Coordinate curvilinee

Versione attuale delle 20:58, 25 dic 2017

Possiamo esprimere tutti gli operatori differenziali in qualsiasi set di variabili; non dimostreremo come si arriva alla formule che andiamo ora ad ottenere. La tabella con i valori di ogni coordinata e parametro è a fine sezione e riportata a fine testo.

Iniziamo dal gradiente di un campo scalare $f$ $f$ :

{\boldsymbol {\nabla }}(f)={\frac {1}{l}}{\frac {\partial f}{\partial u}}{\hat {\mathbf {u} }}+{\frac {1}{m}}{\frac {\partial f}{\partial v}}{\hat {\mathbf {v} }}+{\frac {1}{n}}{\frac {\partial f}{\partial w}}{\hat {\mathbf {w} }}

La divergenza di un campo vettoriale $\mathbf {G}$ $\mathbf {G}$ , invece:

{\boldsymbol {\nabla }}\cdot \mathbf {G} ={\frac {1}{lmn}}\left({\frac {\partial }{\partial u}}(mnG_{u})+{\frac {\partial }{\partial v}}(lnG_{v})+{\frac {\partial }{\partial w}}(lmG_{w})\right)

Il rotore di campo vettoriale:

{\boldsymbol {\nabla }}\times \mathbf {G} ={\frac {1}{mn}}\left({\frac {\partial }{\partial v}}(nG_{w})-{\frac {\partial }{\partial w}}(nG_{v})\right){\hat {\mathbf {u} }}+{\frac {1}{ln}}\left({\frac {\partial }{\partial w}}(lG_{u})-{\frac {\partial }{\partial u}}(nG_{w})\right){\hat {\mathbf {v} }}+{\frac {1}{lm}}\left({\frac {\partial }{\partial u}}(mG_{v})-{\frac {\partial }{\partial v}}(lG_{u})\right){\hat {\mathbf {w} }}

{\boldsymbol {\nabla }}\times \mathbf {G} ={\frac {1}{mn}}\left({\frac {\partial }{\partial v}}(nG_{w})-{\frac {\partial }{\partial w}}(nG_{v})\right){\hat {\mathbf {u} }}+{\frac {1}{ln}}\left({\frac {\partial }{\partial w}}(lG_{u})-{\frac {\partial }{\partial u}}(nG_{w})\right){\hat {\mathbf {v} }}+{\frac {1}{lm}}\left({\frac {\partial }{\partial u}}(mG_{v})-{\frac {\partial }{\partial v}}(lG_{u})\right){\hat {\mathbf {w} }}

Il laplaciano di un campo scalare:

{\boldsymbol {\nabla }}^{2}f={\frac {1}{lmn}}\left[{\frac {\partial }{\partial u}}\left({\frac {mn}{l}}{\frac {\partial f}{\partial u}}\right)+{\frac {\partial }{\partial v}}\left({\frac {ln}{m}}{\frac {\partial f}{\partial v}}\right)+{\frac {\partial }{\partial w}}\left({\frac {lm}{n}}{\frac {\partial f}{\partial w}}\right)\right]

{\boldsymbol {\nabla }}^{2}f={\frac {1}{lmn}}\left[{\frac {\partial }{\partial u}}\left({\frac {mn}{l}}{\frac {\partial f}{\partial u}}\right)+{\frac {\partial }{\partial v}}\left({\frac {ln}{m}}{\frac {\partial f}{\partial v}}\right)+{\frac {\partial }{\partial w}}\left({\frac {lm}{n}}{\frac {\partial f}{\partial w}}\right)\right]

Coordinate	$u\quad v\quad w$ $u\quad v\quad w$	$l\qquad m\qquad n$ $l\qquad m\qquad n$
Cartesiane	$x\quad y\quad z$ $x\quad y\quad z$	$1\qquad 1\qquad 1$ $1\qquad 1\qquad 1$
Sferiche	$r\quad \theta \quad \phi$ $r\quad \theta \quad \phi$	$1\qquad r\quad r\sin \phi$ $1\qquad r\quad r\sin \phi$
Cilindriche	$r\quad \phi \quad z$ $r\quad \phi \quad z$	$1\qquad r\qquad 1$ $1\qquad r\qquad 1$

