# partial derivative

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##### 1: 1.5 Calculus of Two or More Variables
βΊThe function $f(x,y)$ is continuously differentiable if $f$, $\ifrac{\partial f}{\partial x}$, and $\ifrac{\partial f}{\partial y}$ are continuous, and twice-continuously differentiable if also $\ifrac{{\partial}^{2}f}{{\partial x}^{2}}$, $\ifrac{{\partial}^{2}f}{{\partial y}^{2}}$, $\,{\partial}^{2}f/\,\partial x\,\partial y$, and $\,{\partial}^{2}f/\,\partial y\,\partial x$ are continuous. … βΊ
1.5.9 $\frac{\partial}{\partial v}f(x(u,v),y(u,v),z(u,v))=\frac{\partial f}{\partial x% }\frac{\partial x}{\partial v}+\frac{\partial f}{\partial y}\frac{\partial y}{% \partial v}+\frac{\partial f}{\partial z}\frac{\partial z}{\partial v}.$
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1.5.15 $\nabla^{2}f=\frac{{\partial}^{2}f}{{\partial x}^{2}}+\frac{{\partial}^{2}f}{{% \partial y}^{2}}+\frac{{\partial}^{2}f}{{\partial z}^{2}}=\frac{{\partial}^{2}% f}{{\partial r}^{2}}+\frac{1}{r}\frac{\partial f}{\partial r}+\frac{1}{r^{2}}% \frac{{\partial}^{2}f}{{\partial\phi}^{2}}+\frac{{\partial}^{2}f}{{\partial z}% ^{2}}.$
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1.5.38 $\frac{\partial(f,g)}{\partial(x,y)}=\begin{vmatrix}\ifrac{\partial f}{\partial x% }&\ifrac{\partial f}{\partial y}\\ \ifrac{\partial g}{\partial x}&\ifrac{\partial g}{\partial y}\end{vmatrix},$
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1.5.40 $\frac{\partial(f,g,h)}{\partial(x,y,z)}=\begin{vmatrix}\ifrac{\partial f}{% \partial x}&\ifrac{\partial f}{\partial y}&\ifrac{\partial f}{\partial z}\\ \ifrac{\partial g}{\partial x}&\ifrac{\partial g}{\partial y}&\ifrac{\partial g% }{\partial z}\\ \ifrac{\partial h}{\partial x}&\ifrac{\partial h}{\partial y}&\ifrac{\partial h% }{\partial z}\end{vmatrix},$
##### 2: 36.10 Differential Equations
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36.10.6 $\frac{{\partial}^{ln}\Psi_{K}}{{\partial x_{m}}^{ln}}=i^{n(l-m)}\frac{{% \partial}^{mn}\Psi_{K}}{{\partial x_{l}}^{mn}},$ $1\leq m\leq K$, $1\leq l\leq K$.
