# partial

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##### 1: 19.18 Derivatives and Differential Equations
Let $\partial_{j}=\ifrac{\partial}{\partial z_{j}}$, and $\mathbf{e}_{j}$ be an $n$-tuple with 1 in the $j$th place and 0’s elsewhere. … If $n=2$, then elimination of $\partial_{2}v$ between (19.18.11) and (19.18.12), followed by the substitution $(b_{1},b_{2},z_{1},z_{2})=(b,c-b,1-z,1)$, produces the Gauss hypergeometric equation (15.10.1). …
19.18.14 $\frac{{\partial}^{2}w}{{\partial x}^{2}}=\frac{{\partial}^{2}w}{{\partial y}^{% 2}}+\frac{1}{y}\frac{\partial w}{\partial y}.$
19.18.15 $\frac{{\partial}^{2}W}{{\partial t}^{2}}=\frac{{\partial}^{2}W}{{\partial x}^{% 2}}+\frac{{\partial}^{2}W}{{\partial y}^{2}}.$
19.18.16 $\frac{{\partial}^{2}u}{{\partial x}^{2}}+\frac{{\partial}^{2}u}{{\partial y}^{% 2}}+\frac{1}{y}\frac{\partial u}{\partial y}=0,$
##### 2: 36.10 Differential Equations
###### §36.10(ii) Partial Derivatives with Respect to the $x_{n}$
36.10.7 $\frac{{\partial}^{2n}\Psi_{2}}{{\partial x}^{2n}}=i^{n}\frac{{\partial}^{n}% \Psi_{2}}{{\partial y}^{n}}.$
36.10.8 $\frac{{\partial}^{2n}\Psi_{3}}{{\partial x}^{2n}}=i^{n}\frac{{\partial}^{n}% \Psi_{3}}{{\partial y}^{n}},$
36.10.10 $\frac{{\partial}^{3n}\Psi_{3}}{{\partial y}^{3n}}=i^{n}\frac{{\partial}^{2n}% \Psi_{3}}{{\partial z}^{2n}}.$
##### 3: 16.14 Partial Differential Equations
###### §16.14(i) Appell Functions
$x(1-x)\frac{{\partial}^{2}{F_{1}}}{{\partial x}^{2}}+y(1-x)\frac{{\partial}^{2% }{F_{1}}}{\partial x\partial y}+\left(\gamma-(\alpha+\beta+1)x\right)\frac{% \partial{F_{1}}}{\partial x}-\beta y\frac{\partial{F_{1}}}{\partial y}-\alpha% \beta{F_{1}}=0,$
$x(1-x)\frac{{\partial}^{2}{F_{2}}}{{\partial x}^{2}}-xy\frac{{\partial}^{2}{F_% {2}}}{\partial x\partial y}+\left(\gamma-(\alpha+\beta+1)x\right)\frac{% \partial{F_{2}}}{\partial x}-\beta y\frac{\partial{F_{2}}}{\partial y}-\alpha% \beta{F_{2}}=0,$
In addition to the four Appell functions there are $24$ other sums of double series that cannot be expressed as a product of two ${{}_{2}F_{1}}$ functions, and which satisfy pairs of linear partial differential equations of the second order. …
##### 4: 10.38 Derivatives with Respect to Order
10.38.1 $\frac{\partial I_{\pm\nu}\left(z\right)}{\partial\nu}=\pm I_{\pm\nu}\left(z% \right)\ln\left(\tfrac{1}{2}z\right)\mp(\tfrac{1}{2}z)^{\pm\nu}\sum_{k=0}^{% \infty}\frac{\psi\left(k+1\pm\nu\right)}{\Gamma\left(k+1\pm\nu\right)}\frac{(% \frac{1}{4}z^{2})^{k}}{k!},$
10.38.2 $\frac{\partial K_{\nu}\left(z\right)}{\partial\nu}=\tfrac{1}{2}\pi\csc\left(% \nu\pi\right)\*\left(\frac{\partial I_{-\nu}\left(z\right)}{\partial\nu}-\frac% {\partial I_{\nu}\left(z\right)}{\partial\nu}\right)-\pi\cot\left(\nu\pi\right% )K_{\nu}\left(z\right),$ $\nu\notin\mathbb{Z}$.
