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