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36.10.7 $\frac{{\partial}^{2n}\Psi_{2}}{{\partial x}^{2n}}=i^{n}\frac{{\partial}^{n}% \Psi_{2}}{{\partial y}^{n}}.$
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36.10.8 $\frac{{\partial}^{2n}\Psi_{3}}{{\partial x}^{2n}}=i^{n}\frac{{\partial}^{n}% \Psi_{3}}{{\partial y}^{n}},$
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36.10.9 $\frac{{\partial}^{3n}\Psi_{3}}{{\partial x}^{3n}}=(-1)^{n}\frac{{\partial}^{n}% \Psi_{3}}{{\partial z}^{n}},$
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36.10.10 $\frac{{\partial}^{3n}\Psi_{3}}{{\partial y}^{3n}}=i^{n}\frac{{\partial}^{2n}% \Psi_{3}}{{\partial z}^{2n}}.$
##### 3: 19.18 Derivatives and Differential Equations
βΊLet $\partial_{j}=\ifrac{\partial}{\partial z_{j}}$, and $\mathbf{e}_{j}$ be an $n$-tuple with 1 in the $j$th place and 0’s elsewhere. … βΊ
19.18.6 $\left(\frac{\partial}{\partial x}+\frac{\partial}{\partial y}+\frac{\partial}{% \partial z}\right)R_{F}\left(x,y,z\right)=\frac{-1}{2\sqrt{x}\sqrt{y}\sqrt{z}},$
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19.18.14 $\frac{{\partial}^{2}w}{{\partial x}^{2}}=\frac{{\partial}^{2}w}{{\partial y}^{% 2}}+\frac{1}{y}\frac{\partial w}{\partial y}.$
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19.18.15 $\frac{{\partial}^{2}W}{{\partial t}^{2}}=\frac{{\partial}^{2}W}{{\partial x}^{% 2}}+\frac{{\partial}^{2}W}{{\partial y}^{2}}.$
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19.18.16 $\frac{{\partial}^{2}u}{{\partial x}^{2}}+\frac{{\partial}^{2}u}{{\partial y}^{% 2}}+\frac{1}{y}\frac{\partial u}{\partial y}=0,$
##### 4: 10.38 Derivatives with Respect to Order
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10.38.2 $\frac{\partial K_{\nu}\left(z\right)}{\partial\nu}=\tfrac{1}{2}\pi\csc\left(% \nu\pi\right)\*\left(\frac{\partial I_{-\nu}\left(z\right)}{\partial\nu}-\frac% {\partial I_{\nu}\left(z\right)}{\partial\nu}\right)-\pi\cot\left(\nu\pi\right% )K_{\nu}\left(z\right),$ $\nu\notin\mathbb{Z}$.
βΊFor $\ifrac{\partial I_{\nu}\left(z\right)}{\partial\nu}$ at $\nu=-n$ combine (10.38.1), (10.38.2), and (10.38.4). … βΊ
$\left.\frac{\partial I_{\nu}\left(z\right)}{\partial\nu}\right|_{\nu=0}=-K_{0}% \left(z\right),$
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$\left.\frac{\partial K_{\nu}\left(z\right)}{\partial\nu}\right|_{\nu=0}=0.$
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10.38.7 $\left.\frac{\partial K_{\nu}\left(x\right)}{\partial\nu}\right|_{\nu=\pm\frac{% 1}{2}}=\pm\sqrt{\frac{\pi}{2x}}E_{1}\left(2x\right)e^{x}.$
##### 5: 16.14 Partial Differential Equations
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$x(1-x)\frac{{\partial}^{2}{F_{1}}}{{\partial x}^{2}}+y(1-x)\frac{\,{\partial}^% {2}{F_{1}}}{\,\partial x\,\partial y}+\left(\gamma-(\alpha+\beta+1)x\right)% \frac{\partial{F_{1}}}{\partial x}-\beta y\frac{\partial{F_{1}}}{\partial y}-% \alpha\beta{F_{1}}=0,$