For $\ifrac{\partial I_{\nu}\left(z\right)}{\partial\nu}$ at $\nu=-n$ combine (10.38.1), (10.38.2), and (10.38.4).
10.38.4 $\left.\frac{\partial K_{\nu}\left(z\right)}{\partial\nu}\right|_{\nu=n}=\frac{% n!}{2(\frac{1}{2}z)^{n}}\sum_{k=0}^{n-1}\frac{(\frac{1}{2}z)^{k}K_{k}\left(z% \right)}{k!(n-k)}.$
$\left.\frac{\partial K_{\nu}\left(z\right)}{\partial\nu}\right|_{\nu=0}=0.$
##### 5: 1.5 Calculus of Two or More Variables
###### §1.5(i) Partial Derivatives
The 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. … If $F(x,y)$ is continuously differentiable, $F(a,b)=0$, and $\ifrac{\partial F}{\partial y}\not=0$ at $(a,b)$, then in a neighborhood of $(a,b)$, that is, an open disk centered at $a,b$, the equation $F(x,y)=0$ defines a continuously differentiable function $y=g(x)$ such that $F(x,g(x))=0$, $b=g(a)$, and $g^{\prime}(x)=-F_{x}/F_{y}$. … Sufficient conditions for validity are: (a) $f$ and $\ifrac{\partial f}{\partial x}$ are continuous on a rectangle $a\leq x\leq b$, $c\leq y\leq d$; (b) when $x\in[a,b]$ both $\alpha(x)$ and $\beta(x)$ are continuously differentiable and lie in $[c,d]$. … Suppose that $a,b,c$ are finite, $d$ is finite or $+\infty$, and $f(x,y)$, $\ifrac{\partial f}{\partial x}$ are continuous on the partly-closed rectangle or infinite strip $[a,b]\times[c,d)$. …
##### 6: 10.15 Derivatives with Respect to Order
10.15.1 $\frac{\partial J_{\pm\nu}\left(z\right)}{\partial\nu}=\pm J_{\pm\nu}\left(z% \right)\ln\left(\tfrac{1}{2}z\right)\mp(\tfrac{1}{2}z)^{\pm\nu}\sum_{k=0}^{% \infty}(-1)^{k}\frac{\psi\left(k+1\pm\nu\right)}{\Gamma\left(k+1\pm\nu\right)}% \frac{(\tfrac{1}{4}z^{2})^{k}}{k!},$
10.15.2 $\frac{\partial Y_{\nu}\left(z\right)}{\partial\nu}=\cot\left(\nu\pi\right)% \left(\frac{\partial J_{\nu}\left(z\right)}{\partial\nu}-\pi Y_{\nu}\left(z% \right)\right)-\csc\left(\nu\pi\right)\frac{\partial J_{-\nu}\left(z\right)}{% \partial\nu}-\pi J_{\nu}\left(z\right).$
10.15.3 $\left.\frac{\partial J_{\nu}\left(z\right)}{\partial\nu}\right|_{\nu=n}=\frac{% \pi}{2}Y_{n}\left(z\right)+\frac{n!}{2(\tfrac{1}{2}z)^{n}}\sum_{k=0}^{n-1}% \frac{(\tfrac{1}{2}z)^{k}J_{k}\left(z\right)}{k!(n-k)}.$
For $\ifrac{\partial J_{\nu}\left(z\right)}{\partial\nu}$ at $\nu=-n$ combine (10.2.4) and (10.15.3). …
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: 10.73 Physical Applications
