Differenze tra le versioni di "Calcolo differenziale/Funzioni su spazi vettoriali"

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:<math>d\mathbf f (\mathbf x_0)(\Delta \mathbf x) = D\mathbf f(\mathbf x_0) \Delta \mathbf x</math>
</div>
 
==Proprietà del differenziale e della derivata rispetto alle operazione algebriche==
 
===Differenziale e derivata del prodotto per uno scalare===
 
Si vede subito che:
<div style="float:center; width:85%; padding:15px; background: white; border: 1px solid blue;
margin-left:8px; margin-right:8px;margin-bottom:15px; text-align:left">
*<math>d(k \mathbf f) = kd(\mathbf f)</math>
*<math>D(k \mathbf f) = kD(\mathbf f)</math>
</div>
 
===Differenziale e derivata di una somma===
 
Si vede subito che:
 
<div style="float:center; width:85%; padding:15px; background: white; border: 1px solid blue;
margin-left:8px; margin-right:8px;margin-bottom:15px; text-align:left">
*<math>d(\mathbf f_1 + \mathbf f_2) = d(\mathbf f_1)+d(\mathbf f_2)</math>
*<math>D(\mathbf f_1 + \mathbf f_2) = D(\mathbf f_1)+D(\mathbf f_2)</math>
</div>
 
===Differenziale e derivata di un prodotto scalare===
 
Si ha:
<div style="float:center; width:85%; padding:15px; background: white; border: 1px solid blue;
margin-left:8px; margin-right:8px;margin-bottom:15px; text-align:left">
*<math>d<\mathbf f_1, \mathbf f_2> =
<\mathbf f_1, d\mathbf f_2> + <\mathbf f_2,d\mathbf f_1></math>
*<math>D<\mathbf f_1, \mathbf f_2> =
\mathbf f_1^T D\mathbf f_2 + \mathbf f_2^T D\mathbf f_1</math>
*<math>D_{\hat \mathbf v} <\mathbf f_1, \mathbf f_2> =
<\mathbf f_1, D_{\hat \mathbf v} \mathbf f_2> + <\mathbf f_2,D_{\hat \mathbf v} \mathbf f_1></math>
</div>
 
In particolare se <math>\mathbf v</math> è un vettore costante allora:
<div style="float:center; width:85%; padding:15px; background: white; border: 1px solid blue;
margin-left:8px; margin-right:8px;margin-bottom:15px; text-align:left">
*<math>d<\mathbf v, \mathbf f> = <\mathbf v, d\mathbf f></math>
*<math>D<\mathbf v, \mathbf f> = \mathbf v^T D\mathbf f</math>
*<math>D_{\hat \mathbf v} <\mathbf f_1, \mathbf f_2> =
<\mathbf f_1, D_{\hat \mathbf v} \mathbf f_2> + <\mathbf f_2,D_{\hat \mathbf v} \mathbf f_1></math>
</div>
 
 
 
Dimostrazione:
:<math><\mathbf f_1, \mathbf f_2>(\mathbf x_0 + \Delta \mathbf x) =
<\mathbf f_1(\mathbf x_0 + \Delta \mathbf x),\mathbf f_2(\mathbf x_0 + \Delta \mathbf x)> =</math>
:<math><\mathbf f_1(\mathbf x_0) + d\mathbf f_1(\mathbf x_0)(\Delta \mathbf x)+ o(\|\Delta x\|),
\mathbf f_2(\mathbf x_0) + d\mathbf f_2(\mathbf x_0)(\Delta \mathbf x)+ o(\|\Delta x\|)>=</math>
:<math><\mathbf f_1, \mathbf f_2>(\mathbf x_0) +
<\mathbf f_1(\mathbf x_0), d\mathbf f_2(\mathbf x_0)(\Delta \mathbf x)> +
<\mathbf f_2(\mathbf x_0),d\mathbf f_1(\mathbf x_0)(\Delta \mathbf x)> + o(\|\Delta x\|)</math>
 
da cui segue che:
:<math>d<\mathbf f_1, \mathbf f_2>(\mathbf x_0)(\Delta \mathbf x) =
<\mathbf f_1(\mathbf x_0), d\mathbf f_2(\mathbf x_0)(\Delta \mathbf x)> +
<\mathbf f_2(\mathbf x_0),d\mathbf f_1(\mathbf x_0)(\Delta \mathbf x)></math>
 