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$x(1-x)\frac{{\partial}^{2}{F_{2}}}{{\partial x}^{2}}-xy\frac{\,{\partial}^{2}{% F_{2}}}{\,\partial x\,\partial y}+\left(\gamma-(\alpha+\beta+1)x\right)\frac{% \partial{F_{2}}}{\partial x}-\beta y\frac{\partial{F_{2}}}{\partial y}-\alpha% \beta{F_{2}}=0,$
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$y(1-y)\frac{{\partial}^{2}{F_{2}}}{{\partial y}^{2}}-xy\frac{\,{\partial}^{2}{% F_{2}}}{\,\partial x\,\partial y}+\left(\gamma^{\prime}-(\alpha+\beta^{\prime}% +1)y\right)\frac{\partial{F_{2}}}{\partial y}-\beta^{\prime}x\frac{\partial{F_% {2}}}{\partial x}-\alpha\beta^{\prime}{F_{2}}=0,$
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$x(1-x)\frac{{\partial}^{2}{F_{4}}}{{\partial x}^{2}}-2xy\frac{\,{\partial}^{2}% {F_{4}}}{\,\partial x\,\partial y}-y^{2}\frac{{\partial}^{2}{F_{4}}}{{\partial y% }^{2}}+\left(\gamma-(\alpha+\beta+1)x\right)\frac{\partial{F_{4}}}{\partial x}% -(\alpha+\beta+1)y\frac{\partial{F_{4}}}{\partial y}-\alpha\beta{F_{4}}=0,$
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$y(1-y)\frac{{\partial}^{2}{F_{4}}}{{\partial y}^{2}}-2xy\frac{\,{\partial}^{2}% {F_{4}}}{\,\partial x\,\partial y}-x^{2}\frac{{\partial}^{2}{F_{4}}}{{\partial x% }^{2}}+\left(\gamma^{\prime}-(\alpha+\beta+1)y\right)\frac{\partial{F_{4}}}{% \partial y}-(\alpha+\beta+1)x\frac{\partial{F_{4}}}{\partial x}-\alpha\beta{F_% {4}}=0.$
##### 6: 10.15 Derivatives with Respect to Order
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10.15.1 $\frac{\partial J_{\pm\nu}\left(z\right)}{\partial\nu}=\pm J_{\pm\nu}\left(z% \right)\ln\left(\tfrac{1}{2}z\right)\mp(\tfrac{1}{2}z)^{\pm\nu}\sum_{k=0}^{% \infty}(-1)^{k}\frac{\psi\left(k+1\pm\nu\right)}{\Gamma\left(k+1\pm\nu\right)}% \frac{(\tfrac{1}{4}z^{2})^{k}}{k!},$
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10.15.2 $\frac{\partial Y_{\nu}\left(z\right)}{\partial\nu}=\cot\left(\nu\pi\right)% \left(\frac{\partial J_{\nu}\left(z\right)}{\partial\nu}-\pi Y_{\nu}\left(z% \right)\right)-\csc\left(\nu\pi\right)\frac{\partial J_{-\nu}\left(z\right)}{% \partial\nu}-\pi J_{\nu}\left(z\right).$
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10.15.3 $\left.\frac{\partial J_{\nu}\left(z\right)}{\partial\nu}\right|_{\nu=n}=\frac{% \pi}{2}Y_{n}\left(z\right)+\frac{n!}{2(\tfrac{1}{2}z)^{n}}\sum_{k=0}^{n-1}% \frac{(\tfrac{1}{2}z)^{k}J_{k}\left(z\right)}{k!(n-k)}.$
βΊFor $\ifrac{\partial J_{\nu}\left(z\right)}{\partial\nu}$ at $\nu=-n$ combine (10.2.4) and (10.15.3). … βΊ
10.15.5 $\left.\frac{\partial J_{\nu}\left(z\right)}{\partial\nu}\right|_{\nu=0}=\frac{% \pi}{2}Y_{0}\left(z\right),\quad\left.\frac{\partial Y_{\nu}\left(z\right)}{% \partial\nu}\right|_{\nu=0}=-\frac{\pi}{2}J_{0}\left(z\right).$
##### 7: 12.17 Physical Applications
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12.17.2 $\nabla^{2}=\frac{{\partial}^{2}}{{\partial x}^{2}}+\frac{{\partial}^{2}}{{% \partial y}^{2}}+\frac{{\partial}^{2}}{{\partial z}^{2}}$
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12.17.4 $\frac{1}{\xi^{2}+\eta^{2}}\left(\frac{{\partial}^{2}w}{{\partial\xi}^{2}}+% \frac{{\partial}^{2}w}{{\partial\eta}^{2}}\right)+\frac{{\partial}^{2}w}{{% \partial\zeta}^{2}}+k^{2}w=0.$