10.73.1 $\nabla^{2}V=\frac{1}{r}\frac{\partial}{\partial r}\left(r\frac{\partial V}{% \partial r}\right)+\frac{1}{r^{2}}\frac{{\partial}^{2}V}{{\partial\phi}^{2}}+% \frac{{\partial}^{2}V}{{\partial z}^{2}}=0,$
10.73.2 $\nabla^{2}\psi=\frac{1}{c^{2}}\frac{{\partial}^{2}\psi}{{\partial t}^{2}},$
10.73.3 $\nabla^{4}W+\lambda^{2}\frac{{\partial}^{2}W}{{\partial t}^{2}}=0.$
10.73.4 $(\nabla^{2}+k^{2})f=\frac{1}{\rho^{2}}\frac{\partial}{\partial\rho}\left(\rho^% {2}\frac{\partial f}{\partial\rho}\right)+\frac{1}{\rho^{2}\sin\theta}\frac{% \partial}{\partial\theta}\left(\sin\theta\frac{\partial f}{\partial\theta}% \right)+\frac{1}{\rho^{2}{\sin}^{2}\theta}\frac{{\partial}^{2}f}{{\partial\phi% }^{2}}+k^{2}f.$
##### 8: 12.17 Physical Applications
12.17.2 $\nabla^{2}=\frac{{\partial}^{2}}{{\partial x}^{2}}+\frac{{\partial}^{2}}{{% \partial y}^{2}}+\frac{{\partial}^{2}}{{\partial z}^{2}}$
12.17.4 $\frac{1}{\xi^{2}+\eta^{2}}\left(\frac{{\partial}^{2}w}{{\partial\xi}^{2}}+% \frac{{\partial}^{2}w}{{\partial\eta}^{2}}\right)+\frac{{\partial}^{2}w}{{% \partial\zeta}^{2}}+k^{2}w=0.$
##### 9: 28.32 Mathematical Applications
28.32.3 $\frac{{\partial}^{2}V}{{\partial\xi}^{2}}+\frac{{\partial}^{2}V}{{\partial\eta% }^{2}}+\frac{1}{2}c^{2}k^{2}(\cosh\left(2\xi\right)-\cos\left(2\eta\right))V=0.$
28.32.4 $\frac{{\partial}^{2}K}{{\partial z}^{2}}-\frac{{\partial}^{2}K}{{\partial\zeta% }^{2}}=2q\left(\cos\left(2z\right)-\cos\left(2\zeta\right)\right)K.$
28.32.5 $K(z,\zeta)\frac{\mathrm{d}u(\zeta)}{\mathrm{d}\zeta}-u(\zeta)\frac{\partial K(% z,\zeta)}{\partial\zeta}$
##### 10: 1.6 Vectors and Vector-Valued Functions
1.6.20 $\operatorname{grad}f=\nabla f=\frac{\partial f}{\partial x}\mathbf{i}+\frac{% \partial f}{\partial y}\mathbf{j}+\frac{\partial f}{\partial z}\mathbf{k}.$
1.6.21 $\operatorname{div}\mathbf{F}=\nabla\cdot\mathbf{F}=\frac{\partial F_{1}}{% \partial x}+\frac{\partial F_{2}}{\partial y}+\frac{\partial F_{3}}{\partial z}.$
1.6.46 $\mathbf{T}_{u}=\frac{\partial x}{\partial u}(u_{0},v_{0})\mathbf{i}+\frac{% \partial y}{\partial u}(u_{0},v_{0})\mathbf{j}+\frac{\partial z}{\partial u}(u% _{0},v_{0})\mathbf{k}$
Suppose $S$ is an oriented surface with boundary $\partial S$ which is oriented so that its direction is clockwise relative to the normals of $S$. … where $\ifrac{\partial g}{\partial n}=\nabla g\cdot\mathbf{n}$ is the derivative of $g$ normal to the surface outwards from $V$ and $\mathbf{n}$ is the unit outer normal vector. …