Show that a system of equations to find a curve based on its curvature and torsion has a unique solution.
up vote
1
down vote
favorite
1
Quoted from Pressley's Differential Geometry :
How to show that the three mentioned equations have a unique solution with initial conditions?
Since $k=k(s)$ and $t=t(s)$ are functions not constants so any try for solving them leads to even more complicated system of equations.
differential-equations differential-geometry systems-of-equations
asked Nov 16 at 11:44
72D
52616
|
show 4 more comments
up vote
1
down vote
favorite
1
Quoted from Pressley's Differential Geometry :
How to show that the three mentioned equations have a unique solution with initial conditions?
Since $k=k(s)$ and $t=t(s)$ are functions not constants so any try for solving them leads to even more complicated system of equations.
differential-equations differential-geometry systems-of-equations
asked Nov 16 at 11:44
72D
52616
Picard-Lindelöf theorem
– achille hui
Nov 16 at 11:51
@achillehui, there are three functions in not-3-independent-equations) to be solved not one.
– 72D
Nov 16 at 12:08
The $y$ in the theorem can be vector, the wiki book entry for the theorem spell this out explicitly.
– achille hui
Nov 16 at 12:22
@achillehui, I started to learn the one-dim case from an easy textbook now so I hopefully can understand the one in wikibook... Thanks.
– 72D
Nov 16 at 12:52
@achillehui, I understood all thing but one! Let $x(s)=(T_1(s), T_2(s), T_3(s), N_1, ..., B_1, ...)$ and $f(s,x(s))=(k(s)N_1(s), ..., -k(s)T_1+t(s)B_1, ..., -t(s)N_3)$ then $frac{d}{ds}x(s)=f(s,x(s))$ together with existence of $L$ (Lipschitz const) implies that $text{max}_{s in [s_0, s_1]} (k^2(N_1-N'_2)^2+...+(-tN_3+tN'_3)^2) le L^2 text{max}_{s in [s_0, s_1]} (T_1-T'_2)^2+...+(B_3-B'_3)^2)$. But for the smooth functions of $T$, $N$, $B$ in the (differential geo) text (image in OP) how the existence of $L$ is proven/guaranteed? ( ' means y instead of x , not for derivative)
– 72D
Nov 17 at 10:50
|
show 4 more comments
up vote
1
down vote
favorite
1
up vote
1
down vote
favorite
1
1
Quoted from Pressley's Differential Geometry :
How to show that the three mentioned equations have a unique solution with initial conditions?
Since $k=k(s)$ and $t=t(s)$ are functions not constants so any try for solving them leads to even more complicated system of equations.
differential-equations differential-geometry systems-of-equations
asked Nov 16 at 11:44
72D
52616
Quoted from Pressley's Differential Geometry :
How to show that the three mentioned equations have a unique solution with initial conditions?
Since $k=k(s)$ and $t=t(s)$ are functions not constants so any try for solving them leads to even more complicated system of equations.
differential-equations differential-geometry systems-of-equations
differential-equations differential-geometry systems-of-equations
asked Nov 16 at 11:44
72D
52616
asked Nov 16 at 11:44
72D
52616
edited Nov 18 at 9:01
asked Nov 16 at 11:44
72D
52616
asked Nov 16 at 11:44
72D
52616
asked Nov 16 at 11:44
72D
52616
52616
Picard-Lindelöf theorem
– achille hui
Nov 16 at 11:51
@achillehui, there are three functions in not-3-independent-equations) to be solved not one.
– 72D
Nov 16 at 12:08
The $y$ in the theorem can be vector, the wiki book entry for the theorem spell this out explicitly.
– achille hui
Nov 16 at 12:22
@achillehui, I started to learn the one-dim case from an easy textbook now so I hopefully can understand the one in wikibook... Thanks.
– 72D
Nov 16 at 12:52
@achillehui, I understood all thing but one! Let $x(s)=(T_1(s), T_2(s), T_3(s), N_1, ..., B_1, ...)$ and $f(s,x(s))=(k(s)N_1(s), ..., -k(s)T_1+t(s)B_1, ..., -t(s)N_3)$ then $frac{d}{ds}x(s)=f(s,x(s))$ together with existence of $L$ (Lipschitz const) implies that $text{max}_{s in [s_0, s_1]} (k^2(N_1-N'_2)^2+...+(-tN_3+tN'_3)^2) le L^2 text{max}_{s in [s_0, s_1]} (T_1-T'_2)^2+...+(B_3-B'_3)^2)$. But for the smooth functions of $T$, $N$, $B$ in the (differential geo) text (image in OP) how the existence of $L$ is proven/guaranteed? ( ' means y instead of x , not for derivative)
– 72D
Nov 17 at 10:50
|
show 4 more comments
Picard-Lindelöf theorem
– achille hui
Nov 16 at 11:51
@achillehui, there are three functions in not-3-independent-equations) to be solved not one.
– 72D
Nov 16 at 12:08
The $y$ in the theorem can be vector, the wiki book entry for the theorem spell this out explicitly.
– achille hui
Nov 16 at 12:22
@achillehui, I started to learn the one-dim case from an easy textbook now so I hopefully can understand the one in wikibook... Thanks.
