surface integrals — is it “dx” or “ds”?
$begingroup$
I was reading this note about Surface Integrals and came across this paragraph:
Let S be a surface parameterized by $mathbf X : D → mathbf R^3$.
A point $(s_0, t_0) in D$, is mapped to $mathbf X(s_0, t_0) inmathbb R^3$.
An infinitesimal $dx × dt$ parallelogram at $(s_0, t_0) in D$ has area $dx dt$. It’s mapped to an infinitesimal $mathbf T_s(s_0, t_0)ds × mathbf T_t(s_0, t_0)$ rectangle with area $||mathbf T_s × mathbf T_t|| ds dt$, which equals $|mathbf N| ds dt$.
We’ll call this infinitesimal parallelogram the surface area differential, denoted $dS$.
I'm quite new to the topic, and I kind of got confused here. When the note says "An infinitesimal $dx times dt$ parallelogram... has area $dx dt$", are the $dx$'s supposed to be $ds$'s? Otherwise, if $dx$ is right, could someone help me understand why we use $dx$?
calculus integration multivariable-calculus surface-integrals
edited Dec 15 '18 at 17:00
amWhy
1
asked Dec 15 '18 at 15:12
jjhhjjhh
2,13911122
$endgroup$
add a comment |
$begingroup$
I was reading this note about Surface Integrals and came across this paragraph:
Let S be a surface parameterized by $mathbf X : D → mathbf R^3$.
A point $(s_0, t_0) in D$, is mapped to $mathbf X(s_0, t_0) inmathbb R^3$.
An infinitesimal $dx × dt$ parallelogram at $(s_0, t_0) in D$ has area $dx dt$. It’s mapped to an infinitesimal $mathbf T_s(s_0, t_0)ds × mathbf T_t(s_0, t_0)$ rectangle with area $||mathbf T_s × mathbf T_t|| ds dt$, which equals $|mathbf N| ds dt$.
We’ll call this infinitesimal parallelogram the surface area differential, denoted $dS$.
I'm quite new to the topic, and I kind of got confused here. When the note says "An infinitesimal $dx times dt$ parallelogram... has area $dx dt$", are the $dx$'s supposed to be $ds$'s? Otherwise, if $dx$ is right, could someone help me understand why we use $dx$?
calculus integration multivariable-calculus surface-integrals
edited Dec 15 '18 at 17:00
amWhy
1
asked Dec 15 '18 at 15:12
jjhhjjhh
2,13911122
$endgroup$
add a comment |
$begingroup$
I was reading this note about Surface Integrals and came across this paragraph:
Let S be a surface parameterized by $mathbf X : D → mathbf R^3$.
A point $(s_0, t_0) in D$, is mapped to $mathbf X(s_0, t_0) inmathbb R^3$.
An infinitesimal $dx × dt$ parallelogram at $(s_0, t_0) in D$ has area $dx dt$. It’s mapped to an infinitesimal $mathbf T_s(s_0, t_0)ds × mathbf T_t(s_0, t_0)$ rectangle with area $||mathbf T_s × mathbf T_t|| ds dt$, which equals $|mathbf N| ds dt$.
We’ll call this infinitesimal parallelogram the surface area differential, denoted $dS$.
I'm quite new to the topic, and I kind of got confused here. When the note says "An infinitesimal $dx times dt$ parallelogram... has area $dx dt$", are the $dx$'s supposed to be $ds$'s? Otherwise, if $dx$ is right, could someone help me understand why we use $dx$?
calculus integration multivariable-calculus surface-integrals
edited Dec 15 '18 at 17:00
amWhy
1
asked Dec 15 '18 at 15:12
jjhhjjhh
2,13911122
$endgroup$
I was reading this note about Surface Integrals and came across this paragraph:
Let S be a surface parameterized by $mathbf X : D → mathbf R^3$.
A point $(s_0, t_0) in D$, is mapped to $mathbf X(s_0, t_0) inmathbb R^3$.
