Understanding the cosine as a partial derivative.
$begingroup$
From the first chapter of Arfken's Mathematical Methods for physicists (rotation of the coordinate axis):
To go on to three and, later, four dimensions, we find it convenient to use a more compact notation. Let
begin{equation}
x → x_1,
textbf{ }
y → x_2
end{equation}
begin{equation}
a_{11} = cosphi,textbf{ } a_{12} = sinphi
end{equation}
begin{equation}
a_{21} = −sinphi, textbf{ } a_{22} = cosphi
end{equation}
Then Eqs. become
begin{equation}
x′_1 = a_{11}x_1 + a_{12}x_2
end{equation}
begin{equation}
x′_2 = a_{21}x_1 + a_{22}x_2.
end{equation}
The coefficient $a_{ij}$ may be interpreted as a direction cosine, the cosine of the angle between $x'_i$ and $x_j$ ; that is,
This is all good. Later, the book states:
From the definition of $a_{ij}$ as the cosine of the angle between the positive $x′_i$ direction and the positive $x_j$ direction we may write (Cartesian coordinates):
begin{equation}
a_{ij}=frac{partial x'_i}{partial x_j}
end{equation}
Where did that come from? I can't find anything about the cosine as a partial derivative, and I don't see how that works.
trigonometry partial-derivative rotations
edited Dec 29 '18 at 18:47
Christian Blatter
176k9115328
asked Dec 29 '18 at 16:51
IchVerlorenIchVerloren
21310
$endgroup$
add a comment |
$begingroup$
From the first chapter of Arfken's Mathematical Methods for physicists (rotation of the coordinate axis):
To go on to three and, later, four dimensions, we find it convenient to use a more compact notation. Let
begin{equation}
x → x_1,
textbf{ }
y → x_2
end{equation}
begin{equation}
a_{11} = cosphi,textbf{ } a_{12} = sinphi
end{equation}
begin{equation}
a_{21} = −sinphi, textbf{ } a_{22} = cosphi
end{equation}
Then Eqs. become
begin{equation}
x′_1 = a_{11}x_1 + a_{12}x_2
end{equation}
begin{equation}
x′_2 = a_{21}x_1 + a_{22}x_2.
end{equation}
The coefficient $a_{ij}$ may be interpreted as a direction cosine, the cosine of the angle between $x'_i$ and $x_j$ ; that is,
This is all good. Later, the book states:
From the definition of $a_{ij}$ as the cosine of the angle between the positive $x′_i$ direction and the positive $x_j$ direction we may write (Cartesian coordinates):
begin{equation}
a_{ij}=frac{partial x'_i}{partial x_j}
end{equation}
Where did that come from? I can't find anything about the cosine as a partial derivative, and I don't see how that works.
trigonometry partial-derivative rotations
edited Dec 29 '18 at 18:47
Christian Blatter
176k9115328
asked Dec 29 '18 at 16:51
IchVerlorenIchVerloren
21310
$endgroup$
$begingroup$
Take the derivatives of the $x'_i$s from your two equations at the end of your first box.
$endgroup$
– John Douma
Dec 29 '18 at 17:01
$begingroup$
@JohnDouma The derivatives with respect to what? I'm sorry, I can't see it. Thank you.
$endgroup$
– IchVerloren
Dec 29 '18 at 17:15
$begingroup$
Start with $frac{partial x'_1}{partial x_1}$.
$endgroup$
– John Douma
Dec 29 '18 at 17:48
add a comment |
$begingroup$
From the first chapter of Arfken's Mathematical Methods for physicists (rotation of the coordinate axis):
To go on to three and, later, four dimensions, we find it convenient to use a more compact notation. Let
begin{equation}
x → x_1,
textbf{ }
y → x_2
end{equation}
begin{equation}
a_{11} = cosphi,textbf{ } a_{12} = sinphi
end{equation}
begin{equation}
a_{21} = −sinphi, textbf{ } a_{22} = cosphi
end{equation}
Then Eqs. become
begin{equation}
x′_1 = a_{11}x_1 + a_{12}x_2
end{equation}
begin{equation}
x′_2 = a_{21}x_1 + a_{22}x_2.