– 72D
Nov 16 at 12:52
@achillehui, I understood all thing but one! Let $x(s)=(T_1(s), T_2(s), T_3(s), N_1, ..., B_1, ...)$ and $f(s,x(s))=(k(s)N_1(s), ..., -k(s)T_1+t(s)B_1, ..., -t(s)N_3)$ then $frac{d}{ds}x(s)=f(s,x(s))$ together with existence of $L$ (Lipschitz const) implies that $text{max}_{s in [s_0, s_1]} (k^2(N_1-N'_2)^2+...+(-tN_3+tN'_3)^2) le L^2 text{max}_{s in [s_0, s_1]} (T_1-T'_2)^2+...+(B_3-B'_3)^2)$. But for the smooth functions of $T$, $N$, $B$ in the (differential geo) text (image in OP) how the existence of $L$ is proven/guaranteed? ( ' means y instead of x , not for derivative)
– 72D
Nov 17 at 10:50
Picard-Lindelöf theorem
– achille hui
Nov 16 at 11:51
Picard-Lindelöf theorem
– achille hui
Nov 16 at 11:51
@achillehui, there are three functions in not-3-independent-equations) to be solved not one.
– 72D
Nov 16 at 12:08
@achillehui, there are three functions in not-3-independent-equations) to be solved not one.
– 72D
Nov 16 at 12:08
The $y$ in the theorem can be vector, the wiki book entry for the theorem spell this out explicitly.
– achille hui
Nov 16 at 12:22
The $y$ in the theorem can be vector, the wiki book entry for the theorem spell this out explicitly.
– achille hui
Nov 16 at 12:22
@achillehui, I started to learn the one-dim case from an easy textbook now so I hopefully can understand the one in wikibook... Thanks.
– 72D
Nov 16 at 12:52
@achillehui, I started to learn the one-dim case from an easy textbook now so I hopefully can understand the one in wikibook... Thanks.
– 72D
Nov 16 at 12:52
@achillehui, I understood all thing but one! Let $x(s)=(T_1(s), T_2(s), T_3(s), N_1, ..., B_1, ...)$ and $f(s,x(s))=(k(s)N_1(s), ..., -k(s)T_1+t(s)B_1, ..., -t(s)N_3)$ then $frac{d}{ds}x(s)=f(s,x(s))$ together with existence of $L$ (Lipschitz const) implies that $text{max}_{s in [s_0, s_1]} (k^2(N_1-N'_2)^2+...+(-tN_3+tN'_3)^2) le L^2 text{max}_{s in [s_0, s_1]} (T_1-T'_2)^2+...+(B_3-B'_3)^2)$. But for the smooth functions of $T$, $N$, $B$ in the (differential geo) text (image in OP) how the existence of $L$ is proven/guaranteed? ( ' means y instead of x , not for derivative)
– 72D
Nov 17 at 10:50
@achillehui, I understood all thing but one! Let $x(s)=(T_1(s), T_2(s), T_3(s), N_1, ..., B_1, ...)$ and $f(s,x(s))=(k(s)N_1(s), ..., -k(s)T_1+t(s)B_1, ..., -t(s)N_3)$ then $frac{d}{ds}x(s)=f(s,x(s))$ together with existence of $L$ (Lipschitz const) implies that $text{max}_{s in [s_0, s_1]} (k^2(N_1-N'_2)^2+...+(-tN_3+tN'_3)^2) le L^2 text{max}_{s in [s_0, s_1]} (T_1-T'_2)^2+...+(B_3-B'_3)^2)$. But for the smooth functions of $T$, $N$, $B$ in the (differential geo) text (image in OP) how the existence of $L$ is proven/guaranteed? ( ' means y instead of x , not for derivative)
– 72D
Nov 17 at 10:50
|
show 4 more comments
active
oldest
votes
active
oldest
votes
active
oldest
votes
active
oldest
votes
active
oldest
votes
draft saved
draft discarded
Thanks for contributing an answer to Mathematics Stack Exchange!
- Please be sure to answer the question. Provide details and share your research!
But avoid …
- Asking for help, clarification, or responding to other answers.
- Making statements based on opinion; back them up with references or personal experience.
Use MathJax to format equations. MathJax reference.
To learn more, see our tips on writing great answers.
Some of your past answers have not been well-received, and you're in danger of being blocked from answering.
Please pay close attention to the following guidance:
- Please be sure to answer the question. Provide details and share your research!
But avoid …
- Asking for help, clarification, or responding to other answers.
- Making statements based on opinion; back them up with references or personal experience.
To learn more, see our tips on writing great answers.
draft saved
draft discarded
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
StackExchange.ready(
function () {
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3001044%2fshow-that-a-system-of-equations-to-find-a-curve-based-on-its-curvature-and-torsi%23new-answer', 'question_page');
}
);
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Picard-Lindelöf theorem
– achille hui
Nov 16 at 11:51
@achillehui, there are three functions in not-3-independent-equations) to be solved not one.
– 72D
Nov 16 at 12:08
The $y$ in the theorem can be vector, the wiki book entry for the theorem spell this out explicitly.
– achille hui
Nov 16 at 12:22
@achillehui, I started to learn the one-dim case from an easy textbook now so I hopefully can understand the one in wikibook... Thanks.
– 72D
Nov 16 at 12:52
@achillehui, I understood all thing but one! Let $x(s)=(T_1(s), T_2(s), T_3(s), N_1, ..., B_1, ...)$ and $f(s,x(s))=(k(s)N_1(s), ..., -k(s)T_1+t(s)B_1, ..., -t(s)N_3)$ then $frac{d}{ds}x(s)=f(s,x(s))$ together with existence of $L$ (Lipschitz const) implies that $text{max}_{s in [s_0, s_1]} (k^2(N_1-N'_2)^2+...+(-tN_3+tN'_3)^2) le L^2 text{max}_{s in [s_0, s_1]} (T_1-T'_2)^2+...+(B_3-B'_3)^2)$. But for the smooth functions of $T$, $N$, $B$ in the (differential geo) text (image in OP) how the existence of $L$ is proven/guaranteed? ( ' means y instead of x , not for derivative)
– 72D
Nov 17 at 10:50