Laplacià en coordenades polars. Laplaciana en coordenadas polares (Operador laplaciano)



1-Ingredients

    1-Operador Laplacià en coordenades rectangulars.

    `\nabla^2=\delta^2/(\deltax^2)+\delta^2/(\deltay^2)+\delta^2/(\deltaz^2)`



    2-Coordenades rectangulars en coordenades polars.

    `x=rsin \theta cos \varphi`


    `y=rsin\theta\sin\varphi`


    `z=rcos\theta`



    3-Coordenades polars en coordenades rectangulars.

    `r=sqrt(x^2+y^2+z^2)`


    `\theta=arc cos(z/sqrt(x^2+y^2+z^2))`


    `\varphi= arc tan (y/x)`



    4-RESULTAT: Operador Laplaciá en coordenades polars.

    `\nabla^2=1/r^2\delta/(\deltar)(r^2\delta/(\deltar))+1/(r^2sin^2\theta)\delta^2/(\delta\varphi^2)+1/(r^2sin\theta)\delta/(\delta\theta)(sin\theta\delta/(\delta\theta))`


    Si el que necessiteu és la fórmula, ja ho podeu deixar aquí. Si us va la marxa us convidem a continuar. Cal saber calcular derivades senzilles `1/x, sinx, cosx`, derivada del producte, derivada de la funció composta, ..., i relacions trigonomètriques senzilles (i importants) com `sin^2x+cos^2x=1`. Operacions amb matrius, càlcul de la matriu inversa, resolució de sistemes mitjançant matrius.


    Si ho desenvolupem:
    `\nabla^2=2/r\delta/(\deltar) + \delta^2/(\deltar^2)+1/(r^2sin^2\theta)\delta^2/(\delta\varphi^2)+cos\theta/(r^2sin\theta)\delta/(\delta\theta)+1/r^2\delta^2/(\delta\theta^2)`





1'-Càlcul operador Laplacià en coordenades polars 2D

`\nabla^2=\delta^2/(\deltax^2)+\delta^2/(\deltay^2)=\delta^2/(\deltar^2) + 1/r\delta/(\deltar)+ 1/r^2\delta^2/(\delta\theta^2)`


    Entrenament perquè el que ve a continuació no ens ennuegui ;-)



2-Càlculs. 1a derivada


`\nabla=(\delta)/(\deltax)+(\delta)/(\deltay)+(\delta)/(\deltaz)`



`(\delta)/(\delta x)=(\delta)/(\delta r)(\delta r)/(\delta x)+(\delta)/(\delta\theta)(\delta\theta)/(\deltax)+(\delta)/(\delta\varphi)(\delta\varphi)/(\deltax)`


`(\delta)/(\delta y)=(\delta)/(\delta r)(\delta r)/(\delta y)+(\delta)/(\delta\theta)(\delta\theta)/(\deltay)+(\delta)/(\delta\varphi)(\delta\varphi)/(\deltay)`


`(\delta)/(\delta z)=(\delta)/(\delta r)(\delta r)/(\delta z)+(\delta)/(\delta\theta)(\delta\theta)/(\deltaz)+(\delta)/(\delta\varphi)(\delta\varphi)/(\deltaz)`




`(\deltax)/(\deltar)=sin \theta cos \varphi` `(\deltay)/(\deltar)=sin\theta\sin\varphi` `(\deltaz)/(\delta r)=cos\theta`
`(\deltax)/(\delta\theta)=rcos \theta cos \varphi` `(\deltay)/(\delta\theta)=rcos\theta\sin\varphi` `(\deltaz)/(\delta\theta)=-rsin\theta`
`(\deltax)/(delta\varphi)=-rsin \theta sin \varphi` `(\deltay)/(\delta\varphi)=rsin\theta\cos\varphi` `(\deltaz)/(\delta\varphi)=0`




`(\delta)/(\deltar) = (\deltax)/(\deltar)(\delta)/(\deltax)+(\deltay)/(\deltar)(\delta)/(\deltay)+(\deltaz)/(\deltar)(\delta)/(\deltaz) = sin \theta cos \varphi(\delta)/(\deltax)+sin\theta\sin\varphi(\delta)/(\deltay)+cos\theta(\delta)/(\deltaz)`


