The gradient of a scalar functiondotting gradient in spherical coordinates with displacement vectorDivergence in spherical coordinates$operatornamediv$ and $operatornamegrad$ in spherical coordinates. Formula from general relativity goes crazyWhy the gradient of the r vector is the identity map, geometrically speaking?Finding the Gradient of a Tensor FieldFinding the Gradient of a Vector Function by its ComponentsThe gradient on sphereDivergence in spherical coordinates starting from contractionTime derivative of scalar function that takes vector as argument
System.debug(JSON.Serialize(o)) Not longer shows full string
Sort a list by elements of another list
Unreliable Magic - Is it worth it?
Is HostGator storing my password in plaintext?
How do I find the solutions of the following equation?
Is expanding the research of a group into machine learning as a PhD student risky?
What does "I’d sit this one out, Cap," imply or mean in the context?
How to check is there any negative term in a large list?
Integer addition + constant, is it a group?
Why didn't Theresa May consult with Parliament before negotiating a deal with the EU?
How long to clear the 'suck zone' of a turbofan after start is initiated?
Why not increase contact surface when reentering the atmosphere?
Term for the "extreme-extension" version of a straw man fallacy?
Anatomically Correct Strange Women In Ponds Distributing Swords
Increase performance creating Mandelbrot set in python
How did Arya survive the stabbing?
How can a function with a hole (removable discontinuity) equal a function with no hole?
Risk of infection at the gym?
How easy is it to start Magic from scratch?
Where does the Z80 processor start executing from?
How do we know the LHC results are robust?
How to be diplomatic in refusing to write code that breaches the privacy of our users
What happens if you roll doubles 3 times then land on "Go to jail?"
Method to test if a number is a perfect power?
The gradient of a scalar function
dotting gradient in spherical coordinates with displacement vectorDivergence in spherical coordinates$operatornamediv$ and $operatornamegrad$ in spherical coordinates. Formula from general relativity goes crazyWhy the gradient of the r vector is the identity map, geometrically speaking?Finding the Gradient of a Tensor FieldFinding the Gradient of a Vector Function by its ComponentsThe gradient on sphereDivergence in spherical coordinates starting from contractionTime derivative of scalar function that takes vector as argument
$begingroup$
I found this definition of gradient of scalar function $Phi$:
$nabla Phi = (g^ijpartial_j) vecg_i$
And I know:
Metric tensor of spherical coordinates
$g_11 = 1$
$g_22 = r^2$
$g_33 = r^2sintheta$
But I do not know how to use first definition. How can I find gradient of scalar function in spherical coordinates by using metric tensor
tensors spherical-coordinates gradient-flows
asked Mar 17 at 21:11
Ali OzAli Oz
83
$endgroup$
add a comment |
$begingroup$
I found this definition of gradient of scalar function $Phi$:
$nabla Phi = (g^ijpartial_j) vecg_i$
And I know:
Metric tensor of spherical coordinates
$g_11 = 1$
$g_22 = r^2$
$g_33 = r^2sintheta$
But I do not know how to use first definition. How can I find gradient of scalar function in spherical coordinates by using metric tensor
tensors spherical-coordinates gradient-flows
asked Mar 17 at 21:11
Ali OzAli Oz
83
$endgroup$
add a comment |
$begingroup$
I found this definition of gradient of scalar function $Phi$:
$nabla Phi = (g^ijpartial_j) vecg_i$
And I know:
Metric tensor of spherical coordinates
$g_11 = 1$
$g_22 = r^2$
$g_33 = r^2sintheta$
But I do not know how to use first definition. How can I find gradient of scalar function in spherical coordinates by using metric tensor
tensors spherical-coordinates gradient-flows
asked Mar 17 at 21:11
Ali OzAli Oz
83
$endgroup$
I found this definition of gradient of scalar function $Phi$:
$nabla Phi = (g^ijpartial_j) vecg_i$
And I know:
Metric tensor of spherical coordinates
$g_11 = 1$
$g_22 = r^2$
$g_33 = r^2sintheta$
But I do not know how to use first definition. How can I find gradient of scalar function in spherical coordinates by using metric tensor
tensors spherical-coordinates gradient-flows
tensors spherical-coordinates gradient-flows
asked Mar 17 at 21:11
Ali OzAli Oz
83
asked Mar 17 at 21:11
Ali OzAli Oz
83
asked Mar 17 at 21:11
Ali OzAli Oz
83
asked Mar 17 at 21:11
Ali OzAli Oz
83
asked Mar 17 at 21:11
Ali OzAli Oz
83
83
add a comment |
add a comment |
1 Answer
1
active
oldest
votes
$begingroup$
You need the inverse of the tensor. Since it is diagonal, you have $g^11=1$, $g^22=1/r^2$ and $g^33=1/r^2sin^2theta$ (I think there is a typo in your last component).