##### 8: 10.73 Physical Applications
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10.73.1 $\nabla^{2}V=\frac{1}{r}\frac{\partial}{\partial r}\left(r\frac{\partial V}{% \partial r}\right)+\frac{1}{r^{2}}\frac{{\partial}^{2}V}{{\partial\phi}^{2}}+% \frac{{\partial}^{2}V}{{\partial z}^{2}}=0,$
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10.73.2 $\nabla^{2}\psi=\frac{1}{c^{2}}\frac{{\partial}^{2}\psi}{{\partial t}^{2}},$
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10.73.3 $\nabla^{4}W+\lambda^{2}\frac{{\partial}^{2}W}{{\partial t}^{2}}=0.$
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10.73.4 $(\nabla^{2}+k^{2})f=\frac{1}{\rho^{2}}\frac{\partial}{\partial\rho}\left(\rho^% {2}\frac{\partial f}{\partial\rho}\right)+\frac{1}{\rho^{2}\sin\theta}\frac{% \partial}{\partial\theta}\left(\sin\theta\frac{\partial f}{\partial\theta}% \right)+\frac{1}{\rho^{2}{\sin}^{2}\theta}\frac{{\partial}^{2}f}{{\partial\phi% }^{2}}+k^{2}f.$
##### 9: 20.13 Physical Applications
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20.13.1 $\ifrac{\partial\theta(z|\tau)}{\partial\tau}=\kappa\ifrac{{\partial}^{2}\theta% (z|\tau)}{{\partial z}^{2}},$
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20.13.2 $\ifrac{\partial\theta}{\partial t}=\alpha\ifrac{{\partial}^{2}\theta}{{% \partial z}^{2}},$
##### 10: 30.13 Wave Equation in Prolate Spheroidal Coordinates
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30.13.3 $h_{\xi}^{2}=\left(\frac{\partial x}{\partial\xi}\right)^{2}+\left(\frac{% \partial y}{\partial\xi}\right)^{2}+\left(\frac{\partial z}{\partial\xi}\right% )^{2}=\frac{c^{2}(\xi^{2}-\eta^{2})}{\xi^{2}-1},$
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30.13.4 $h_{\eta}^{2}=\left(\frac{\partial x}{\partial\eta}\right)^{2}+\left(\frac{% \partial y}{\partial\eta}\right)^{2}+\left(\frac{\partial z}{\partial\eta}% \right)^{2}=\frac{c^{2}(\xi^{2}-\eta^{2})}{1-\eta^{2}},$
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30.13.5 $h_{\phi}^{2}=\left(\frac{\partial x}{\partial\phi}\right)^{2}+\left(\frac{% \partial y}{\partial\phi}\right)^{2}+\left(\frac{\partial z}{\partial\phi}% \right)^{2}=c^{2}(\xi^{2}-1)(1-\eta^{2}).$
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30.13.6 $\nabla^{2}=\frac{1}{h_{\xi}h_{\eta}h_{\phi}}\left(\frac{\partial}{\partial\xi}% \left(\frac{h_{\eta}h_{\phi}}{h_{\xi}}\frac{\partial}{\partial\xi}\right)+% \frac{\partial}{\partial\eta}\left(\frac{h_{\xi}h_{\phi}}{h_{\eta}}\frac{% \partial}{\partial\eta}\right)+\frac{\partial}{\partial\phi}\left(\frac{h_{\xi% }h_{\eta}}{h_{\phi}}\frac{\partial}{\partial\phi}\right)\right)=\frac{1}{c^{2}% (\xi^{2}-\eta^{2})}\left(\frac{\partial}{\partial\xi}\left((\xi^{2}-1)\frac{% \partial}{\partial\xi}\right)+\frac{\partial}{\partial\eta}\left((1-\eta^{2})% \frac{\partial}{\partial\eta}\right)+\frac{\xi^{2}-\eta^{2}}{(\xi^{2}-1)(1-% \eta^{2})}\frac{{\partial}^{2}}{{\partial\phi}^{2}}\right).$