Passando alle derivate:
:<math><D<\mathbf f_1, \mathbf f_2>(\mathbf x_0),\Delta \mathbf x> =
<\mathbf f_1(\mathbf x_0), D\mathbf f_2(\mathbf x_0)\Delta \mathbf x> +
<\mathbf f_2(\mathbf x_0),D\mathbf f_1(\mathbf x_0)\Delta \mathbf x> =</math>
:<math><\mathbf f_1^T (x_0)D\mathbf f_2(\mathbf x_0), \Delta \mathbf x> +
<\mathbf f_2^T(\mathbf x_0) D\mathbf f_1(\mathbf x_0),\Delta \mathbf x> =
<\mathbf f_1(\mathbf x_0)D\mathbf f_2(\mathbf x_0)+\mathbf f_2(x_0)D\mathbf f_1(\mathbf x_0),\Delta \mathbf x></math>
 
da cui:
:<math>D<\mathbf f_1, \mathbf f_2>(\mathbf x_0) =
\mathbf f_1^T(\mathbf x_0) D\mathbf f_2(\mathbf x_0)+ \mathbf f_2^T(x_0)D\mathbf f_1(\mathbf x_0)</math>
 
Inoltre si ha:
:<math>D_{\hat \mathbf v} <\mathbf f_1, \mathbf f_2>(\mathbf x_0) =
d<\mathbf f_1, \mathbf f_2>(\mathbf x_0)(\hat \mathbf v) =
<\mathbf f_1(\mathbf x_0), d\mathbf f_2(\mathbf x_0)(\hat \mathbf v)> +
<\mathbf f_2(\mathbf x_0),d\mathbf f_1(\mathbf x_0)(\hat \mathbf v)>=</math>
:<math><\mathbf f_1(\mathbf x_0), D_{\hat \mathbf v}\mathbf f_2(\mathbf x_0)> +
<\mathbf f_2(\mathbf x_0),D_{\hat \mathbf v}\mathbf f_1(\mathbf x_0)></math>
 
===Differenziale e derivata di una funzione composta===
 
Si ha:
<div style="float:center; width:85%; padding:15px; background: white; border: 1px solid blue;
margin-left:8px; margin-right:8px;margin-bottom:15px; text-align:left">
*<math>d(\mathbf f \circ \mathbf g)(\mathbf x_0)(\Delta \mathbf x) =
d\mathbf f (\mathbf g(\mathbf x_0))(d\mathbf g(\mathbf x_0)(\Delta \mathbf x))</math>
*<math>D(\mathbf f \circ \mathbf g)(\mathbf x_0) =
D\mathbf f (\mathbf g(\mathbf x_0))D\mathbf g(\mathbf x_0)</math>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;ovvero&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<math>D(\mathbf f \circ \mathbf g) = (D\mathbf f \circ \mathbf g)D\mathbf g</math>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(regola della catena)
</div>
 
Dimostrazione:
:<math>(\mathbf f \circ \mathbf g)(\mathbf x_0 + \Delta \mathbf x) = \mathbf f (\mathbf g(\mathbf x_0 + \Delta x)) =
\mathbf f (\mathbf g(\mathbf x_0) + d\mathbf g(\mathbf x_0)(\Delta \mathbf x) + o(\|\Delta x\|)) =</math>
:<math>\mathbf f (\mathbf g(\mathbf x_0)) + d\mathbf f (\mathbf g(\mathbf x_0))(d\mathbf g(\mathbf x_0)(\Delta \mathbf x)) +
o(\|\Delta x\|))</math>
 
da cui segue che:
:<math>d(\mathbf f \circ \mathbf g)(\mathbf x_0)(\Delta \mathbf x) =
d\mathbf f (\mathbf g(\mathbf x_0))(d\mathbf g(\mathbf x_0)(\Delta \mathbf x))</math>
 