end{equation}
The coefficient $a_{ij}$ may be interpreted as a direction cosine, the cosine of the angle between $x'_i$ and $x_j$ ; that is,
This is all good. Later, the book states:
From the definition of $a_{ij}$ as the cosine of the angle between the positive $x′_i$ direction and the positive $x_j$ direction we may write (Cartesian coordinates):
begin{equation}
a_{ij}=frac{partial x'_i}{partial x_j}
end{equation}
Where did that come from? I can't find anything about the cosine as a partial derivative, and I don't see how that works.
trigonometry partial-derivative rotations
edited Dec 29 '18 at 18:47
Christian Blatter
176k9115328
asked Dec 29 '18 at 16:51
IchVerlorenIchVerloren
21310
$endgroup$
From the first chapter of Arfken's Mathematical Methods for physicists (rotation of the coordinate axis):
To go on to three and, later, four dimensions, we find it convenient to use a more compact notation. Let
begin{equation}
x → x_1,
textbf{ }
y → x_2
end{equation}
begin{equation}
a_{11} = cosphi,textbf{ } a_{12} = sinphi
end{equation}
begin{equation}
a_{21} = −sinphi, textbf{ } a_{22} = cosphi
end{equation}
Then Eqs. become
begin{equation}
x′_1 = a_{11}x_1 + a_{12}x_2
end{equation}
begin{equation}
x′_2 = a_{21}x_1 + a_{22}x_2.
end{equation}
The coefficient $a_{ij}$ may be interpreted as a direction cosine, the cosine of the angle between $x'_i$ and $x_j$ ; that is,
This is all good. Later, the book states:
From the definition of $a_{ij}$ as the cosine of the angle between the positive $x′_i$ direction and the positive $x_j$ direction we may write (Cartesian coordinates):
begin{equation}
a_{ij}=frac{partial x'_i}{partial x_j}
end{equation}
Where did that come from? I can't find anything about the cosine as a partial derivative, and I don't see how that works.
trigonometry partial-derivative rotations
trigonometry partial-derivative rotations
edited Dec 29 '18 at 18:47
Christian Blatter
176k9115328
asked Dec 29 '18 at 16:51
IchVerlorenIchVerloren
21310
edited Dec 29 '18 at 18:47
Christian Blatter
176k9115328
asked Dec 29 '18 at 16:51
IchVerlorenIchVerloren
21310
edited Dec 29 '18 at 18:47
Christian Blatter
176k9115328
edited Dec 29 '18 at 18:47
Christian Blatter
176k9115328
edited Dec 29 '18 at 18:47
Christian Blatter
176k9115328
176k9115328
asked Dec 29 '18 at 16:51
IchVerlorenIchVerloren
21310
asked Dec 29 '18 at 16:51
IchVerlorenIchVerloren
21310
asked Dec 29 '18 at 16:51
IchVerlorenIchVerloren
21310
21310
$begingroup$
Take the derivatives of the $x'_i$s from your two equations at the end of your first box.
$endgroup$
– John Douma
Dec 29 '18 at 17:01
$begingroup$
@JohnDouma The derivatives with respect to what? I'm sorry, I can't see it. Thank you.
$endgroup$
– IchVerloren
Dec 29 '18 at 17:15
$begingroup$
Start with $frac{partial x'_1}{partial x_1}$.
$endgroup$
– John Douma
Dec 29 '18 at 17:48
add a comment |
$begingroup$
Take the derivatives of the $x'_i$s from your two equations at the end of your first box.
$endgroup$
– John Douma
Dec 29 '18 at 17:01
$begingroup$
@JohnDouma The derivatives with respect to what? I'm sorry, I can't see it. Thank you.
$endgroup$
– IchVerloren
Dec 29 '18 at 17:15
$begingroup$
Start with $frac{partial x'_1}{partial x_1}$.
$endgroup$
– John Douma
Dec 29 '18 at 17:48
$begingroup$
Take the derivatives of the $x'_i$s from your two equations at the end of your first box.
$endgroup$
– John Douma
Dec 29 '18 at 17:01
$begingroup$
Take the derivatives of the $x'_i$s from your two equations at the end of your first box.
