14.3: Identidades matemáticas
- Page ID
- 125804
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)\ begin {alineado}\ overline {\ mathrm {A}} &=\ hat {x}\ mathrm {A} _ {\ mathrm {x}} +\ hat {y}\ mathrm {A} _ {\ mathrm {y}} +\ hat {z}\ mathrm {A} _ {\ mathrm {z}}\\ [4pt]\ overline {\ mathrm {A}}\ bullet\ overline {\ mathrm {B}} &=\ mathrm {A} _ {\ mathrm {x}}\ mathrm {B} _ {\ mathrm {x}} +\ mathrm {A} _ {\ mathrm {y}}\ mathrm {B} _ {\ mathrm {y}} +\ mathrm {A} _ { \ mathrm {z}}\ mathrm {B} _ {\ mathrm {z}} =\ sombrero {\ mathrm {a}}\ veces\ sombrero {\ mathrm {b}} |\ overline {\ mathrm {A}} ||\ overline {\ mathrm {B}} |\ cos\ theta\ [4pt]\ overline {\ mathrm {A}}\ veces\ overline {\ mathrm {B}} &=\ nombreoperador {det}\ izquierda|\ begin {array} {ccc}\ hat {x} &\ hat {y} &\ hat {z}\\ [4pt]\ mathrm {A} _ {\ mathrm {x}} & amp;\ mathrm {A} _ {\ mathrm {y}} &\ mathrm {A} _ {\ mathrm {z}}\\ [4pt]\ mathrm {B} _ {\ mathrm {x}} &\ mathrm {B} _ _ {\ mathrm {y}} &\ mathrm {B} _ {\ mathrm {z}}\ end {array}\ right|\\ [4pt] &=\ hat {x}\ left (\ mathrm {A} _ {\ mathrm {y}}\ mathrm {B} _ {\ mathrm {z}} -\ mathrm {A} _ {\ mathrm {z}}\ mathrm {B} _ _ {\ mathrm {y}}\ derecha) +\ sombrero {y}\ izquierda (\ mathrm {A} _ {\ mathrm {z}}\ mathrm {B} _ {\ mathrm {x}} -\ mathrm {A} _ {\ mathrm {x}}\ mathrm {B} _ {\ mathrm {z}}\ derecha) +\ sombrero {2}\ izquierda (\ mathrm {A} _ {\ mathrm {x}}\ mathrm {B} {\ mathrm {y}} -\ mathrm {A} _ {\ mathrm {y}}\ mathrm {B} _ {\ mathrm {x}}\ derecha)\\ [4pt] &=\ hat {\ mathrm {a}}\ veces\ hat {\ mathrm {b}} |\ overline {\ mathrm {A}} ||\ mathrm {\ overline B} |\ sin\ theta\ [4pt]\ overline {\ mathrm {A}}\ bullet (\ overline {\ mathrm {B}}\ times\ overline {\ mathrm {C}}) &=\ overline {\ mathrm {B}}\ bullet (\ overline {\ mathrm {C}}\ times\ overline {\ mathrm {A}}) =\ overline {\ mathrm {C}}\ bullet (\ overline {\ mathrm {A}}\ veces\ overline {\ mathrm {B}})\\ [4pt]\ overline {\ mathrm {A}}\ times (\ overline {\ mathrm {B}}\ times\ overline {\ mathrm {C}}) &= (\ overline {\ mathrm {A}}\ bullet\ overline {\ mathrm {C}})\ overline {\ mathrm {B}} - (\ overline {\ mathrm {A}}\ bullet\ overline {\ mathrm {B}})\ overline {\ mathrm {C}}\\ [4pt] (\ overline {\ mathrm {A}}\ veces\ overline {\ mathrm {B}})\ bullet (\ overline {\ mathrm {C}}\ veces\ overline {\ mathrm {D}}) &= (\ overline {\ mathrm {A}}\ bullet\ overline {\ mathrm {C}}) (\ overline {\ mathrm {B}}\ bullet\ overline {\ mathrm {D}}) - (\ overline {\ mathrm {A}}\ bullet\ overline {\ mathrm {D}}) (\ overline {\ mathrm {B}}\ bullet\ overline {\ mathrm {C}})\\ [4pt]\ nabla\ veces\ nabla\ Psi &=0\\ [4pt]\ nabla\ bullet (\ nabla \ veces\ overline {\ mathrm {A}}) &=0\\ [4pt]\ nabla\ veces (\ nabla\ veces\ overline {\ mathrm {A}}) &=\ nabla (\ nabla\ bullet\ overline {\ mathrm {A}}) -\ nabla^ {2}\ overline {\ mathrm {A}}\ [4pt] -\ overline {\ mathrm {A}}\ times (\ nabla\ times\ overline {\ mathrm {A}}) &= (\ overline {\ mathrm {A}}\ bullet\ nabla)\ overline {\ mathrm {A}} -\ frac {1} {2}\ nabla (\ overline {\ mathrm {A}}\ bullet\ overline {\ mathrm {A}})\\ [4pt]\ nabla (\ Psi\ Phi) &=\ Psi\ nabla\ Phi+\ Phi\ nabla\ Psi\ Psi\ Psi\ {\ mathrm {A}}) &=\ overline {\ mathrm {A}}\ bullet\ nabla\ Psi+\ Psi\ nabla\ bullet\ overline {\ mathrm {A}}\\ [4pt]\ nabla\ veces (\ Psi\ overline {\ mathrm {A}}) &=\ nabla\ Psi\ veces\ overline {\ mathrm {A}} +\ Psi\ nabla\ veces\ overline {\ mathrm {A}}\\ [4pt]\ nabla^ {2}\ Psi &=\ nabla\ bala\ nabla\ nabla\ Psi\\ Psi\ [4pt]\ nabla (\ overline {\ mathrm {A}}\ bullet\ overline {\ mathrm {B}}) &= (\ overline {\ mathrm {A}}\ bullet\ nabla)\ overline {\ mathrm {B}} + (\ overline {\ mathrm {B}}\ bullet\ nabla)\ overline {\ mathrm {A}} +\ overline {\ mathrm {A}}\ times (\ nabla\ times\ overline {\ mathrm {B}}) +\ overline {\ mathrm {B}}\ times (\ nabla\ times\ overline {\ mathrm {A}})\ [4pt]\ nabla\ bullet (\ overline {\ mathrm {A}}\ times\ overline {\ mathrm {B}}) &=\ overline {\ mathrm {B}}\ bullet (\ nabla\ times\ overline {\ mathrm {A}}) -\ overline {\ mathrm {A}}\ bullet (\ nabla\ times\ overline {\ mathrm {B}})\\ [4pt]\ nabla\ times (\ overline {\ mathrm {A}}\ times\ overline {\ mathrm {B}}) &=\ overline {\ mathrm {A}} (\ nabla\ bullet\ overline {\ mathrm {B}}) -\ overline {\ mathrm {B}} (\ nabla\ bullet\ overline {\ mathrm {A}}) + (\ overline {\ mathrm {B}}\ bullet\ nabla)\ overline {\ mathrm {A}} - (\ overline {\ mathrm {A}}\ bullet\ nabla)\ overline {\ mathrm {B}}\ end {alineado}
Coordenadas cartesianas (x, y, z):
\ [\ begin {alineado}
\ nabla\ Psi &=\ hat {x}\ frac {\ parcial\ Psi} {\ parcial\ mathrm {x}} +\ hat {y}\ frac {\ parcial\ Psi} {\ parcial\ mathrm {y}} +\ hat {z}\ frac {\ parcial\ Psi} {\ parcial\ mathrm z {}}\\ [4pt]
\ nabla\ bullet\ overline {\ mathrm {A}} &=\ frac {\ parcial\ mathrm {A} _ {\ mathrm {x}}} {\ parcial\ mathrm {x}} +\ frac {\ parcial\ mathrm {A} _ {\ mathrm {y}}} {\ parcial\ mathrm {y}} +\ frac {\ parcial\ mathrm {A} _ {\ mathrm {z}}} {\ parcial\ mathrm {z}}\\ [4pt]
