Calculating the cross product of a cross product
up vote
1
down vote
favorite
so I really can't see what I am doing wrong. I want to use this formula:
$atimes (btimes c) = b(acdot c) - c(acdot b)$
Calculate the rotation of $v(x,y,z)=(x,y,z)^T times omega$ with $omega in mathbb R^3$
Solution:
$acdot c=nablacdot omega=0$
and
$acdot b = nabla cdot (x,y,z)^T= partial_x x + partial_y y + partial_z z = 3$
so we get $-omega 3=-3omega$
The actual solution (which I do get by direct calculation) is: $-2omega$
cross-product curl
add a comment |
up vote
1
down vote
favorite
so I really can't see what I am doing wrong. I want to use this formula:
$atimes (btimes c) = b(acdot c) - c(acdot b)$
Calculate the rotation of $v(x,y,z)=(x,y,z)^T times omega$ with $omega in mathbb R^3$
Solution:
$acdot c=nablacdot omega=0$
and
$acdot b = nabla cdot (x,y,z)^T= partial_x x + partial_y y + partial_z z = 3$
so we get $-omega 3=-3omega$
The actual solution (which I do get by direct calculation) is: $-2omega$
cross-product curl
1
You can't just replace $a$ with $nabla$ like that, you also need to keep track of what is integrated. See here: math.stackexchange.com/questions/836041/…
– Hans Lundmark
Oct 23 at 18:32
add a comment |
up vote
1
down vote
favorite
up vote
1
down vote
favorite
so I really can't see what I am doing wrong. I want to use this formula:
$atimes (btimes c) = b(acdot c) - c(acdot b)$
Calculate the rotation of $v(x,y,z)=(x,y,z)^T times omega$ with $omega in mathbb R^3$
Solution:
$acdot c=nablacdot omega=0$
and
$acdot b = nabla cdot (x,y,z)^T= partial_x x + partial_y y + partial_z z = 3$
so we get $-omega 3=-3omega$
The actual solution (which I do get by direct calculation) is: $-2omega$
cross-product curl
so I really can't see what I am doing wrong. I want to use this formula:
$atimes (btimes c) = b(acdot c) - c(acdot b)$
Calculate the rotation of $v(x,y,z)=(x,y,z)^T times omega$ with $omega in mathbb R^3$
Solution:
$acdot c=nablacdot omega=0$
and
$acdot b = nabla cdot (x,y,z)^T= partial_x x + partial_y y + partial_z z = 3$
so we get $-omega 3=-3omega$
The actual solution (which I do get by direct calculation) is: $-2omega$
cross-product curl
cross-product curl
asked Oct 23 at 18:16
xotix
36829
36829
1
You can't just replace $a$ with $nabla$ like that, you also need to keep track of what is integrated. See here: math.stackexchange.com/questions/836041/…
– Hans Lundmark
Oct 23 at 18:32
add a comment |
1
You can't just replace $a$ with $nabla$ like that, you also need to keep track of what is integrated. See here: math.stackexchange.com/questions/836041/…
– Hans Lundmark
Oct 23 at 18:32
1
1
You can't just replace $a$ with $nabla$ like that, you also need to keep track of what is integrated. See here: math.stackexchange.com/questions/836041/…
– Hans Lundmark
Oct 23 at 18:32
You can't just replace $a$ with $nabla$ like that, you also need to keep track of what is integrated. See here: math.stackexchange.com/questions/836041/…
– Hans Lundmark
Oct 23 at 18:32
add a comment |
2 Answers
2
active
oldest
votes
up vote
1
down vote
accepted
If you want to generalise formulae with vectors having commuting components, make sure to write the usual formula so as to preserve the order in products. For example, here the $i$th component of the RHS is $sum_j a_j (b_i c_j-b_j c_i)$, once we impose the $abc$ order of the LHS. Now you know the generalisation off the top of your head. For $a_i=partial_i$, the result's $i$th component is $$sum_j partial_j (b_i c_j-b_j c_i)=b_inablacdot c+ccdotnabla b_i-(nablacdot b) c-bcdotnabla c_i.$$The vector, in other words, is $$vec{b}(nablacdotvec{c})-(nablacdotvec{b})vec{c}+(vec{c}cdotnabla)vec{b}-(vec{b}cdotnabla)vec{c}.$$(I've swapped the middle terms to mirror @md2perpe's quoted result, but the letters $b,,c$ still need to be changed to $A,,B$).