An infinitesimal $dx × dt$ parallelogram at $(s_0, t_0) in D$ has area $dx dt$. It’s mapped to an infinitesimal $mathbf T_s(s_0, t_0)ds × mathbf T_t(s_0, t_0)$ rectangle with area $||mathbf T_s × mathbf T_t|| ds dt$, which equals $|mathbf N| ds dt$.
We’ll call this infinitesimal parallelogram the surface area differential, denoted $dS$.
I'm quite new to the topic, and I kind of got confused here. When the note says "An infinitesimal $dx times dt$ parallelogram... has area $dx dt$", are the $dx$'s supposed to be $ds$'s? Otherwise, if $dx$ is right, could someone help me understand why we use $dx$?
calculus integration multivariable-calculus surface-integrals
calculus integration multivariable-calculus surface-integrals
edited Dec 15 '18 at 17:00
amWhy
1
asked Dec 15 '18 at 15:12
jjhhjjhh
2,13911122
edited Dec 15 '18 at 17:00
amWhy
1
asked Dec 15 '18 at 15:12
jjhhjjhh
2,13911122
edited Dec 15 '18 at 17:00
amWhy
1
edited Dec 15 '18 at 17:00
amWhy
1
edited Dec 15 '18 at 17:00
amWhy
1
1
asked Dec 15 '18 at 15:12
jjhhjjhh
2,13911122
asked Dec 15 '18 at 15:12
jjhhjjhh
2,13911122
asked Dec 15 '18 at 15:12
jjhhjjhh
2,13911122
2,13911122
add a comment |
add a comment |
1 Answer
1
active
oldest
votes
$begingroup$
I believe $dx$ should be $ds$. I believe also $T_s(s_0, t_0)ds times T_t(s_0, t_0)$ should be $T_s(s_0, t_0)ds times T_t(s_0, t_0)dt$.
answered Dec 15 '18 at 15:28
Lorenzo B.Lorenzo B.
1,8402520
$endgroup$
add a comment |
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%2f3041588%2fsurface-integrals-is-it-dx-or-ds%23new-answer', 'question_page');
}
);
Post as a guest
Required, but never shown
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
$begingroup$
I believe $dx$ should be $ds$. I believe also $T_s(s_0, t_0)ds times T_t(s_0, t_0)$ should be $T_s(s_0, t_0)ds times T_t(s_0, t_0)dt$.
answered Dec 15 '18 at 15:28
Lorenzo B.Lorenzo B.
1,8402520
$endgroup$
add a comment |
$begingroup$
I believe $dx$ should be $ds$. I believe also $T_s(s_0, t_0)ds times T_t(s_0, t_0)$ should be $T_s(s_0, t_0)ds times T_t(s_0, t_0)dt$.
answered Dec 15 '18 at 15:28
Lorenzo B.Lorenzo B.
1,8402520
$endgroup$
add a comment |
$begingroup$
I believe $dx$ should be $ds$. I believe also $T_s(s_0, t_0)ds times T_t(s_0, t_0)$ should be $T_s(s_0, t_0)ds times T_t(s_0, t_0)dt$.
answered Dec 15 '18 at 15:28
Lorenzo B.Lorenzo B.
1,8402520
$endgroup$
I believe $dx$ should be $ds$. I believe also $T_s(s_0, t_0)ds times T_t(s_0, t_0)$ should be $T_s(s_0, t_0)ds times T_t(s_0, t_0)dt$.
answered Dec 15 '18 at 15:28
Lorenzo B.Lorenzo B.
1,8402520
answered Dec 15 '18 at 15:28
Lorenzo B.Lorenzo B.
1,8402520
answered Dec 15 '18 at 15:28
Lorenzo B.Lorenzo B.
1,8402520
answered Dec 15 '18 at 15:28
Lorenzo B.Lorenzo B.
1,8402520
1,8402520
add a comment |
add a comment |
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.
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%2f3041588%2fsurface-integrals-is-it-dx-or-ds%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
