Differential of a map $mathbb{T}^2 rightarrow S^2$
up vote
1
down vote
favorite
I want to calculate the differential of a map from the 2-torus $mathbb{T}^2$ to the 2-sphere $S^2$. The map is given in the form $$f: (x,y) mapstobegin{pmatrix} x \ y \ Phi end{pmatrix} frac{1}{sqrt{x^2 + y^2 +Phi^2}},$$ defined in a neighbourhood $Sigma$ of a point, say around the point $(0,0)$. What is the procedure to calculate the differential at a point $xi in Sigma$, i.e. $df: T_ximathbb{T}^2 rightarrow T_{f(xi)}S^2?$ I guess for this one should switch to local coordinate charts (how does this look like in this concrete example?)?
derivatives differential-geometry
asked Nov 21 at 20:58
Breaking Mad
581416
add a comment |
up vote
1
down vote
favorite
I want to calculate the differential of a map from the 2-torus $mathbb{T}^2$ to the 2-sphere $S^2$. The map is given in the form $$f: (x,y) mapstobegin{pmatrix} x \ y \ Phi end{pmatrix} frac{1}{sqrt{x^2 + y^2 +Phi^2}},$$ defined in a neighbourhood $Sigma$ of a point, say around the point $(0,0)$. What is the procedure to calculate the differential at a point $xi in Sigma$, i.e. $df: T_ximathbb{T}^2 rightarrow T_{f(xi)}S^2?$ I guess for this one should switch to local coordinate charts (how does this look like in this concrete example?)?
derivatives differential-geometry
asked Nov 21 at 20:58
Breaking Mad
581416
2
I don't see how this defines a map on the torus, $x=0$ and $x=2pi$ should give the same image.
– Christoph
Nov 21 at 21:10
Sorry, I guess it is only defined in a neighbourhood of some point on the torus.
– Breaking Mad
Nov 21 at 22:01
add a comment |
up vote
1
down vote
favorite
up vote
1
down vote
favorite
I want to calculate the differential of a map from the 2-torus $mathbb{T}^2$ to the 2-sphere $S^2$. The map is given in the form $$f: (x,y) mapstobegin{pmatrix} x \ y \ Phi end{pmatrix} frac{1}{sqrt{x^2 + y^2 +Phi^2}},$$ defined in a neighbourhood $Sigma$ of a point, say around the point $(0,0)$. What is the procedure to calculate the differential at a point $xi in Sigma$, i.e. $df: T_ximathbb{T}^2 rightarrow T_{f(xi)}S^2?$ I guess for this one should switch to local coordinate charts (how does this look like in this concrete example?)?
derivatives differential-geometry
asked Nov 21 at 20:58
Breaking Mad
581416
I want to calculate the differential of a map from the 2-torus $mathbb{T}^2$ to the 2-sphere $S^2$. The map is given in the form $$f: (x,y) mapstobegin{pmatrix} x \ y \ Phi end{pmatrix} frac{1}{sqrt{x^2 + y^2 +Phi^2}},$$ defined in a neighbourhood $Sigma$ of a point, say around the point $(0,0)$. What is the procedure to calculate the differential at a point $xi in Sigma$, i.e. $df: T_ximathbb{T}^2 rightarrow T_{f(xi)}S^2?$ I guess for this one should switch to local coordinate charts (how does this look like in this concrete example?)?
derivatives differential-geometry
derivatives differential-geometry
asked Nov 21 at 20:58
Breaking Mad
581416
asked Nov 21 at 20:58
Breaking Mad
581416
edited Nov 21 at 22:02
asked Nov 21 at 20:58
Breaking Mad
581416
asked Nov 21 at 20:58
Breaking Mad
581416
asked Nov 21 at 20:58
Breaking Mad
581416
581416
2
I don't see how this defines a map on the torus, $x=0$ and $x=2pi$ should give the same image.
– Christoph
Nov 21 at 21:10
Sorry, I guess it is only defined in a neighbourhood of some point on the torus.
– Breaking Mad
Nov 21 at 22:01
add a comment |
2
I don't see how this defines a map on the torus, $x=0$ and $x=2pi$ should give the same image.
– Christoph
Nov 21 at 21:10
Sorry, I guess it is only defined in a neighbourhood of some point on the torus.
– Breaking Mad
Nov 21 at 22:01
2
2
I don't see how this defines a map on the torus, $x=0$ and $x=2pi$ should give the same image.
– Christoph
Nov 21 at 21:10
I don't see how this defines a map on the torus, $x=0$ and $x=2pi$ should give the same image.
– Christoph
Nov 21 at 21:10
Sorry, I guess it is only defined in a neighbourhood of some point on the torus.
– Breaking Mad
Nov 21 at 22:01
Sorry, I guess it is only defined in a neighbourhood of some point on the torus.
– Breaking Mad
Nov 21 at 22:01
add a comment |
active
oldest
votes
active
oldest
votes
active
oldest
votes
active
oldest
votes
active
oldest
votes
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.
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%2f3008351%2fdifferential-of-a-map-mathbbt2-rightarrow-s2%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
2
I don't see how this defines a map on the torus, $x=0$ and $x=2pi$ should give the same image.
– Christoph
Nov 21 at 21:10
Sorry, I guess it is only defined in a neighbourhood of some point on the torus.
– Breaking Mad
Nov 21 at 22:01