`(\delta)/(\delta\theta) = (\deltax)/(\delta\theta)(\delta)/(\deltax)+(\deltay)/(\delta\theta)(\delta)/(\deltay)+(\deltaz)/(\delta\theta)(\delta)/(\deltaz) = rcos \theta cos \varphi(\delta)/(\deltax)+rcos\theta\sin\varphi(\delta)/(\deltay)-rsin\theta(\delta)/(\deltaz)`


`(\delta)/(\delta\varphi) = (\deltax)/(\delta\varphi)(\delta)/(\deltax)+(\deltay)/(\delta\varphi)(\delta)/(\deltay)+(\deltaz)/(\delta\varphi)(\delta)/(\deltaz) = -rsin \theta sin \varphi (\delta)/(\deltax)+rsin\theta\cos\varphi(\delta)/(\deltay)`



Això ho podem veure com un sistema de 3 equacions amb 3 incògnites, `(\delta)/(\deltax)`, `(\delta)/(\deltay)`, `(\delta)/(\deltaz)`. Ho posem en notació matricial.


$$ \begin{pmatrix} \frac{\delta}{\delta r}\\\ \frac{\delta}{\delta\theta}\\\ \frac{\delta}{\delta\varphi}\end{pmatrix}= \begin{pmatrix} sin \theta cos \varphi & sin\theta\sin\varphi & cos\theta\\\ rcos \theta cos \varphi & rcos \theta sin \varphi & -rsin\theta\\\ -rsin \theta sin \varphi & rsin\theta\cos\varphi & 0\end{pmatrix} · \begin{pmatrix} \frac{\delta}{\delta x}\\\ \frac{\delta}{\delta y}\\\ \frac{\delta}{\delta z}\end{pmatrix} $$


Ho girem per posar-ho de la forma tradicional, termes independents a la dreta.


$$ \begin{pmatrix} sin \theta cos \varphi & sin\theta\sin\varphi & cos\theta\\\ rcos \theta cos \varphi & rcos \theta sin \varphi & -rsin\theta\\\ -rsin \theta sin \varphi & rsin\theta\cos\varphi & 0\end{pmatrix} · \begin{pmatrix} \frac{\delta}{\delta x}\\\ \frac{\delta}{\delta y}\\\ \frac{\delta}{\delta z}\end{pmatrix}= \begin{pmatrix} \frac{\delta}{\delta r}\\\ \frac{\delta}{\delta\theta}\\\ \frac{\delta}{\delta\varphi}\end{pmatrix} $$


Calculem la matriu inversa de la matriu del sistema i ho multipliquem per l'esquerra.


$$ M · \begin{pmatrix} \frac{\delta}{\delta x}\\\ \frac{\delta}{\delta y}\\\ \frac{\delta}{\delta z}\end{pmatrix}= \begin{pmatrix} \frac{\delta}{\delta r}\\\ \frac{\delta}{\delta\theta}\\\ \frac{\delta}{\delta\varphi}\end{pmatrix} $$
$$ \begin{pmatrix} \frac{\delta}{\delta x}\\\ \frac{\delta}{\delta y}\\\ \frac{\delta}{\delta z}\end{pmatrix}= M^{-1} · \begin{pmatrix} \frac{\delta}{\delta r}\\\ \frac{\delta}{\delta\theta}\\\ \frac{\delta}{\delta\varphi}\end{pmatrix} $$

Per calcular `M^(-1)` en primer lloc calculem `det(M)`

$$ \begin{vmatrix} sin \theta cos \varphi & sin\theta\sin\varphi & cos\theta\\\ rcos \theta cos \varphi & rcos \theta sin \varphi & -rsin\theta\\\ -rsin \theta sin \varphi & rsin\theta\cos\varphi & 0\end{vmatrix}= $$
`0+r^2sin^3\theta sin^2 \varphi + r^2cos^2\theta sin\theta cos^2\varphi + r^2cos^2\theta sin\thetasin^2\varphi + r^2sin^3\theta cos^2\varphi-0=`


`r^2sin^3\theta sin^2 \varphi + r^2sin^3\theta cos^2\varphi+ r^2cos^2\theta sin\theta cos^2\varphi + r^2cos^2\theta sin\thetasin^2\varphi=`


`r^2sin^3\theta (sin^2 \varphi + cos^2\varphi) + r^2cos^2\theta sin\theta (cos^2\varphi + sin^2\varphi)=`


