Jacobian and area differential
$begingroup$
A transformation T (u, v) is said to be a conformal transformation if its
Jacobian matrix preserves angles between tangent vectors. Consider that the
vector $langle 0,1rangle$ is parallel to the line $r=pi$ and that the vector $langle 1,1 rangle$ is parallel to the line $r=theta$. Also, notice that $r=pi$ and $r=theta$ intersects at $(r,theta)=(pi,pi)$ at a $45^{circ}$ angle. For $J(r,theta)$ for polar coordinates, i want to calculate $v=J(pi,pi)bigg[begin{matrix}0\1end{matrix}bigg]$ and $w=J(pi,pi)bigg[begin{matrix}1\1end{matrix}bigg]$
Is the angle between v and w a $45^{circ}$ angle? Is the polar coordinates transformation conformal?
I also want to find the jacobian and repeat the previous exercise for the transformation
$T(rho,theta)=langle e^{rho}cos{(theta)},e^{rho}sin{(theta)}rangle$
I also want to calculate the area differential of the transformation $T(rho,theta)=langle e^{rho}cos{(theta)},e^{rho}sin{(theta)}rangle$ both computationally and geometrically.
matrices trigonometry self-learning transformation jacobian
asked Jan 3 at 9:36
Dhamnekar WinodDhamnekar Winod
429514
$endgroup$
add a comment |
$begingroup$
A transformation T (u, v) is said to be a conformal transformation if its
Jacobian matrix preserves angles between tangent vectors. Consider that the
vector $langle 0,1rangle$ is parallel to the line $r=pi$ and that the vector $langle 1,1 rangle$ is parallel to the line $r=theta$. Also, notice that $r=pi$ and $r=theta$ intersects at $(r,theta)=(pi,pi)$ at a $45^{circ}$ angle. For $J(r,theta)$ for polar coordinates, i want to calculate $v=J(pi,pi)bigg[begin{matrix}0\1end{matrix}bigg]$ and $w=J(pi,pi)bigg[begin{matrix}1\1end{matrix}bigg]$
Is the angle between v and w a $45^{circ}$ angle? Is the polar coordinates transformation conformal?
I also want to find the jacobian and repeat the previous exercise for the transformation
$T(rho,theta)=langle e^{rho}cos{(theta)},e^{rho}sin{(theta)}rangle$
I also want to calculate the area differential of the transformation $T(rho,theta)=langle e^{rho}cos{(theta)},e^{rho}sin{(theta)}rangle$ both computationally and geometrically.
matrices trigonometry self-learning transformation jacobian
asked Jan 3 at 9:36
Dhamnekar WinodDhamnekar Winod
429514
$endgroup$
$begingroup$
$r=pi$ is a circle of radius $pi$ . can you explain what it means for a vector to be parallel to a circle?
$endgroup$
– user617446
Jan 3 at 9:44
add a comment |
$begingroup$
A transformation T (u, v) is said to be a conformal transformation if its
Jacobian matrix preserves angles between tangent vectors. Consider that the
vector $langle 0,1rangle$ is parallel to the line $r=pi$ and that the vector $langle 1,1 rangle$ is parallel to the line $r=theta$. Also, notice that $r=pi$ and $r=theta$ intersects at $(r,theta)=(pi,pi)$ at a $45^{circ}$ angle. For $J(r,theta)$ for polar coordinates, i want to calculate $v=J(pi,pi)bigg[begin{matrix}0\1end{matrix}bigg]$ and $w=J(pi,pi)bigg[begin{matrix}1\1end{matrix}bigg]$
Is the angle between v and w a $45^{circ}$ angle? Is the polar coordinates transformation conformal?
I also want to find the jacobian and repeat the previous exercise for the transformation
$T(rho,theta)=langle e^{rho}cos{(theta)},e^{rho}sin{(theta)}rangle$
I also want to calculate the area differential of the transformation $T(rho,theta)=langle e^{rho}cos{(theta)},e^{rho}sin{(theta)}rangle$ both computationally and geometrically.
matrices trigonometry self-learning transformation jacobian
asked Jan 3 at 9:36
Dhamnekar WinodDhamnekar Winod
429514
$endgroup$
A transformation T (u, v) is said to be a conformal transformation if its
Jacobian matrix preserves angles between tangent vectors. Consider that the
vector $langle 0,1rangle$ is parallel to the line $r=pi$ and that the vector $langle 1,1 rangle$ is parallel to the line $r=theta$. Also, notice that $r=pi$ and $r=theta$ intersects at $(r,theta)=(pi,pi)$ at a $45^{circ}$ angle. For $J(r,theta)$ for polar coordinates, i want to calculate $v=J(pi,pi)bigg[begin{matrix}0\1end{matrix}bigg]$ and $w=J(pi,pi)bigg[begin{matrix}1\1end{matrix}bigg]$
Is the angle between v and w a $45^{circ}$ angle? Is the polar coordinates transformation conformal?
I also want to find the jacobian and repeat the previous exercise for the transformation
$T(rho,theta)=langle e^{rho}cos{(theta)},e^{rho}sin{(theta)}rangle$
I also want to calculate the area differential of the transformation $T(rho,theta)=langle e^{rho}cos{(theta)},e^{rho}sin{(theta)}rangle$ both computationally and geometrically.
matrices trigonometry self-learning transformation jacobian
matrices trigonometry self-learning transformation jacobian
asked Jan 3 at 9:36
Dhamnekar WinodDhamnekar Winod
429514
asked Jan 3 at 9:36
Dhamnekar WinodDhamnekar Winod
429514
edited Feb 24 at 14:00
Dhamnekar Winod
asked Jan 3 at 9:36
Dhamnekar WinodDhamnekar Winod
429514
asked Jan 3 at 9:36
Dhamnekar WinodDhamnekar Winod
429514
asked Jan 3 at 9:36
Dhamnekar WinodDhamnekar Winod
429514
429514
$begingroup$
$r=pi$ is a circle of radius $pi$ . can you explain what it means for a vector to be parallel to a circle?
$endgroup$
– user617446
Jan 3 at 9:44
add a comment |
$begingroup$
$r=pi$ is a circle of radius $pi$ . can you explain what it means for a vector to be parallel to a circle?
$endgroup$
– user617446
Jan 3 at 9:44
$begingroup$
$r=pi$ is a circle of radius $pi$ . can you explain what it means for a vector to be parallel to a circle?
$endgroup$
– user617446
Jan 3 at 9:44
$begingroup$
$r=pi$ is a circle of radius $pi$ . can you explain what it means for a vector to be parallel to a circle?
$endgroup$
– user617446
Jan 3 at 9:44
add a comment |
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$begingroup$
$r=pi$ is a circle of radius $pi$ . can you explain what it means for a vector to be parallel to a circle?
$endgroup$
– user617446
Jan 3 at 9:44