\ nabla\ veces\ overline {\ mathrm {A}} &=\ hat {x}\ left (\ frac {\ parcial\ mathrm {A} _ {\ mathrm {z}}} {\ parcial\ mathrm {y}} -\ frac {\ parcial\ mathrm {A} _ {\ mathrm {y}}} {\ parcial\ mathrm {z}}\ derecha) +\ hat {y}\ izquierda (\ frac {\ parcial\ mathrm {A} _ {\ mathrm {x}}} {\ parcial\ mathrm {z}} -\ frac {\ parcial\ mathrm {A} _ {\ mathrm {z}}} {\ parcial\ mathrm {x}\ derecha) +\ sombrero {z}\ izquierda (\ frac {\ parcial\ mathrm {A} _ {\ mathrm {y}}} {\ parcial\ mathrm {x}} -\ frac {\ parcial\ mathrm {A} _ {\ mathrm {x}}} {\ parcial\ mathrm {y }}\ derecha)\\ [4pt]
\ nabla^ {2}\ Psi &=\ frac {\ parcial^ {2}\ Psi} {\ parcial\ mathrm {x} ^ {2}} +\ frac {\ parcial^ {2}\ Psi} {\ parcial\ mathrm {y} ^ {2}} +\ frac {\ parcial^ {2}\ Psi} {\ parcial\ mathrm {z} ^ {2}}
\ final {alineado}\]
Coordenadas cilíndricas (r, φ, z):
\ [\ begin {alineado}
\ nabla\ Psi &=\ hat {\ rho}\ frac {\ parcial\ Psi} {\ parcial\ mathrm {r}} +\ hat {\ phi}\ frac {1} {\ mathrm {r}}\ frac {\ parcial\ Psi} {\ parcial\ mathrm {y}} +\ hat {z}\ frac {\ parcial\ Psi\} {\ parcial\ mathrm {z}}\\ [4pt]
\ nabla\ bullet\ overline {\ mathrm {A}} &=\ frac {1} {\ mathrm {r}} \ frac {\ parcial\ izquierda (\ mathrm {r}\ mathrm {A} _ {\ mathrm {r}}\ derecha)} {\ parcial\ mathrm {r}} +\ frac {1} {\ mathrm {r}}\ frac {\ parcial\ mathrm {A} _ {\ phi}} {\ parcial\ phi} +\ frac {\ parcial\ mathrm {A} _ {\ mathrm {z}}} {\ parcial\ mathrm {z}}\\ [4pt]
\ nabla\ veces\ overline {\ mathrm {A}} =&\ hat {r}\ left (\ frac {1} {\ mathrm {r}} \ frac {\ parcial\ mathrm {A} _ {\ mathrm {z}}} {\ parcial\ phi} -\ frac {\ parcial\ mathrm {A} _ {\ phi}} {\ parcial\ mathrm {z}}\ derecha) +\ hat {\ phi}\ izquierda (\ frac {\ parcial\ mathrm {A} _ {\ mathrm {r}} {parcial\ mathrm {z}} -\ frac {\ parcial\ mathrm {A} _ {\ mathrm {z}}} {\ parcial\ mathrm {r}}\ derecha) +\ hat {z}\ frac {1} {\ mathrm {r}}\ izquierda (\ frac {\ parcial\ izquierda (\ mathrm {r}\ mathrm {A} _ {\ phi}\ derecha)} {\ parcial\ mathrm {r}} -\ frac {\ parcial\ mathrm {A} _ {\ mathrm {r}}} {\ parcial\ phi}\ derecha) =\ frac {1} {\ mathrm {r}}\ nombreoperador {det}\ izquierda|\ comenzar array} {ccc}
\ hat {r}/\ parcial\ mathrm {r} &\ parcial/\ parcial\ phi &\ parcial (\ parcial\ mathrm {z}\ mid\\ [4pt]
\ mathrm {A} _ {\ mathrm {r}} &\ mathrm {r}\ mathrm {A} _ {\ phi} &\ mathrm {A} _ {\ mathrm {z}}
\ end {array}\ derecha|\\ [4pt]
\ nabla^ {2}\ Psi &=\ frac {1} {\ mathrm {r}}\ frac {parcial\} {\ parcial\ mathrm {r}}\ izquierda (\ mathrm {r}\ frac {\ parcial\ Psi} {\ parcial\ mathrm {r}}\ derecha) +\ frac {1 } {\ mathrm {r} ^ {2}}\ frac {\ parcial^ {2}\ Psi} {\ parcial\ phi^ {2}} +\ frac {\ parcial^ {2}\ Psi} {\ parcial\ mathrm {z} ^ {2}}
