Differential of inverse function to a tubular neighborhood
Suppose $S$ is a (regular) compact differentiable surface embedded in $mathbb{R}^3$ so that tubular neighborhoods exist. Consider the diffeomorphism to one of them:
$$F:Stimes(-epsilon,epsilon)rightarrow N_{epsilon}(S)$$
$$F(p,t)=p+tN_p$$
where $N$ is the Gauss map, that is invertible. I have to calculate the differential of its inverse, however I think I don't fully understand how am I supposed to express it in general. I know the inverse function $F^{-1}$ takes a point in the neighborhood to a point of the space that has for its two first coordinates the orthogonal projection onto the surface and for the third one the oriented distance from said surface. Every time I try to calculate the inverse I get stuck in some circular reasoning (for example I know how to express the distance in function of the projection and vice versa but I cannot explicitly write both of them). Any help is greatly appreciated.
differential-geometry surfaces inverse-function inverse-function-theorem
asked Jan 3 at 22:29
Renato Faraone
2,32911627
add a comment |
Suppose $S$ is a (regular) compact differentiable surface embedded in $mathbb{R}^3$ so that tubular neighborhoods exist. Consider the diffeomorphism to one of them:
$$F:Stimes(-epsilon,epsilon)rightarrow N_{epsilon}(S)$$
$$F(p,t)=p+tN_p$$
where $N$ is the Gauss map, that is invertible. I have to calculate the differential of its inverse, however I think I don't fully understand how am I supposed to express it in general. I know the inverse function $F^{-1}$ takes a point in the neighborhood to a point of the space that has for its two first coordinates the orthogonal projection onto the surface and for the third one the oriented distance from said surface. Every time I try to calculate the inverse I get stuck in some circular reasoning (for example I know how to express the distance in function of the projection and vice versa but I cannot explicitly write both of them). Any help is greatly appreciated.
differential-geometry surfaces inverse-function inverse-function-theorem
asked Jan 3 at 22:29
Renato Faraone
2,32911627
Differential of the inverse map is just the inverse of the differential of the original map. If you are able to compute $dF$, then it is immediate to compute $dF^{-1}$ just by inverting the differential pointwise.
– cjackal
Jan 4 at 0:40
@cjackal but the inverse of $dF$ in $p$ is the differential of $F^{-1}$ in $F(p)$ right? So for the differential of $F^{-1}$ at x I need to calculate explicitly $F^{-1}(x)$ or no?
– Renato Faraone
2 days ago
You can (with a little bit of differential geometry) compute the differential of $F$ at $(p,t)$ and so, inverting (abstractly), you'll have the differential of $F^{-1}$ at $x=F(p,t)$. You won't see it naturally in the Euclidean coordinates on $Bbb R^3$. You'll see it more naturally in terms of a basis consisting of the principal directions at $p$ and $N_p$.
– Ted Shifrin
2 days ago
add a comment |
Suppose $S$ is a (regular) compact differentiable surface embedded in $mathbb{R}^3$ so that tubular neighborhoods exist. Consider the diffeomorphism to one of them:
$$F:Stimes(-epsilon,epsilon)rightarrow N_{epsilon}(S)$$
$$F(p,t)=p+tN_p$$
where $N$ is the Gauss map, that is invertible. I have to calculate the differential of its inverse, however I think I don't fully understand how am I supposed to express it in general. I know the inverse function $F^{-1}$ takes a point in the neighborhood to a point of the space that has for its two first coordinates the orthogonal projection onto the surface and for the third one the oriented distance from said surface. Every time I try to calculate the inverse I get stuck in some circular reasoning (for example I know how to express the distance in function of the projection and vice versa but I cannot explicitly write both of them). Any help is greatly appreciated.