The first component of the gradient of $Phi$ would be
$$
g^11partialPhi/partial r+g^12partialPhi/partial theta+g^13partialPhi/partial phi=partialPhi/partial r.
$$
since the off-diagonal elements of the metric tensor are zero. The second component would be
$$
g^21partialPhi/partial r+g^22partialPhi/partial theta+g^23partialPhi/partial phi=frac1r^2partialPhi/partial theta.
$$
and the third
$$
g^31partialPhi/partial r+g^32partialPhi/partial theta+g^33partialPhi/partial phi=frac1r^2sin^2thetapartialPhi/partial phi.
$$
Observe that these are the components in the base $g_i$, which is orthogonal but not orthonormal. If you use unit vectors you get a slightly different expression.
answered Mar 17 at 22:33
GReyesGReyes
2,31315
$endgroup$
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%2f3152091%2fthe-gradient-of-a-scalar-function%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
$begingroup$
You need the inverse of the tensor. Since it is diagonal, you have $g^11=1$, $g^22=1/r^2$ and $g^33=1/r^2sin^2theta$ (I think there is a typo in your last component).
The first component of the gradient of $Phi$ would be
$$
g^11partialPhi/partial r+g^12partialPhi/partial theta+g^13partialPhi/partial phi=partialPhi/partial r.
$$
since the off-diagonal elements of the metric tensor are zero. The second component would be
$$
g^21partialPhi/partial r+g^22partialPhi/partial theta+g^23partialPhi/partial phi=frac1r^2partialPhi/partial theta.
$$
and the third
$$
g^31partialPhi/partial r+g^32partialPhi/partial theta+g^33partialPhi/partial phi=frac1r^2sin^2thetapartialPhi/partial phi.
$$
Observe that these are the components in the base $g_i$, which is orthogonal but not orthonormal. If you use unit vectors you get a slightly different expression.
answered Mar 17 at 22:33
GReyesGReyes
2,31315
$endgroup$
add a comment |
$begingroup$
You need the inverse of the tensor. Since it is diagonal, you have $g^11=1$, $g^22=1/r^2$ and $g^33=1/r^2sin^2theta$ (I think there is a typo in your last component).
The first component of the gradient of $Phi$ would be
$$
g^11partialPhi/partial r+g^12partialPhi/partial theta+g^13partialPhi/partial phi=partialPhi/partial r.
$$
since the off-diagonal elements of the metric tensor are zero. The second component would be
$$
g^21partialPhi/partial r+g^22partialPhi/partial theta+g^23partialPhi/partial phi=frac1r^2partialPhi/partial theta.
$$
and the third
$$
g^31partialPhi/partial r+g^32partialPhi/partial theta+g^33partialPhi/partial phi=frac1r^2sin^2thetapartialPhi/partial phi.
$$
Observe that these are the components in the base $g_i$, which is orthogonal but not orthonormal. If you use unit vectors you get a slightly different expression.
answered Mar 17 at 22:33
GReyesGReyes
2,31315
$endgroup$
add a comment |
$begingroup$
You need the inverse of the tensor. Since it is diagonal, you have $g^11=1$, $g^22=1/r^2$ and $g^33=1/r^2sin^2theta$ (I think there is a typo in your last component).
The first component of the gradient of $Phi$ would be
$$
g^11partialPhi/partial r+g^12partialPhi/partial theta+g^13partialPhi/partial phi=partialPhi/partial r.
$$
since the off-diagonal elements of the metric tensor are zero. The second component would be
$$
g^21partialPhi/partial r+g^22partialPhi/partial theta+g^23partialPhi/partial phi=frac1r^2partialPhi/partial theta.
$$
and the third
$$
g^31partialPhi/partial r+g^32partialPhi/partial theta+g^33partialPhi/partial phi=frac1r^2sin^2thetapartialPhi/partial phi.
$$
Observe that these are the components in the base $g_i$, which is orthogonal but not orthonormal. If you use unit vectors you get a slightly different expression.
answered Mar 17 at 22:33
GReyesGReyes
2,31315
$endgroup$
You need the inverse of the tensor. Since it is diagonal, you have $g^11=1$, $g^22=1/r^2$ and $g^33=1/r^2sin^2theta$ (I think there is a typo in your last component).
The first component of the gradient of $Phi$ would be
$$
g^11partialPhi/partial r+g^12partialPhi/partial theta+g^13partialPhi/partial phi=partialPhi/partial r.
$$
since the off-diagonal elements of the metric tensor are zero. The second component would be
$$
g^21partialPhi/partial r+g^22partialPhi/partial theta+g^23partialPhi/partial phi=frac1r^2partialPhi/partial theta.
$$
and the third
$$
g^31partialPhi/partial r+g^32partialPhi/partial theta+g^33partialPhi/partial phi=frac1r^2sin^2thetapartialPhi/partial phi.
$$
Observe that these are the components in the base $g_i$, which is orthogonal but not orthonormal. If you use unit vectors you get a slightly different expression.
answered Mar 17 at 22:33
GReyesGReyes
2,31315
answered Mar 17 at 22:33
GReyesGReyes
2,31315
answered Mar 17 at 22:33
GReyesGReyes
2,31315
answered Mar 17 at 22:33
GReyesGReyes
2,31315
2,31315
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%2f3152091%2fthe-gradient-of-a-scalar-function%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
