# partial

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##### 1: 1.5 Calculus of Two or More Variables
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###### §1.5(i) Partial Derivatives
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ΊSufficient 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 $[c,d]$. … βΊSuppose 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 $[a,b]\times[c,d)$. Suppose also that $\int^{d}_{c}f(x,y)\,\mathrm{d}y$ converges and $\int^{d}_{c}(\ifrac{\partial f}{\partial x})\,\mathrm{d}y$ converges uniformly on $a\leq x\leq b$, that is, given any positive number $\epsilon$, however small, we can find a number $c_{0}\in[c,d)$ that is independent of $x$ and is such that …
##### 2: 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ΊIf $n=2$, 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). … βΊ
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 $\frac{{\partial}^{2}u}{{\partial x}^{2}}+\frac{{\partial}^{2}u}{{\partial y}^{% 2}}+\frac{1}{y}\frac{\partial u}{\partial y}=0,$
##### 3: 36.10 Differential Equations
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###### §36.10(ii) Partial Derivatives with Respect to the $x_{n}$
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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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36.10.10 $\frac{{\partial}^{3n}\Psi_{3}}{{\partial y}^{3n}}=i^{n}\frac{{\partial}^{2n}% \Psi_{3}}{{\partial z}^{2n}}.$
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##### 4: 16.14 Partial Differential Equations
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###### §16.14(i) Appell Functions
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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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$y(1-y)\frac{{\partial}^{2}{F_{1}}}{{\partial y}^{2}}+x(1-y)\frac{\,{\partial}^% {2}{F_{1}}}{\,\partial x\,\partial y}+\left(\gamma-(\alpha+\beta^{\prime}+1)y% \right)\frac{\partial{F_{1}}}{\partial y}-\beta^{\prime}x\frac{\partial{F_{1}}% }{\partial x}-\alpha\beta^{\prime}{F_{1}}=0,$
βΊIn 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. …
##### 5: 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
10.38.4 $\left.\frac{\partial K_{\nu}\left(z\right)}{\partial\nu}\right|_{\nu=n}=\frac{% n!}{2(\frac{1}{2}z)^{n}}\sum_{k=0}^{n-1}\frac{(\frac{1}{2}z)^{k}K_{k}\left(z% \right)}{k!(n-k)}.$
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$\left.\frac{\partial I_{\nu}\left(z\right)}{\partial\nu}\right|_{\nu=0}=-K_{0}% \left(z\right),$
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10.38.7 $\left.\frac{\partial K_{\nu}\left(x\right)}{\partial\nu}\right|_{\nu=\pm\frac{% 1}{2}}=\pm\sqrt{\frac{\pi}{2x}}E_{1}\left(2x\right)e^{x}.$
##### 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: 23.21 Physical Applications
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###### §23.21(ii) Nonlinear Evolution Equations
βΊAirault et al. (1977) applies the function $\wp$ to an integrable classical many-body problem, and relates the solutions to nonlinear partial differential equations. … βΊ
23.21.2 $(\eta-\zeta)(\zeta-\xi)(\xi-\eta)\nabla^{2}=(\zeta-\eta)f(\xi)f^{\prime}(\xi)% \frac{\partial}{\partial\xi}+(\xi-\zeta)f(\eta)f^{\prime}(\eta)\frac{\partial}% {\partial\eta}+(\eta-\xi)f(\zeta)f^{\prime}(\zeta)\frac{\partial}{\partial% \zeta},$
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23.21.5 $\left(\wp\left(v\right)-\wp\left(w\right)\right)\left(\wp\left(w\right)-\wp% \left(u\right)\right)\left(\wp\left(u\right)-\wp\left(v\right)\right)\nabla^{2% }=\left(\wp\left(w\right)-\wp\left(v\right)\right)\frac{{\partial}^{2}}{{% \partial u}^{2}}+\left(\wp\left(u\right)-\wp\left(w\right)\right)\frac{{% \partial}^{2}}{{\partial v}^{2}}+\left(\wp\left(v\right)-\wp\left(u\right)% \right)\frac{{\partial}^{2}}{{\partial w}^{2}}.$
##### 10: 36.4 Bifurcation Sets
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36.4.1 $\frac{\partial}{\partial t}\Phi_{K}\left(t_{j}(\mathbf{x});\mathbf{x}\right)=0.$
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$\frac{\partial}{\partial s}\Phi^{(\mathrm{U})}\left(s_{j}(\mathbf{x}),t_{j}(% \mathbf{x});\mathbf{x}\right)=0,$
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$\frac{\partial}{\partial t}\Phi^{(\mathrm{U})}\left(s_{j}(\mathbf{x}),t_{j}(% \mathbf{x});\mathbf{x}\right)=0.$
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36.4.3 $\frac{{\partial}^{2}}{{\partial t}^{2}}\Phi_{K}\left(t;\mathbf{x}\right)=0.$
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36.4.4 $\frac{{\partial}^{2}}{{\partial s}^{2}}\Phi^{(\mathrm{U})}\left(s,t;\mathbf{x}% \right)\frac{{\partial}^{2}}{{\partial t}^{2}}\Phi^{(\mathrm{U})}\left(s,t;% \mathbf{x}\right)-\left(\frac{\,{\partial}^{2}}{\,\partial s\,\partial t}\Phi^% {(\mathrm{U})}\left(s,t;\mathbf{x}\right)\right)^{2}=0.$