`r^2sin^3\theta + r^2cos^2\theta sin\theta =`


`r^2sin^3\theta + r^2(1-sin^2\theta) sin\theta =`


`r^2sin^3\theta + r^2sin\theta - r^2sin^3\theta =`


`r^2sin\theta`


Calculem la matriu d'adjunts `M^\star`

$$ M=\begin{vmatrix} sin \theta cos \varphi & sin\theta\sin\varphi & cos\theta\\\ rcos \theta cos \varphi & rcos \theta sin \varphi & -rsin\theta\\\ -rsin \theta sin \varphi & rsin\theta\cos\varphi & 0\end{vmatrix} $$
$$ M^\star=\begin{pmatrix} \begin{vmatrix} rcos \theta sin \varphi & -rsin\theta\\\ rsin\theta\cos\varphi & 0\ \end{vmatrix}& -\begin{vmatrix} rcos \theta cos \varphi & -rsin\theta\\\ -rsin \theta sin \varphi & 0\end{vmatrix} & \begin{vmatrix} rcos \theta cos \varphi & rcos \theta sin \varphi\\\ -rsin \theta sin \varphi & rsin\theta\cos\varphi \end{vmatrix}\\\ -\begin{vmatrix} sin\theta\sin\varphi & cos\theta\\\ rsin\theta\cos\varphi & 0\end{vmatrix} & \begin{vmatrix}sin \theta cos \varphi & cos\theta\\\ -rsin \theta sin \varphi & 0 \end{vmatrix} & -\begin{vmatrix}sin \theta cos \varphi & sin\theta\sin\varphi\\\ -rsin \theta sin \varphi & rsin\theta\cos\varphi \end{vmatrix}\\\ \begin{vmatrix} sin\theta\sin\varphi & cos\theta\\\ rcos \theta sin \varphi & -rsin\theta\end{vmatrix} & -\begin{vmatrix} sin \theta cos \varphi & cos\theta\\\ rcos \theta cos \varphi & -rsin\theta\end{vmatrix} & \begin{vmatrix} sin \theta cos \varphi & sin\theta\sin\varphi\\\ rcos \theta cos \varphi & rcos \theta sin \varphi \end{vmatrix}\end{pmatrix} $$
$$ M^\star=\begin{pmatrix} r^2sin^2\theta\cos\varphi & r^2sin^2\theta\sin\varphi & r^2cos\theta sin\theta cos^2 \varphi + r^2cos \theta sin\theta sin^2 \varphi\\\ rsin\theta\cos\theta\cos\varphi & rsin\theta\cos\theta\sin\varphi & -rsin^2\theta\cos^2\varphi-rsin^2\theta\sin^2\varphi\\\ -rsin^2\theta sin\varphi -rcos^2\theta sin\varphi & rsin^2\theta cos\varphi +rcos^2\theta cos\varphi & rsin\theta cos\theta cos\varphi sin\varphi - rcos\theta sin\theta cos\varphi sin\varphi \end{pmatrix} $$
$$ M^\star=\begin{pmatrix} r^2sin^2\theta\cos\varphi & r^2sin^2\theta\sin\varphi & r^2cos \theta sin\theta\\\ rsin\theta\cos\theta\cos\varphi & rsin\theta\cos\theta\sin\varphi & -rsin^2\theta\\\ -rsin\varphi & rcos\varphi & 0 \end{pmatrix} $$

Calculem la matriu transposta `(M^\star)^t`

$$ (M^\star)^t=\begin{pmatrix} r^2sin^2\theta\cos\varphi & rsin\theta\cos\theta\cos\varphi & -rsin\varphi\\\ r^2sin^2\theta\sin\varphi & rsin\theta\cos\theta\sin\varphi & rcos\varphi\\\ r^2cos \theta sin\theta & -rsin^2\theta & 0 \end{pmatrix} $$

