3.2: Differentiation of Bivariate Functions
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)Let us briefly consider differentiation of bivariate functions. For simplicity, we restrict ourselves to estimation of the first derivative, i.e., the gradient.
Example 3.2.1 first-order gradient
Following the procedure for estimating the first derivative of univariate functions, we first construct a polynomial interpolant and then evaluate its derivative. In particular, we consider a linear interpolant on a triangle. Recall the second form of linear interpolant, \[(\mathcal{I} f)(\boldsymbol{x})=f\left(\bar{x}^{\mathbf{1}}\right)+b^{\prime}\left(x-\bar{x}^{1}\right)+c^{\prime}\left(y-\bar{y}^{1}\right) .\] The partial derivative in the \(x\)-direction is \[\frac{\partial(\mathcal{I} f)}{\partial x}=b^{\prime}=\frac{1}{A}\left[\left(f\left(\overline{\boldsymbol{x}}^{2}\right)-f\left(\overline{\boldsymbol{x}}^{1}\right)\right)\left(\bar{y}^{3}-\bar{y}^{1}\right)-\left(f\left(\overline{\boldsymbol{x}}^{3}\right)-f\left(\overline{\boldsymbol{x}}^{1}\right)\right)\left(\bar{y}^{2}-\bar{y}^{1}\right)\right],\] where we recall that \(A\) is twice the area of the triangle, i.e., \(A=\left(x_{2}-x_{1}\right)\left(y_{3}-y_{1}\right)-\left(x_{3}-x_{1}\right)\left(y_{2}-y_{1}\right)\). Similarly, the derivative in the \(y\)-direction is \[\frac{\partial(\mathcal{I} f)}{\partial y}=c^{\prime}=\frac{1}{A}\left[\left(f\left(\overline{\boldsymbol{x}}^{3}\right)-f\left(\overline{\boldsymbol{x}}^{1}\right)\right)\left(\bar{x}^{2}-\bar{x}^{1}\right)-\left(f\left(\overline{\boldsymbol{x}}^{2}\right)-f\left(\overline{\boldsymbol{x}}^{1}\right)\right)\left(\bar{x}^{3}-\bar{x}^{1}\right)\right] .\] In general, a directional derivative in the direction \(s=\left(s_{x}, s_{y}\right)\) is \[\frac{\partial(\mathcal{I} f)}{\partial s}=s_{x} \frac{\partial(\mathcal{I} f)}{\partial x}+s_{y} \frac{\partial(\mathcal{I} f)}{\partial y}=s_{x} b^{\prime}+s_{y} c^{\prime}\] Because the gradient approximation is constructed from a linear function, the gradient estimate is constant over the triangle. The approximation is first-order accurate.