Passando alle derivate:
:<math>D(\mathbf f \circ \mathbf g)(\mathbf x_0)\Delta \mathbf x =
D\mathbf f (\mathbf g(\mathbf x_0))D\mathbf g(\mathbf x_0)\Delta \mathbf x</math>
 
da cui:
:<math>D(\mathbf f \circ \mathbf g)(\mathbf x_0) =
D\mathbf f (\mathbf g(\mathbf x_0))D\mathbf g(\mathbf x_0)</math>
 
 
==Derivata e differenziale direzionali==
:<math>d = <\nabla, d\mathbf x></math>
</div>
 
==Proprietà del differenziale e della derivata rispetto alle operazione algebriche==
 
===Differenziale e derivata del prodotto per uno scalare===
 
Si vede subito che:
<div style="float:center; width:85%; padding:15px; background: white; border: 1px solid blue;
margin-left:8px; margin-right:8px;margin-bottom:15px; text-align:left">
*<math>d(k \mathbf f) = kd(\mathbf f)</math>
*<math>D(k \mathbf f) = kD(\mathbf f)</math>
</div>
 
===Differenziale e derivata di una somma===
 
Si vede subito che:
 
<div style="float:center; width:85%; padding:15px; background: white; border: 1px solid blue;
margin-left:8px; margin-right:8px;margin-bottom:15px; text-align:left">
*<math>d(\mathbf f_1 + \mathbf f_2) = d(\mathbf f_1)+d(\mathbf f_2)</math>
*<math>D(\mathbf f_1 + \mathbf f_2) = D(\mathbf f_1)+D(\mathbf f_2)</math>
</div>
 
===Differenziale e derivata di un prodotto scalare===
 
Si ha:
<div style="float:center; width:85%; padding:15px; background: white; border: 1px solid blue;
margin-left:8px; margin-right:8px;margin-bottom:15px; text-align:left">
*<math>d<\mathbf f_1, \mathbf f_2> =
<\mathbf f_1, d\mathbf f_2> + <\mathbf f_2,d\mathbf f_1></math>
*<math>D<\mathbf f_1, \mathbf f_2> =
\mathbf f_1^T D\mathbf f_2 + \mathbf f_2^T D\mathbf f_1</math>
*<math>D_{\hat \mathbf v} <\mathbf f_1, \mathbf f_2> =
<\mathbf f_1, D_{\hat \mathbf v} \mathbf f_2> + <\mathbf f_2,D_{\hat \mathbf v} \mathbf f_1></math>
</div>
 
 
Dimostrazione:
:<math><\mathbf f_1, \mathbf f_2>(\mathbf x_0 + \Delta \mathbf x) =
<\mathbf f_1(\mathbf x_0 + \Delta \mathbf x),\mathbf f_2(\mathbf x_0 + \Delta \mathbf x)> =</math>
:<math><\mathbf f_1(\mathbf x_0) + d\mathbf f_1(\mathbf x_0)(\Delta \mathbf x)+ o(\|\Delta x\|),
\mathbf f_2(\mathbf x_0) + d\mathbf f_2(\mathbf x_0)(\Delta \mathbf x)+ o(\|\Delta x\|)>=</math>
:<math><\mathbf f_1, \mathbf f_2>(\mathbf x_0) +
<\mathbf f_1(\mathbf x_0), d\mathbf f_2(\mathbf x_0)(\Delta \mathbf x)> +
<\mathbf f_2(\mathbf x_0),d\mathbf f_1(\mathbf x_0)(\Delta \mathbf x)> + o(\|\Delta x\|)</math>
 
da cui segue che:
:<math>d<\mathbf f_1, \mathbf f_2>(\mathbf x_0)(\Delta \mathbf x) =
<\mathbf f_1(\mathbf x_0), d\mathbf f_2(\mathbf x_0)(\Delta \mathbf x)> +
<\mathbf f_2(\mathbf x_0),d\mathbf f_1(\mathbf x_0)(\Delta \mathbf x)></math>
 