Finalment per trobar la inversa cal dividir pel determinant `M^(-1)=(M^\star)^t/(det M)`

$$ M^{-1}=\begin{pmatrix} \frac{r^2sin^2\theta\cos\varphi}{r^2sin\theta} & \frac{rsin\theta\cos\theta\cos\varphi}{r^2sin\theta} & \frac{-rsin\varphi}{r^2sin\theta} \\\ \frac{r^2sin^2\theta\sin\varphi}{r^2sin\theta} & \frac{rsin\theta\cos\theta\sin\varphi}{r^2sin\theta} & \frac{rcos\varphi}{r^2sin\theta} \\\ \frac{r^2cos \theta sin\theta}{r^2sin\theta} & \frac{-rsin^2\theta}{r^2sin\theta} & 0 \end{pmatrix} $$

$$ M^{-1}=\begin{pmatrix} sin\theta\cos\varphi & \frac{\cos\theta\cos\varphi}{r} & \frac{-sin\varphi}{rsin\theta} \\\ sin\theta\sin\varphi & \frac{\cos\theta\sin\varphi}{r} & \frac{cos\varphi}{rsin\theta} \\\ cos \theta & \frac{-sin\theta}{r} & 0 \end{pmatrix} $$

Finalment

$$ \begin{pmatrix} \frac{\delta}{\delta x}\\\ \frac{\delta}{\delta y}\\\ \frac{\delta}{\delta z}\end{pmatrix}= M^{-1} · \begin{pmatrix} \frac{\delta}{\delta r}\\\ \frac{\delta}{\delta\theta}\\\ \frac{\delta}{\delta\varphi}\end{pmatrix} $$

$$ \begin{pmatrix} \frac{\delta}{\delta x}\\\ \frac{\delta}{\delta y}\\\ \frac{\delta}{\delta z}\end{pmatrix}= \begin{pmatrix} sin\theta\cos\varphi & \frac{\cos\theta\cos\varphi}{r} & \frac{-sin\varphi}{rsin\theta} \\\ sin\theta\sin\varphi & \frac{\cos\theta\sin\varphi}{r} & \frac{cos\varphi}{rsin\theta} \\\ cos \theta & \frac{-sin\theta}{r} & 0 \end{pmatrix} · \begin{pmatrix} \frac{\delta}{\delta r}\\\ \frac{\delta}{\delta\theta}\\\ \frac{\delta}{\delta\varphi}\end{pmatrix} $$

$$ \begin{pmatrix} \frac{\delta}{\delta x}\\\ \frac{\delta}{\delta y}\\\ \frac{\delta}{\delta z}\end{pmatrix}= \begin{pmatrix} sin\theta\cos\varphi\frac{\delta}{\delta r} & \frac{\cos\theta\cos\varphi}{r}\frac{\delta}{\delta\theta} & \frac{-sin\varphi}{rsin\theta}\frac{\delta}{\delta\varphi} \\\ sin\theta\sin\varphi\frac{\delta}{\delta r} & \frac{\cos\theta\sin\varphi}{r}\frac{\delta}{\delta\theta} & \frac{cos\varphi}{rsin\theta}\frac{\delta}{\delta\varphi} \\\ cos \theta\frac{\delta}{\delta r} & \frac{-sin\theta}{r}\frac{\delta}{\delta\theta} & 0 \end{pmatrix} · $$

En definitiva: $$ \frac{\delta}{\delta x}=sin\theta\cos\varphi\frac{\delta}{\delta r} + \frac{\cos\theta\cos\varphi}{r}\frac{\delta}{\delta\theta} + \frac{-sin\varphi}{rsin\theta}\frac{\delta}{\delta\varphi} $$

$$ \frac{\delta}{\delta y}=sin\theta\sin\varphi\frac{\delta}{\delta r} + \frac{\cos\theta\sin\varphi}{r}\frac{\delta}{\delta\theta} + \frac{cos\varphi}{rsin\theta}\frac{\delta}{\delta\varphi} $$

$$ \frac{\delta}{\delta z}=cos \theta\frac{\delta}{\delta r} + \frac{-sin\theta}{r}\frac{\delta}{\delta\theta} $$

3-Càlculs. 2a derivada



`\nabla^2=\delta^2/(\deltax^2)+\delta^2/(\deltay^2)+\delta^2/(\deltaz^2)=`




`\delta/(\deltax)(\delta/(\deltax))+\delta/(\deltay)(\delta/(\deltay))+\delta/(\deltaz)(\delta/(\deltaz))=`




`\delta/(\deltax)(sin\thetacos\varphi\delta/(\deltar)+(cos\thetacos\varphi)/r\delta/(\delta\theta)-sin\varphi/(rsin\theta)\delta/(\delta\varphi))+`