\ final {alineado}\]
Coordenadas esféricas (r, θ, φ):
\[\begin{aligned} \nabla \Psi &=\hat{r} \frac{\partial \Psi}{\partial \mathrm{r}}+\hat{\theta} \frac{1}{\mathrm{r}} \frac{\partial \Psi}{\partial \theta}+\hat{\phi} \frac{1}{\mathrm{r} \sin \theta} \frac{\partial \Psi}{\partial \phi} \\[4pt] \nabla \bullet \overline{\mathrm{A}} &=\frac{1}{\mathrm{r}^{2}} \frac{\partial\left(\mathrm{r}^{2} \mathrm{A}_{\mathrm{r}}\right)}{\partial \mathrm{r}}+\frac{1}{\mathrm{r} \sin \theta} \frac{\partial\left(\sin \theta \mathrm{A}_{\theta}\right)}{\partial \theta}+\frac{1}{\mathrm{r} \sin \theta} \frac{\partial \mathrm{A}_{\phi}}{\partial \phi} \\[4pt] \nabla \times \overline{\mathrm{A}} &=\hat{r} \frac{1}{\mathrm{r} \sin \theta}\left(\frac{\partial\left(\mathrm{r} \sin \theta \mathrm{A}_{\phi}\right)}{\partial \theta}-\frac{\partial \mathrm{A}_{\theta}}{\partial \phi}\right)+\hat{\theta}\left(\frac{1}{\mathrm{r} \sin \theta} \frac{\partial \mathrm{A}_{\mathrm{r}}}{\partial \phi}-\frac{1}{\mathrm{r}} \frac{\partial\left(\mathrm{r} \mathrm{A}_{\phi}\right)}{\partial \mathrm{r}}\right)+\hat{\phi} \frac{1}{\mathrm{r}}\left(\frac{\partial\left(\mathrm{r} \mathrm{A}_{\theta}\right)}{\partial \mathrm{r}}-\frac{\partial \mathrm{A}_{\mathrm{r}}}{\partial \theta}\right) \\[4pt] &=\frac{1}{\mathrm{r}^{2} \sin \theta} \operatorname{det}\left|\begin{array}{ccc} \hat{r} & r \hat{\theta} & \mathrm{r} \sin \theta \hat{\phi} \\[4pt] \partial / \partial \mathrm{r} & \partial / \partial \theta & \partial / \partial \phi \\[4pt] \mathrm{A}_{\mathrm{r}} & \mathrm{r} \mathrm{A}_{\theta} & \mathrm{r} \sin \theta \mathrm{A}_{\phi} \end{array}\right| \\[4pt] \nabla^{2} \Psi &=\mathrm{\frac{1}{r^{2}} \frac{\partial}{\partial r}\left(r^{2} \frac{\partial \Psi}{\partial r}\right)+\frac{1}{r^{2} \sin \theta} \frac{\partial}{\partial \theta}\left(\sin \theta \frac{\partial \Psi}{\partial \theta}\right)+\frac{1}{r^{2} \sin ^{2} \theta} \frac{\partial^{2} \Psi}{\partial \phi^{2}}} \end{aligned}\]
Teorema de Divergencia de Gauss:
\[\int_{V} \nabla \bullet \overline{G} d v=\oint_{A} \overline{G} \bullet \hat{n} d a \nonumber\]
Teorema de Stokes:
\[ \mathrm{\int_{A}(\nabla \times \overline{G}) \bullet \hat{n} \ d a=\oint_{C} \overline{G} \bullet d} \overline{\ell} \nonumber\]
Transformadas de Fourier para señales de pulso h (t):
\[\begin{aligned} \mathrm{\underline{H}(f)} &= \mathrm{\int_{-\infty}^{\infty} h(t) e^{-j 2 \pi t} d t} \\[4pt] \mathrm{h(t)} &= \mathrm{\int_{-\infty}^{\infty} \underline{H}(f) e^{+j 2 \pi f t} d f} \end{aligned}\]