Passando alle derivate:
:<math><D<\mathbf f_1, \mathbf f_2>(\mathbf x_0),\Delta \mathbf x> =
<\mathbf f_1(\mathbf x_0), D\mathbf f_2(\mathbf x_0)\Delta \mathbf x> +
<\mathbf f_2(\mathbf x_0),D\mathbf f_1(\mathbf x_0)\Delta \mathbf x> =</math>
:<math><\mathbf f_1^T (x_0)D\mathbf f_2(\mathbf x_0), \Delta \mathbf x> +
<\mathbf f_2^T(\mathbf x_0) D\mathbf f_1(\mathbf x_0),\Delta \mathbf x> =
<\mathbf f_1(\mathbf x_0)D\mathbf f_2(\mathbf x_0)+\mathbf f_2(x_0)D\mathbf f_1(\mathbf x_0),\Delta \mathbf x></math>
 
da cui:
:<math>D<\mathbf f_1, \mathbf f_2>(\mathbf x_0) =
\mathbf f_1^T(\mathbf x_0) D\mathbf f_2(\mathbf x_0)+ \mathbf f_2^T(x_0)D\mathbf f_1(\mathbf x_0)</math>
 
Inoltre si ha:
:<math>D_{\hat \mathbf v} <\mathbf f_1, \mathbf f_2>(\mathbf x_0) =
d<\mathbf f_1, \mathbf f_2>(\mathbf x_0)(\hat \mathbf v) =
<\mathbf f_1(\mathbf x_0), d\mathbf f_2(\mathbf x_0)(\hat \mathbf v)> +
<\mathbf f_2(\mathbf x_0),d\mathbf f_1(\mathbf x_0)(\hat \mathbf v)>=</math>
:<math><\mathbf f_1(\mathbf x_0), D_{\hat \mathbf v}\mathbf f_2(\mathbf x_0)> +
<\mathbf f_2(\mathbf x_0),D_{\hat \mathbf v}\mathbf f_1(\mathbf x_0)></math>
 
===Differenziale e derivata di una funzione composta===
 
Si ha:
<div style="float:center; width:85%; padding:15px; background: white; border: 1px solid blue;
margin-left:8px; margin-right:8px;margin-bottom:15px; text-align:left">
*<math>d(\mathbf f \circ \mathbf g)(\mathbf x_0)(\Delta \mathbf x) =
d\mathbf f (\mathbf g(\mathbf x_0))(d\mathbf g(\mathbf x_0)(\Delta \mathbf x))</math>
*<math>D(\mathbf f \circ \mathbf g)(\mathbf x_0) =
D\mathbf f (\mathbf g(\mathbf x_0))D\mathbf g(\mathbf x_0)</math>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;ovvero&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<math>D(\mathbf f \circ \mathbf g) = (D\mathbf f \circ \mathbf g)D\mathbf g</math>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(regola della catena)
</div>
 
Dimostrazione:
:<math>(\mathbf f \circ \mathbf g)(\mathbf x_0 + \Delta \mathbf x) = \mathbf f (\mathbf g(\mathbf x_0 + \Delta x)) =
\mathbf f (\mathbf g(\mathbf x_0) + d\mathbf g(\mathbf x_0)(\Delta \mathbf x) + o(\|\Delta x\|)) =</math>
:<math>\mathbf f (\mathbf g(\mathbf x_0)) + d\mathbf f (\mathbf g(\mathbf x_0))(d\mathbf g(\mathbf x_0)(\Delta \mathbf x)) +
o(\|\Delta x\|))</math>
 
da cui segue che:
:<math>d(\mathbf f \circ \mathbf g)(\mathbf x_0)(\Delta \mathbf x) =
d\mathbf f (\mathbf g(\mathbf x_0))(d\mathbf g(\mathbf x_0)(\Delta \mathbf x))</math>
 
Passando alle derivate:
:<math>D(\mathbf f \circ \mathbf g)(\mathbf x_0)\Delta \mathbf x =
D\mathbf f (\mathbf g(\mathbf x_0))D\mathbf g(\mathbf x_0)\Delta \mathbf x</math>
 
da cui:
:<math>D(\mathbf f \circ \mathbf g)(\mathbf x_0) =
D\mathbf f (\mathbf g(\mathbf x_0))D\mathbf g(\mathbf x_0)</math>
 
 
 
 
 
 

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