`\delta/(\deltay)(sin\thetasin\varphi\delta/(\deltar)+(cos\thetasin\varphi)/r\delta/(\delta\theta)+cos\varphi/(rsin\theta)\delta/(\delta\varphi))+`

`\delta/(\deltaz)(cos\theta\delta/(\deltar)-sin\theta/r\delta/(\delta\theta))=`




`(sin\thetacos\varphi\delta/(\deltar)+(cos\thetacos\varphi)/r\delta/(\delta\theta)-sin\varphi/(rsin\theta)\delta/(\delta\varphi))(sin\thetacos\varphi\delta/(\deltar)+(cos\thetacos\varphi)/r\delta/(\delta\theta)-sin\varphi/(rsin\theta)\delta/(\delta\varphi))+`

`(sin\thetasin\varphi\delta/(\deltar)+(cos\thetasin\varphi)/r\delta/(\delta\theta)+cos\varphi/(rsin\theta)\delta/(\delta\varphi))(sin\thetasin\varphi\delta/(\deltar)+(cos\thetasin\varphi)/r\delta/(\delta\theta)+cos\varphi/(rsin\theta)\delta/(\delta\varphi))+`

`(cos\theta\delta/(\deltar)-sin\theta/r\delta/(\delta\theta))(cos\theta\delta/(\deltar)-sin\theta/r\delta/(\delta\theta))=`




`(sin\thetacos\varphi\delta/(\deltar))(sin\thetacos\varphi\delta/(\deltar)) + (sin\thetacos\varphi\delta/(\deltar))((cos\thetacos\varphi)/r\delta/(\delta\theta)) - (sin\thetacos\varphi\delta/(\deltar))(sin\varphi/(rsin\theta)\delta/(\delta\varphi))+`

`((cos\thetacos\varphi)/r\delta/(\delta\theta))(sin\thetacos\varphi\delta/(\deltar)) + ((cos\thetacos\varphi)/r\delta/(\delta\theta))((cos\thetacos\varphi)/r\delta/(\delta\theta)) - ((cos\thetacos\varphi)/r\delta/(\delta\theta))(sin\varphi/(rsin\theta)\delta/(\delta\varphi))+`

`(-sin\varphi/(rsin\theta)\delta/(\delta\varphi))(sin\thetacos\varphi\delta/(\deltar)) + (-sin\varphi/(rsin\theta)\delta/(\delta\varphi))((cos\thetacos\varphi)/r\delta/(\delta\theta)) - (-sin\varphi/(rsin\theta)\delta/(\delta\varphi))(sin\varphi/(rsin\theta)\delta/(\delta\varphi))+`


`(sin\thetasin\varphi\delta/(\deltar))(sin\thetasin\varphi\delta/(\deltar))+(sin\thetasin\varphi\delta/(\deltar))((cos\thetasin\varphi)/r\delta/(\delta\theta))+(sin\thetasin\varphi\delta/(\deltar))(cos\varphi/(rsin\theta)\delta/(\delta\varphi))+`

`((cos\thetasin\varphi)/r\delta/(\delta\theta))(sin\thetasin\varphi\delta/(\deltar))+((cos\thetasin\varphi)/r\delta/(\delta\theta))((cos\thetasin\varphi)/r\delta/(\delta\theta))+((cos\thetasin\varphi)/r\delta/(\delta\theta))(cos\varphi/(rsin\theta)\delta/(\delta\varphi))+`

`(cos\varphi/(rsin\theta)\delta/(\delta\varphi))(sin\thetasin\varphi\delta/(\deltar))+(cos\varphi/(rsin\theta)\delta/(\delta\varphi))((cos\thetasin\varphi)/r\delta/(\delta\theta))+(cos\varphi/(rsin\theta)\delta/(\delta\varphi))(cos\varphi/(rsin\theta)\delta/(\delta\varphi))+`


`(cos\theta\delta/(\deltar))(cos\theta\delta/(\deltar))-(cos\theta\delta/(\deltar))(sin\theta/r\delta/(\delta\theta))+`

`-(sin\theta/r\delta/(\delta\theta))(cos\theta\delta/(\deltar))+(sin\theta/r\delta/(\delta\theta))(sin\theta/r\delta/(\delta\theta))=`