differential-geometry surfaces inverse-function inverse-function-theorem
asked Jan 3 at 22:29
Renato Faraone
2,32911627
Suppose $S$ is a (regular) compact differentiable surface embedded in $mathbb{R}^3$ so that tubular neighborhoods exist. Consider the diffeomorphism to one of them:
$$F:Stimes(-epsilon,epsilon)rightarrow N_{epsilon}(S)$$
$$F(p,t)=p+tN_p$$
where $N$ is the Gauss map, that is invertible. I have to calculate the differential of its inverse, however I think I don't fully understand how am I supposed to express it in general. I know the inverse function $F^{-1}$ takes a point in the neighborhood to a point of the space that has for its two first coordinates the orthogonal projection onto the surface and for the third one the oriented distance from said surface. Every time I try to calculate the inverse I get stuck in some circular reasoning (for example I know how to express the distance in function of the projection and vice versa but I cannot explicitly write both of them). Any help is greatly appreciated.
differential-geometry surfaces inverse-function inverse-function-theorem
differential-geometry surfaces inverse-function inverse-function-theorem
asked Jan 3 at 22:29
Renato Faraone
2,32911627
asked Jan 3 at 22:29
Renato Faraone
2,32911627
asked Jan 3 at 22:29
Renato Faraone
2,32911627
asked Jan 3 at 22:29
Renato Faraone
2,32911627
asked Jan 3 at 22:29
Renato Faraone
2,32911627
2,32911627
Differential of the inverse map is just the inverse of the differential of the original map. If you are able to compute $dF$, then it is immediate to compute $dF^{-1}$ just by inverting the differential pointwise.
– cjackal
Jan 4 at 0:40
@cjackal but the inverse of $dF$ in $p$ is the differential of $F^{-1}$ in $F(p)$ right? So for the differential of $F^{-1}$ at x I need to calculate explicitly $F^{-1}(x)$ or no?
– Renato Faraone
2 days ago
You can (with a little bit of differential geometry) compute the differential of $F$ at $(p,t)$ and so, inverting (abstractly), you'll have the differential of $F^{-1}$ at $x=F(p,t)$. You won't see it naturally in the Euclidean coordinates on $Bbb R^3$. You'll see it more naturally in terms of a basis consisting of the principal directions at $p$ and $N_p$.
– Ted Shifrin
2 days ago
add a comment |
Differential of the inverse map is just the inverse of the differential of the original map. If you are able to compute $dF$, then it is immediate to compute $dF^{-1}$ just by inverting the differential pointwise.
– cjackal
Jan 4 at 0:40
@cjackal but the inverse of $dF$ in $p$ is the differential of $F^{-1}$ in $F(p)$ right? So for the differential of $F^{-1}$ at x I need to calculate explicitly $F^{-1}(x)$ or no?
– Renato Faraone
2 days ago
You can (with a little bit of differential geometry) compute the differential of $F$ at $(p,t)$ and so, inverting (abstractly), you'll have the differential of $F^{-1}$ at $x=F(p,t)$. You won't see it naturally in the Euclidean coordinates on $Bbb R^3$. You'll see it more naturally in terms of a basis consisting of the principal directions at $p$ and $N_p$.
– Ted Shifrin
2 days ago
Differential of the inverse map is just the inverse of the differential of the original map. If you are able to compute $dF$, then it is immediate to compute $dF^{-1}$ just by inverting the differential pointwise.
– cjackal
Jan 4 at 0:40
Differential of the inverse map is just the inverse of the differential of the original map. If you are able to compute $dF$, then it is immediate to compute $dF^{-1}$ just by inverting the differential pointwise.
– cjackal
Jan 4 at 0:40
@cjackal but the inverse of $dF$ in $p$ is the differential of $F^{-1}$ in $F(p)$ right? So for the differential of $F^{-1}$ at x I need to calculate explicitly $F^{-1}(x)$ or no?
– Renato Faraone
2 days ago
@cjackal but the inverse of $dF$ in $p$ is the differential of $F^{-1}$ in $F(p)$ right? So for the differential of $F^{-1}$ at x I need to calculate explicitly $F^{-1}(x)$ or no?
