Integral of a vector difference
Not sure how to deal with this problem:
$$dfrac {doverrightarrow {h}}{dt}=overrightarrow {r_1}times left( -dfrac {m}{left( overrightarrow {r_2}-overrightarrow {r_1}right) ^{2}}right)$$
where $r_2$ and $r_1$ are vectors. The parameter $t$ represents time. $h$ is the specific angular momentum. The vector direction of the right side term, I suppose, would be $(r_2-r_1)$.
This is as far as I get. But I’m not sure if I’m headed in the right direction:
$$dfrac {doverrightarrow {h}}{dt}=overrightarrow {r_1}times left( -dfrac {m}{left( overrightarrow {r_2}-overrightarrow {r_1}right) ^{3}}left( overrightarrow {r_2}-overrightarrow {r_1}right) right)$$
$$int dfrac {doverrightarrow {h}}{dt} dt=int -dfrac {m}{left( overrightarrow {r_2}-overrightarrow {r_1}right) ^{3}}r_2r_1sin theta dt$$
$$h=int dfrac {mr_2r_1sin theta }{left( left( r_{2}cos theta _{2}-r_{1}cos theta _{1}right) +left( r_{2}sin theta _{2}-r_{1}sin theta _{1}right) right) ^{frac {3}{2}}} dt$$
$$h=dfrac {mr_{2}r_{1}sin theta }{left( overrightarrow {r}_{2}-overrightarrow {r}_{1}right) ^{3}}t+C$$
calculus
asked Nov 29 '18 at 3:17
alexmarison
62
add a comment |
Not sure how to deal with this problem:
$$dfrac {doverrightarrow {h}}{dt}=overrightarrow {r_1}times left( -dfrac {m}{left( overrightarrow {r_2}-overrightarrow {r_1}right) ^{2}}right)$$
where $r_2$ and $r_1$ are vectors. The parameter $t$ represents time. $h$ is the specific angular momentum. The vector direction of the right side term, I suppose, would be $(r_2-r_1)$.
This is as far as I get. But I’m not sure if I’m headed in the right direction:
$$dfrac {doverrightarrow {h}}{dt}=overrightarrow {r_1}times left( -dfrac {m}{left( overrightarrow {r_2}-overrightarrow {r_1}right) ^{3}}left( overrightarrow {r_2}-overrightarrow {r_1}right) right)$$
$$int dfrac {doverrightarrow {h}}{dt} dt=int -dfrac {m}{left( overrightarrow {r_2}-overrightarrow {r_1}right) ^{3}}r_2r_1sin theta dt$$
$$h=int dfrac {mr_2r_1sin theta }{left( left( r_{2}cos theta _{2}-r_{1}cos theta _{1}right) +left( r_{2}sin theta _{2}-r_{1}sin theta _{1}right) right) ^{frac {3}{2}}} dt$$
$$h=dfrac {mr_{2}r_{1}sin theta }{left( overrightarrow {r}_{2}-overrightarrow {r}_{1}right) ^{3}}t+C$$
calculus
asked Nov 29 '18 at 3:17
alexmarison
62
add a comment |
Not sure how to deal with this problem:
$$dfrac {doverrightarrow {h}}{dt}=overrightarrow {r_1}times left( -dfrac {m}{left( overrightarrow {r_2}-overrightarrow {r_1}right) ^{2}}right)$$
where $r_2$ and $r_1$ are vectors. The parameter $t$ represents time. $h$ is the specific angular momentum. The vector direction of the right side term, I suppose, would be $(r_2-r_1)$.
This is as far as I get. But I’m not sure if I’m headed in the right direction:
$$dfrac {doverrightarrow {h}}{dt}=overrightarrow {r_1}times left( -dfrac {m}{left( overrightarrow {r_2}-overrightarrow {r_1}right) ^{3}}left( overrightarrow {r_2}-overrightarrow {r_1}right) right)$$
$$int dfrac {doverrightarrow {h}}{dt} dt=int -dfrac {m}{left( overrightarrow {r_2}-overrightarrow {r_1}right) ^{3}}r_2r_1sin theta dt$$
$$h=int dfrac {mr_2r_1sin theta }{left( left( r_{2}cos theta _{2}-r_{1}cos theta _{1}right) +left( r_{2}sin theta _{2}-r_{1}sin theta _{1}right) right) ^{frac {3}{2}}} dt$$
$$h=dfrac {mr_{2}r_{1}sin theta }{left( overrightarrow {r}_{2}-overrightarrow {r}_{1}right) ^{3}}t+C$$
calculus
asked Nov 29 '18 at 3:17
alexmarison
62
Not sure how to deal with this problem:
$$dfrac {doverrightarrow {h}}{dt}=overrightarrow {r_1}times left( -dfrac {m}{left( overrightarrow {r_2}-overrightarrow {r_1}right) ^{2}}right)$$
where $r_2$ and $r_1$ are vectors. The parameter $t$ represents time. $h$ is the specific angular momentum. The vector direction of the right side term, I suppose, would be $(r_2-r_1)$.
This is as far as I get. But I’m not sure if I’m headed in the right direction:
$$dfrac {doverrightarrow {h}}{dt}=overrightarrow {r_1}times left( -dfrac {m}{left( overrightarrow {r_2}-overrightarrow {r_1}right) ^{3}}left( overrightarrow {r_2}-overrightarrow {r_1}right) right)$$
$$int dfrac {doverrightarrow {h}}{dt} dt=int -dfrac {m}{left( overrightarrow {r_2}-overrightarrow {r_1}right) ^{3}}r_2r_1sin theta dt$$
$$h=int dfrac {mr_2r_1sin theta }{left( left( r_{2}cos theta _{2}-r_{1}cos theta _{1}right) +left( r_{2}sin theta _{2}-r_{1}sin theta _{1}right) right) ^{frac {3}{2}}} dt$$
$$h=dfrac {mr_{2}r_{1}sin theta }{left( overrightarrow {r}_{2}-overrightarrow {r}_{1}right) ^{3}}t+C$$
calculus
calculus
asked Nov 29 '18 at 3:17
alexmarison
62
asked Nov 29 '18 at 3:17
alexmarison
62
edited Nov 29 '18 at 23:38
asked Nov 29 '18 at 3:17
alexmarison
62
asked Nov 29 '18 at 3:17
alexmarison
62
asked Nov 29 '18 at 3:17
alexmarison
62
62
add a comment |
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%2f3018109%2fintegral-of-a-vector-difference%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%2f3018109%2fintegral-of-a-vector-difference%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