`sin^2\thetacos^2\varphi\delta^2/(\deltar^2)-(sin\thetacos\thetacos^2\varphi)/r^2\delta/(\delta\theta)+(sin\thetacos\thetacos^2\varphi)/r\delta^2/(\deltar\delta\theta)+(cos\varphisin\varphi)/r^2\delta/(\delta\varphi)-(cos\varphisin\varphi)/r\delta^2/(\deltar\delta\varphi)+`

`(cos^2\thetacos^2\varphi)/r\delta/(\deltar)+(cos\thetasin\thetacos^2\varphi)/r\delta^2/(\delta\theta\deltar)-(cos\thetasin\thetacos^2\varphi)/r^2\delta/(\delta\theta)+(cos^2\thetacos^2\varphi)/r^2\delta^2/(\delta\theta^2)+(cos^2\thetacos\varphisin\varphi)/(r^2sin^2\theta)\delta/(\delta\varphi)-(cos\thetacos\varphisin\varphi)/(r^2sin\theta)\delta^2/(\delta\theta\delta\varphi)+`

`(sin^2\varphi)/r\delta/(\deltar)-(sin\varphicos\varphi)/r\delta^2/(\delta\varphi\deltar)+(sin^2\varphicos\theta)/(r^2sin\theta)\delta/(\delta\theta)-(sin\varphicos\thetacos\varphi)/(r^2sin\theta)\delta^2/(\delta\varphi\delta\theta)+(sin\varphicos\varphi)/(r^2sin^2\theta)\delta/(\delta\varphi)+sin^2\varphi/(r^2sin^2\theta)\delta^2/(\delta\varphi^2)+`


`sin^2\thetasin^2\varphi\delta^2/(\deltar^2)-(sin\thetacos\thetasin^2\varphi)/r^2\delta/(\delta\theta)+(sin\thetacos\thetasin^2\varphi)/r\delta^2/(\deltar\delta\theta)-(sin\varphicos\varphi)/r^2\delta/(\delta\varphi)+(sin\varphicos\varphi)/r\delta^2/(\deltar\delta\varphi)`

`(cos^2\thetasin^2\varphi)/r\delta/(\deltar)+(cos\thetasin\thetasin^2\varphi)/r\delta^2/(\delta\theta\deltar)-(cos\thetasin\thetasin^2\varphi)/r^2\delta/(\delta\theta)+(cos^2\thetasin^2\varphi)/r^2\delta^2/(\delta\theta^2)-(cos^2\thetasin\varphicos\varphi)/(r^2sin^2\theta)\delta/(\delta\varphi)+(cos\thetasin\varphicos\varphi)/(r^2sin\theta)\delta^2/(\delta\theta\delta\varphi)+`

`(cos^2\varphi)/r\delta/(\deltar)+(cos\varphisin\varphi)/r\delta^2/(\delta\varphi\deltar)+(cos^2\varphicos\theta)/(r^2sin\theta)\delta/(\delta\theta)+(cos\varphicos\thetasin\varphi)/(r^2sin\theta)\delta^2/(\delta\varphi\delta\theta)-(cos\varphisin\varphi)/(r^2sin^2\theta)\delta/(\delta\varphi)+cos^2\varphi/(r^2sin^2\theta)\delta^2/(\delta\varphi^2)+`


`cos^2\theta\delta^2/(\deltar^2)+(cos\thetasin\theta)/r^2\delta/(\delta\theta)-(cos\thetasin\theta)/r\delta^2/(\deltar\delta\theta)+`

`sin^2\theta/r\delta/(\deltar)-(sin\thetacos\theta)/r\delta^2/(\delta\theta\deltar)+(sin\thetacos\theta)/r^2\delta/(\delta\theta)+sin^2\theta/r^2\delta^2/(\delta\theta^2)=`







`2/r\delta/(\deltar) + \delta^2/(\deltar^2) + 1/(r^2sin^2\theta)\delta^2/(\delta\varphi^2) + cos\theta/(r^2sin\theta)\delta/(\delta\theta) + 1/r^2\delta^2/(\delta\theta^2) => `



`\nabla^2=1/r^2\delta/(\deltar)(r^2\delta/(\deltar))+1/(r^2sin^2\theta)\delta^2/(\delta\varphi^2)+1/(r^2sin\theta)\delta/(\delta\theta)(sin\theta\delta/(\delta\theta))`

(Q.E.D.) Tal com volíem demostrar