@@ Riga 3: / Riga 3: @@
 Iniziamo dal '''gradiente''' di un campo scalare <math>f</math>:
-<math display="block"> \mathbf{\nabla} (f)= \frac{1}{l} \frac{\partial f}{\partial u} \hat{u} + \frac{1}{m} \frac{\partial f}{\partial v} \hat{v} + \frac{1}{n} \frac{\partial f}{\partial w} \hat{w}</math>
+<math display="block"> \boldsymbol{\nabla} (f)= \frac{1}{l} \frac{\partial f}{\partial u} \hat{\mathbf{u}} + \frac{1}{m} \frac{\partial f}{\partial v} \hat{\mathbf{v}} + \frac{1}{n} \frac{\partial f}{\partial w} \hat{\mathbf{w}}</math>
 La '''divergenza''' di un campo vettoriale <math>\mathbf{G}</math>, invece:
-<math display="block">\mathbf{\nabla} \cdot \mathbf{G} = \frac{1}{lmn} \left( \frac{\partial}{\partial u} (mn G_u) + \frac{\partial}{\partial v} (ln G_v) + \frac{\partial}{\partial w} (lm G_w) \right)</math>
+<math display="block">\boldsymbol{\nabla} \cdot \mathbf{G} = \frac{1}{lmn} \left( \frac{\partial}{\partial u} (mn G_u) + \frac{\partial}{\partial v} (ln G_v) + \frac{\partial}{\partial w} (lm G_w) \right)</math>
 Il '''rotore''' di campo vettoriale:
-<math display="block"> \mathbf{\nabla}\times \mathbf{G} = \frac{1}{mn} \left( \frac{\partial}{\partial v} (n G_W) - \frac{\partial}{\partial w} (n G_v) \right) \hat{u}+ \frac{1}{ln} \left( \frac{\partial }{\partial w} (l G_u) - \frac{\partial}{\partial u} (n G_W) \right) \hat{v} + \frac{1}{lm} \left(\frac{\partial}{\partial u} (mG_v) - \frac{\partial}{\partial v} (l A_u) \right)\hat{w}</math>
+<math display="block"> \boldsymbol{\nabla}\times \mathbf{G} = \frac{1}{mn} \left( \frac{\partial}{\partial v} (n G_w) - \frac{\partial}{\partial w} (n G_v) \right) \hat{\mathbf{u}}+ \frac{1}{ln} \left( \frac{\partial }{\partial w} (l G_u) - \frac{\partial}{\partial u} (n G_w) \right) \hat{\mathbf{v}} + \frac{1}{lm} \left(\frac{\partial}{\partial u} (mG_v) - \frac{\partial}{\partial v} (l G_u) \right)\hat{\mathbf{w}}</math>
 Il '''laplaciano''' di un campo scalare:
-<math display="block"> \nabla^2 f = \frac{1}{lmb} \left[ \frac{\partial}{\partial u} \left(\frac{mn}{l} \frac{\partial f}{\partial u} \right) + \frac{\partial}{\partial v} \left( \frac{ln}{m} \frac{\partial f}{\partial v} \right) + \frac{\partial}{\partial w} \left( \frac{lm}{n} \frac{\partial f}{\partial w} \right)\right]</math>
+<math display="block"> \boldsymbol{\nabla}^2 f = \frac{1}{lmn} \left[ \frac{\partial}{\partial u} \left(\frac{mn}{l} \frac{\partial f}{\partial u} \right) + \frac{\partial}{\partial v} \left( \frac{ln}{m} \frac{\partial f}{\partial v} \right) + \frac{\partial}{\partial w} \left( \frac{lm}{n} \frac{\partial f}{\partial w} \right)\right]</math>
+{| class="wikitable"
+!Coordinate
+!<math>u \quad v \quad w</math>
+!<math>l \qquad m \qquad n</math>
+|-
+|Cartesiane
+|<math>x \quad y \quad z</math>
+|<math>1 \qquad 1 \qquad 1</math>
+|-
+|Sferiche
+|<math>r \quad \theta \quad \phi</math>
+|<math>1 \qquad r \quad r \sin \phi</math>
+|-
+|Cilindriche
+|<math>r \quad \phi \quad z</math>
+|<math>1 \qquad r \qquad 1</math>
+|}