add a comment |
up vote
1
down vote
On Wikipedia you can see that the formula for the curl of a cross product is given by
$$
nabla times (mathbf{A} times mathbf{B}) = mathbf{A} (nabla cdot mathbf{B}) - mathbf{B} (nabla cdot mathbf{A}) + (mathbf{B} cdot nabla) mathbf{A} - (mathbf{A} cdot nabla) mathbf{B}
.
$$
Applying this on your case gives
$$begin{align}
nabla times (mathbf{x} times mathbf{omega})
&= mathbf{x} (nabla cdot mathbf{omega}) - mathbf{omega} (nabla cdot mathbf{x}) + (mathbf{omega} cdot nabla) mathbf{x} - (mathbf{x} cdot nabla) mathbf{omega} \
&= mathbf{x} 0 - mathbf{omega} 3 + mathbf{omega} - mathbf{0}
= -2 mathbf{omega}
.
end{align}$$
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',
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%2f2968008%2fcalculating-the-cross-product-of-a-cross-product%23new-answer', 'question_page');
}
);
Post as a guest
Required, but never shown
2 Answers
2
active
oldest
votes
2 Answers
2
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
1
down vote
accepted
If you want to generalise formulae with vectors having commuting components, make sure to write the usual formula so as to preserve the order in products. For example, here the $i$th component of the RHS is $sum_j a_j (b_i c_j-b_j c_i)$, once we impose the $abc$ order of the LHS. Now you know the generalisation off the top of your head. For $a_i=partial_i$, the result's $i$th component is $$sum_j partial_j (b_i c_j-b_j c_i)=b_inablacdot c+ccdotnabla b_i-(nablacdot b) c-bcdotnabla c_i.$$The vector, in other words, is $$vec{b}(nablacdotvec{c})-(nablacdotvec{b})vec{c}+(vec{c}cdotnabla)vec{b}-(vec{b}cdotnabla)vec{c}.$$(I've swapped the middle terms to mirror @md2perpe's quoted result, but the letters $b,,c$ still need to be changed to $A,,B$).
add a comment |
up vote
1
down vote
accepted
If you want to generalise formulae with vectors having commuting components, make sure to write the usual formula so as to preserve the order in products. For example, here the $i$th component of the RHS is $sum_j a_j (b_i c_j-b_j c_i)$, once we impose the $abc$ order of the LHS. Now you know the generalisation off the top of your head. For $a_i=partial_i$, the result's $i$th component is $$sum_j partial_j (b_i c_j-b_j c_i)=b_inablacdot c+ccdotnabla b_i-(nablacdot b) c-bcdotnabla c_i.$$The vector, in other words, is $$vec{b}(nablacdotvec{c})-(nablacdotvec{b})vec{c}+(vec{c}cdotnabla)vec{b}-(vec{b}cdotnabla)vec{c}.$$(I've swapped the middle terms to mirror @md2perpe's quoted result, but the letters $b,,c$ still need to be changed to $A,,B$).
add a comment |
up vote
1
down vote
accepted
up vote
1
down vote
accepted
If you want to generalise formulae with vectors having commuting components, make sure to write the usual formula so as to preserve the order in products. For example, here the $i$th component of the RHS is $sum_j a_j (b_i c_j-b_j c_i)$, once we impose the $abc$ order of the LHS. Now you know the generalisation off the top of your head. For $a_i=partial_i$, the result's $i$th component is $$sum_j partial_j (b_i c_j-b_j c_i)=b_inablacdot c+ccdotnabla b_i-(nablacdot b) c-bcdotnabla c_i.$$The vector, in other words, is $$vec{b}(nablacdotvec{c})-(nablacdotvec{b})vec{c}+(vec{c}cdotnabla)vec{b}-(vec{b}cdotnabla)vec{c}.$$(I've swapped the middle terms to mirror @md2perpe's quoted result, but the letters $b,,c$ still need to be changed to $A,,B$).
If you want to generalise formulae with vectors having commuting components, make sure to write the usual formula so as to preserve the order in products. For example, here the $i$th component of the RHS is $sum_j a_j (b_i c_j-b_j c_i)$, once we impose the $abc$ order of the LHS. Now you know the generalisation off the top of your head. For $a_i=partial_i$, the result's $i$th component is $$sum_j partial_j (b_i c_j-b_j c_i)=b_inablacdot c+ccdotnabla b_i-(nablacdot b) c-bcdotnabla c_i.$$The vector, in other words, is $$vec{b}(nablacdotvec{c})-(nablacdotvec{b})vec{c}+(vec{c}cdotnabla)vec{b}-(vec{b}cdotnabla)vec{c}.$$(I've swapped the middle terms to mirror @md2perpe's quoted result, but the letters $b,,c$ still need to be changed to $A,,B$).
edited Nov 22 at 19:40
answered Oct 23 at 20:23
J.G.