– Renato Faraone
2 days ago
You can (with a little bit of differential geometry) compute the differential of $F$ at $(p,t)$ and so, inverting (abstractly), you'll have the differential of $F^{-1}$ at $x=F(p,t)$. You won't see it naturally in the Euclidean coordinates on $Bbb R^3$. You'll see it more naturally in terms of a basis consisting of the principal directions at $p$ and $N_p$.
– Ted Shifrin
2 days ago
You can (with a little bit of differential geometry) compute the differential of $F$ at $(p,t)$ and so, inverting (abstractly), you'll have the differential of $F^{-1}$ at $x=F(p,t)$. You won't see it naturally in the Euclidean coordinates on $Bbb R^3$. You'll see it more naturally in terms of a basis consisting of the principal directions at $p$ and $N_p$.
– Ted Shifrin
2 days ago
add a comment |
0
active
oldest
votes
Your Answer
StackExchange.ifUsing("editor", function () {
return StackExchange.using("mathjaxEditing", function () {
StackExchange.MarkdownEditor.creationCallbacks.add(function (editor, postfix) {
StackExchange.mathjaxEditing.prepareWmdForMathJax(editor, postfix, [["$", "$"], ["\\(","\\)"]]);
});
});
}, "mathjax-editing");
StackExchange.ready(function() {
var channelOptions = {
tags: "".split(" "),
id: "69"
};
initTagRenderer("".split(" "), "".split(" "), channelOptions);
StackExchange.using("externalEditor", function() {
// Have to fire editor after snippets, if snippets enabled
if (StackExchange.settings.snippets.snippetsEnabled) {
StackExchange.using("snippets", function() {
createEditor();
});
}
else {
createEditor();
}
});
function createEditor() {
StackExchange.prepareEditor({
heartbeatType: 'answer',
autoActivateHeartbeat: false,
convertImagesToLinks: true,
noModals: true,
showLowRepImageUploadWarning: true,
reputationToPostImages: 10,
bindNavPrevention: true,
postfix: "",
imageUploader: {
brandingHtml: "Powered by u003ca class="icon-imgur-white" href="https://imgur.com/"u003eu003c/au003e",
contentPolicyHtml: "User contributions licensed under u003ca href="https://creativecommons.org/licenses/by-sa/3.0/"u003ecc by-sa 3.0 with attribution requiredu003c/au003e u003ca href="https://stackoverflow.com/legal/content-policy"u003e(content policy)u003c/au003e",
allowUrls: true
},
noCode: true, onDemand: true,
discardSelector: ".discard-answer"
,immediatelyShowMarkdownHelp:true
});
}
});
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%2f3061089%2fdifferential-of-inverse-function-to-a-tubular-neighborhood%23new-answer', 'question_page');
}
);
Post as a guest
Required, but never shown
0
active
oldest
votes
0
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%2f3061089%2fdifferential-of-inverse-function-to-a-tubular-neighborhood%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
Differential of the inverse map is just the inverse of the differential of the original map. If you are able to compute $dF$, then it is immediate to compute $dF^{-1}$ just by inverting the differential pointwise.
– cjackal
Jan 4 at 0:40
@cjackal but the inverse of $dF$ in $p$ is the differential of $F^{-1}$ in $F(p)$ right? So for the differential of $F^{-1}$ at x I need to calculate explicitly $F^{-1}(x)$ or no?
– Renato Faraone
2 days ago
You can (with a little bit of differential geometry) compute the differential of $F$ at $(p,t)$ and so, inverting (abstractly), you'll have the differential of $F^{-1}$ at $x=F(p,t)$. You won't see it naturally in the Euclidean coordinates on $Bbb R^3$. You'll see it more naturally in terms of a basis consisting of the principal directions at $p$ and $N_p$.
– Ted Shifrin
2 days ago