Mixing
If the gradient of a vector is zero, does that imply that the laplacian of the vector is a null vector?
$begingroup$
Suposse $nabla cdot vec{u} = 0$
Does that imply that $Delta vec{u} = vec{0}$
Thank you!
vector-analysis laplacian divergence
asked Jan 9 at 12:40
L CL C
1
$endgroup$
add a comment |
$begingroup$
Suposse $nabla cdot vec{u} = 0$
Does that imply that $Delta vec{u} = vec{0}$
Thank you!
vector-analysis laplacian divergence
asked Jan 9 at 12:40
L CL C
1
$endgroup$
$begingroup$
Careful, the gradient operator only applies to scalar fields. It looks like you have specified that the divergence of the vector field $vec{u}$ (judging by the arrow and the dot after the $nabla$) is zero.
$endgroup$
– E-mu
Jan 9 at 13:00
add a comment |
$begingroup$
Suposse $nabla cdot vec{u} = 0$
Does that imply that $Delta vec{u} = vec{0}$
Thank you!
vector-analysis laplacian divergence
asked Jan 9 at 12:40
L CL C
1
$endgroup$
Suposse $nabla cdot vec{u} = 0$
Does that imply that $Delta vec{u} = vec{0}$
Thank you!
vector-analysis laplacian divergence
vector-analysis laplacian divergence
asked Jan 9 at 12:40
L CL C
1
asked Jan 9 at 12:40
L CL C
1
asked Jan 9 at 12:40
L CL C
1
asked Jan 9 at 12:40
L CL C
1
asked Jan 9 at 12:40
L CL C
1
1
$begingroup$
Careful, the gradient operator only applies to scalar fields. It looks like you have specified that the divergence of the vector field $vec{u}$ (judging by the arrow and the dot after the $nabla$) is zero.
$endgroup$
– E-mu
Jan 9 at 13:00
add a comment |
$begingroup$
Careful, the gradient operator only applies to scalar fields. It looks like you have specified that the divergence of the vector field $vec{u}$ (judging by the arrow and the dot after the $nabla$) is zero.
$endgroup$
– E-mu
Jan 9 at 13:00
$begingroup$
Careful, the gradient operator only applies to scalar fields. It looks like you have specified that the divergence of the vector field $vec{u}$ (judging by the arrow and the dot after the $nabla$) is zero.
$endgroup$
– E-mu
Jan 9 at 13:00
$begingroup$
Careful, the gradient operator only applies to scalar fields. It looks like you have specified that the divergence of the vector field $vec{u}$ (judging by the arrow and the dot after the $nabla$) is zero.
$endgroup$
– E-mu
Jan 9 at 13:00
add a comment |
1 Answer
1
active
oldest
votes
$begingroup$
To show that it is not true (note that the Laplacian has to be taken component wise for this question to even make sense), take a look at the vector field
$$vec{u}(x,y,z)= begin{pmatrix} y^2\0\0 end{pmatrix}$$
and show that the divergence vanishes but the Laplacian does not.
answered Jan 9 at 12:57
FabianFabian
19.7k3674
$endgroup$
$begingroup$
Thank you for the example!
$endgroup$
– L C
Jan 9 at 13:08
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%2f3067406%2fif-the-gradient-of-a-vector-is-zero-does-that-imply-that-the-laplacian-of-the-v%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
$begingroup$
To show that it is not true (note that the Laplacian has to be taken component wise for this question to even make sense), take a look at the vector field
$$vec{u}(x,y,z)= begin{pmatrix} y^2\0\0 end{pmatrix}$$
and show that the divergence vanishes but the Laplacian does not.
answered Jan 9 at 12:57
FabianFabian
19.7k3674
$endgroup$
$begingroup$
Thank you for the example!
$endgroup$
– L C
Jan 9 at 13:08
add a comment |
$begingroup$
To show that it is not true (note that the Laplacian has to be taken component wise for this question to even make sense), take a look at the vector field
$$vec{u}(x,y,z)= begin{pmatrix} y^2\0\0 end{pmatrix}$$
and show that the divergence vanishes but the Laplacian does not.
answered Jan 9 at 12:57
FabianFabian
19.7k3674
$endgroup$
$begingroup$
Thank you for the example!
$endgroup$
– L C
Jan 9 at 13:08
add a comment |
$begingroup$
To show that it is not true (note that the Laplacian has to be taken component wise for this question to even make sense), take a look at the vector field
$$vec{u}(x,y,z)= begin{pmatrix} y^2\0\0 end{pmatrix}$$
and show that the divergence vanishes but the Laplacian does not.
answered Jan 9 at 12:57
FabianFabian
19.7k3674
$endgroup$
To show that it is not true (note that the Laplacian has to be taken component wise for this question to even make sense), take a look at the vector field
$$vec{u}(x,y,z)= begin{pmatrix} y^2\0\0 end{pmatrix}$$
and show that the divergence vanishes but the Laplacian does not.
answered Jan 9 at 12:57
FabianFabian
19.7k3674
answered Jan 9 at 12:57
FabianFabian
19.7k3674
answered Jan 9 at 12:57
FabianFabian
19.7k3674
answered Jan 9 at 12:57
FabianFabian
19.7k3674
19.7k3674
$begingroup$
Thank you for the example!
$endgroup$
– L C
Jan 9 at 13:08
add a comment |
$begingroup$
Thank you for the example!
$endgroup$
– L C
Jan 9 at 13:08
$begingroup$
Thank you for the example!
$endgroup$
– L C
Jan 9 at 13:08
$begingroup$
Thank you for the example!
$endgroup$
– L C
Jan 9 at 13:08
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%2f3067406%2fif-the-gradient-of-a-vector-is-zero-does-that-imply-that-the-laplacian-of-the-v%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
$begingroup$
Careful, the gradient operator only applies to scalar fields. It looks like you have specified that the divergence of the vector field $vec{u}$ (judging by the arrow and the dot after the $nabla$) is zero.
$endgroup$
– E-mu
Jan 9 at 13:00