21.1k21933
21.1k21933
add a comment |
add a comment |
up vote
1
down vote
On Wikipedia you can see that the formula for the curl of a cross product is given by
$$
nabla times (mathbf{A} times mathbf{B}) = mathbf{A} (nabla cdot mathbf{B}) - mathbf{B} (nabla cdot mathbf{A}) + (mathbf{B} cdot nabla) mathbf{A} - (mathbf{A} cdot nabla) mathbf{B}
.
$$
Applying this on your case gives
$$begin{align}
nabla times (mathbf{x} times mathbf{omega})
&= mathbf{x} (nabla cdot mathbf{omega}) - mathbf{omega} (nabla cdot mathbf{x}) + (mathbf{omega} cdot nabla) mathbf{x} - (mathbf{x} cdot nabla) mathbf{omega} \
&= mathbf{x} 0 - mathbf{omega} 3 + mathbf{omega} - mathbf{0}
= -2 mathbf{omega}
.
end{align}$$
add a comment |
up vote
1
down vote
On Wikipedia you can see that the formula for the curl of a cross product is given by
$$
nabla times (mathbf{A} times mathbf{B}) = mathbf{A} (nabla cdot mathbf{B}) - mathbf{B} (nabla cdot mathbf{A}) + (mathbf{B} cdot nabla) mathbf{A} - (mathbf{A} cdot nabla) mathbf{B}
.
$$
Applying this on your case gives
$$begin{align}
nabla times (mathbf{x} times mathbf{omega})
&= mathbf{x} (nabla cdot mathbf{omega}) - mathbf{omega} (nabla cdot mathbf{x}) + (mathbf{omega} cdot nabla) mathbf{x} - (mathbf{x} cdot nabla) mathbf{omega} \
&= mathbf{x} 0 - mathbf{omega} 3 + mathbf{omega} - mathbf{0}
= -2 mathbf{omega}
.
end{align}$$
add a comment |
up vote
1
down vote
up vote
1
down vote
On Wikipedia you can see that the formula for the curl of a cross product is given by
$$
nabla times (mathbf{A} times mathbf{B}) = mathbf{A} (nabla cdot mathbf{B}) - mathbf{B} (nabla cdot mathbf{A}) + (mathbf{B} cdot nabla) mathbf{A} - (mathbf{A} cdot nabla) mathbf{B}
.
$$
Applying this on your case gives
$$begin{align}
nabla times (mathbf{x} times mathbf{omega})
&= mathbf{x} (nabla cdot mathbf{omega}) - mathbf{omega} (nabla cdot mathbf{x}) + (mathbf{omega} cdot nabla) mathbf{x} - (mathbf{x} cdot nabla) mathbf{omega} \
&= mathbf{x} 0 - mathbf{omega} 3 + mathbf{omega} - mathbf{0}
= -2 mathbf{omega}
.
end{align}$$
On Wikipedia you can see that the formula for the curl of a cross product is given by
$$
nabla times (mathbf{A} times mathbf{B}) = mathbf{A} (nabla cdot mathbf{B}) - mathbf{B} (nabla cdot mathbf{A}) + (mathbf{B} cdot nabla) mathbf{A} - (mathbf{A} cdot nabla) mathbf{B}
.
$$
Applying this on your case gives
$$begin{align}
nabla times (mathbf{x} times mathbf{omega})
&= mathbf{x} (nabla cdot mathbf{omega}) - mathbf{omega} (nabla cdot mathbf{x}) + (mathbf{omega} cdot nabla) mathbf{x} - (mathbf{x} cdot nabla) mathbf{omega} \
&= mathbf{x} 0 - mathbf{omega} 3 + mathbf{omega} - mathbf{0}
= -2 mathbf{omega}
.
end{align}$$
answered Oct 23 at 20:07
md2perpe
7,31311027
7,31311027
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.
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%2f2968008%2fcalculating-the-cross-product-of-a-cross-product%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
1
You can't just replace $a$ with $nabla$ like that, you also need to keep track of what is integrated. See here: math.stackexchange.com/questions/836041/…
– Hans Lundmark
Oct 23 at 18:32