The derivative of the flow is a homothecy
$begingroup$
Let $a,b in mathbb{R}^n.$
$Φ_{a,t} (x) = a + (x − a)e^t$ is the flow corresponding to the vector field $X(x) = x-a$.
Calculate $Z(x, t) = (T_x Φ_{a,t} )^{−1} (X_b (Φ_{a,t} (x)))$.
I don't understand the solution, it states that:
$T_x Φ_{a,t}$ is the homothecy of ratio $e^{-t}$. Why is that? I know that $Φ_{a,t}$ is the homothecy of ratio $e^{-t}$ and center $a$ but I can't go further.
Thank you for your help.
differential-geometry vector-fields
asked Jan 7 at 19:04
PerelManPerelMan
760414
$endgroup$
add a comment |
$begingroup$
Let $a,b in mathbb{R}^n.$
$Φ_{a,t} (x) = a + (x − a)e^t$ is the flow corresponding to the vector field $X(x) = x-a$.
Calculate $Z(x, t) = (T_x Φ_{a,t} )^{−1} (X_b (Φ_{a,t} (x)))$.
I don't understand the solution, it states that:
$T_x Φ_{a,t}$ is the homothecy of ratio $e^{-t}$. Why is that? I know that $Φ_{a,t}$ is the homothecy of ratio $e^{-t}$ and center $a$ but I can't go further.
Thank you for your help.
differential-geometry vector-fields
asked Jan 7 at 19:04
PerelManPerelMan
760414
$endgroup$
add a comment |
$begingroup$
Let $a,b in mathbb{R}^n.$
$Φ_{a,t} (x) = a + (x − a)e^t$ is the flow corresponding to the vector field $X(x) = x-a$.
Calculate $Z(x, t) = (T_x Φ_{a,t} )^{−1} (X_b (Φ_{a,t} (x)))$.
I don't understand the solution, it states that:
$T_x Φ_{a,t}$ is the homothecy of ratio $e^{-t}$. Why is that? I know that $Φ_{a,t}$ is the homothecy of ratio $e^{-t}$ and center $a$ but I can't go further.
Thank you for your help.
differential-geometry vector-fields
asked Jan 7 at 19:04
PerelManPerelMan
760414
$endgroup$
Let $a,b in mathbb{R}^n.$
$Φ_{a,t} (x) = a + (x − a)e^t$ is the flow corresponding to the vector field $X(x) = x-a$.
Calculate $Z(x, t) = (T_x Φ_{a,t} )^{−1} (X_b (Φ_{a,t} (x)))$.
I don't understand the solution, it states that:
$T_x Φ_{a,t}$ is the homothecy of ratio $e^{-t}$. Why is that? I know that $Φ_{a,t}$ is the homothecy of ratio $e^{-t}$ and center $a$ but I can't go further.
Thank you for your help.
differential-geometry vector-fields
differential-geometry vector-fields
asked Jan 7 at 19:04
PerelManPerelMan
760414
asked Jan 7 at 19:04
PerelManPerelMan
760414
asked Jan 7 at 19:04
PerelManPerelMan
760414
asked Jan 7 at 19:04
PerelManPerelMan
760414
asked Jan 7 at 19:04
PerelManPerelMan
760414
760414
add a comment |
add a comment |
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