Mixing
Cartesian product of two sets with smooth boundary
0
$begingroup$
Let $A,B subset mathbb{R}^n$ open and bounded sets with smooth boundary. Does the open and bounded set $A times B$ have (piecewise) smooth boundary? If so can one write the outer unit normal of $A times B$ as $(n_A,n_B)$?
differential-geometry
asked Jan 27 at 9:08
user33user33
13
$endgroup$
add a comment |
0
$begingroup$
Let $A,B subset mathbb{R}^n$ open and bounded sets with smooth boundary. Does the open and bounded set $A times B$ have (piecewise) smooth boundary? If so can one write the outer unit normal of $A times B$ as $(n_A,n_B)$?
differential-geometry
asked Jan 27 at 9:08
user33user33
13
$endgroup$
$begingroup$
So what happens when $A$ and $B$ are open intervals in $mathbb{R}$?
$endgroup$
– Amitai Yuval
Jan 27 at 9:50
$begingroup$
then it works since i only want $A times B$ to have a piecewise smooth boundary.
$endgroup$
– user33
Jan 27 at 11:50
$begingroup$
This depends on what you mean by "smooth" boundary. If you want it to be a smooth embedding of the boundary into $Bbb R^{2n}$, then no. Just consider $A = B = (0,1)$. The boundary of $(0,1)times(0,1)$ is well-known for having four points where it is not smoothly embedded.
$endgroup$
– Paul Sinclair
Jan 27 at 17:59
add a comment |
0
0
0
$begingroup$
Let $A,B subset mathbb{R}^n$ open and bounded sets with smooth boundary. Does the open and bounded set $A times B$ have (piecewise) smooth boundary? If so can one write the outer unit normal of $A times B$ as $(n_A,n_B)$?
differential-geometry
asked Jan 27 at 9:08
user33user33
13
$endgroup$
Let $A,B subset mathbb{R}^n$ open and bounded sets with smooth boundary. Does the open and bounded set $A times B$ have (piecewise) smooth boundary? If so can one write the outer unit normal of $A times B$ as $(n_A,n_B)$?
differential-geometry
differential-geometry
asked Jan 27 at 9:08
user33user33
13
asked Jan 27 at 9:08
user33user33
13
edited Jan 27 at 18:52
user33
asked Jan 27 at 9:08
user33user33
13
asked Jan 27 at 9:08
user33user33
13
asked Jan 27 at 9:08
user33user33
13
13
$begingroup$
So what happens when $A$ and $B$ are open intervals in $mathbb{R}$?
$endgroup$
– Amitai Yuval
Jan 27 at 9:50
$begingroup$
then it works since i only want $A times B$ to have a piecewise smooth boundary.
$endgroup$
– user33
Jan 27 at 11:50
$begingroup$
This depends on what you mean by "smooth" boundary. If you want it to be a smooth embedding of the boundary into $Bbb R^{2n}$, then no. Just consider $A = B = (0,1)$. The boundary of $(0,1)times(0,1)$ is well-known for having four points where it is not smoothly embedded.
$endgroup$
– Paul Sinclair
Jan 27 at 17:59
add a comment |
$begingroup$
So what happens when $A$ and $B$ are open intervals in $mathbb{R}$?
$endgroup$
– Amitai Yuval
Jan 27 at 9:50
$begingroup$
then it works since i only want $A times B$ to have a piecewise smooth boundary.
$endgroup$
– user33
Jan 27 at 11:50
$begingroup$
This depends on what you mean by "smooth" boundary. If you want it to be a smooth embedding of the boundary into $Bbb R^{2n}$, then no. Just consider $A = B = (0,1)$. The boundary of $(0,1)times(0,1)$ is well-known for having four points where it is not smoothly embedded.
$endgroup$
– Paul Sinclair
Jan 27 at 17:59
$begingroup$
So what happens when $A$ and $B$ are open intervals in $mathbb{R}$?
$endgroup$
– Amitai Yuval
Jan 27 at 9:50
$begingroup$
So what happens when $A$ and $B$ are open intervals in $mathbb{R}$?
$endgroup$
– Amitai Yuval
Jan 27 at 9:50
$begingroup$
then it works since i only want $A times B$ to have a piecewise smooth boundary.
$endgroup$
– user33
Jan 27 at 11:50
$begingroup$
then it works since i only want $A times B$ to have a piecewise smooth boundary.
$endgroup$
– user33
Jan 27 at 11:50
$begingroup$
This depends on what you mean by "smooth" boundary. If you want it to be a smooth embedding of the boundary into $Bbb R^{2n}$, then no. Just consider $A = B = (0,1)$. The boundary of $(0,1)times(0,1)$ is well-known for having four points where it is not smoothly embedded.
$endgroup$
– Paul Sinclair
Jan 27 at 17:59
$begingroup$
This depends on what you mean by "smooth" boundary. If you want it to be a smooth embedding of the boundary into $Bbb R^{2n}$, then no. Just consider $A = B = (0,1)$. The boundary of $(0,1)times(0,1)$ is well-known for having four points where it is not smoothly embedded.
$endgroup$
– Paul Sinclair
Jan 27 at 17:59
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
});
}
});
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%2f3089312%2fcartesian-product-of-two-sets-with-smooth-boundary%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
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.
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%2f3089312%2fcartesian-product-of-two-sets-with-smooth-boundary%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
$begingroup$
So what happens when $A$ and $B$ are open intervals in $mathbb{R}$?
$endgroup$
– Amitai Yuval
Jan 27 at 9:50
$begingroup$
then it works since i only want $A times B$ to have a piecewise smooth boundary.
$endgroup$
– user33
Jan 27 at 11:50
$begingroup$
This depends on what you mean by "smooth" boundary. If you want it to be a smooth embedding of the boundary into $Bbb R^{2n}$, then no. Just consider $A = B = (0,1)$. The boundary of $(0,1)times(0,1)$ is well-known for having four points where it is not smoothly embedded.
$endgroup$
– Paul Sinclair
Jan 27 at 17:59