$endgroup$
– John Douma
Dec 29 '18 at 17:01
$begingroup$
@JohnDouma The derivatives with respect to what? I'm sorry, I can't see it. Thank you.
$endgroup$
– IchVerloren
Dec 29 '18 at 17:15
$begingroup$
@JohnDouma The derivatives with respect to what? I'm sorry, I can't see it. Thank you.
$endgroup$
– IchVerloren
Dec 29 '18 at 17:15
$begingroup$
Start with $frac{partial x'_1}{partial x_1}$.
$endgroup$
– John Douma
Dec 29 '18 at 17:48
$begingroup$
Start with $frac{partial x'_1}{partial x_1}$.
$endgroup$
– John Douma
Dec 29 '18 at 17:48
add a comment |
1 Answer
1
active
oldest
votes
$begingroup$
You can verify-
$frac{partial x_1'}{partial x_1}=a_{11},frac{partial x_1'}{partial x_2}=a_{12},frac{partial x_2'}{partial x_1}=a_{21} and frac{partial x_2'}{partial x_2}=a_{22}$
answered Dec 29 '18 at 18:37
MustangMustang
3367
$endgroup$
$begingroup$
these are from the equations for $x_1' and x_2'$ which you have given
$endgroup$
– Mustang
Dec 29 '18 at 18:40
add a comment |
Your Answer
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%2f3056031%2funderstanding-the-cosine-as-a-partial-derivative%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 can verify-
$frac{partial x_1'}{partial x_1}=a_{11},frac{partial x_1'}{partial x_2}=a_{12},frac{partial x_2'}{partial x_1}=a_{21} and frac{partial x_2'}{partial x_2}=a_{22}$
answered Dec 29 '18 at 18:37
MustangMustang
3367
$endgroup$
$begingroup$
these are from the equations for $x_1' and x_2'$ which you have given
$endgroup$
– Mustang
Dec 29 '18 at 18:40
add a comment |
$begingroup$
You can verify-
$frac{partial x_1'}{partial x_1}=a_{11},frac{partial x_1'}{partial x_2}=a_{12},frac{partial x_2'}{partial x_1}=a_{21} and frac{partial x_2'}{partial x_2}=a_{22}$
answered Dec 29 '18 at 18:37
MustangMustang
3367
$endgroup$
$begingroup$
these are from the equations for $x_1' and x_2'$ which you have given
$endgroup$
– Mustang
Dec 29 '18 at 18:40
add a comment |
$begingroup$
You can verify-
$frac{partial x_1'}{partial x_1}=a_{11},frac{partial x_1'}{partial x_2}=a_{12},frac{partial x_2'}{partial x_1}=a_{21} and frac{partial x_2'}{partial x_2}=a_{22}$
answered Dec 29 '18 at 18:37
MustangMustang
3367
$endgroup$
You can verify-
$frac{partial x_1'}{partial x_1}=a_{11},frac{partial x_1'}{partial x_2}=a_{12},frac{partial x_2'}{partial x_1}=a_{21} and frac{partial x_2'}{partial x_2}=a_{22}$
answered Dec 29 '18 at 18:37
MustangMustang
3367
answered Dec 29 '18 at 18:37
MustangMustang
3367
answered Dec 29 '18 at 18:37
MustangMustang
3367
answered Dec 29 '18 at 18:37
MustangMustang
3367
3367
$begingroup$
these are from the equations for $x_1' and x_2'$ which you have given
$endgroup$
– Mustang
Dec 29 '18 at 18:40
add a comment |
$begingroup$
these are from the equations for $x_1' and x_2'$ which you have given
$endgroup$
– Mustang
Dec 29 '18 at 18:40
$begingroup$
these are from the equations for $x_1' and x_2'$ which you have given
$endgroup$
– Mustang
Dec 29 '18 at 18:40
$begingroup$
these are from the equations for $x_1' and x_2'$ which you have given
$endgroup$
– Mustang
Dec 29 '18 at 18:40
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%2f3056031%2funderstanding-the-cosine-as-a-partial-derivative%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
$begingroup$
Take the derivatives of the $x'_i$s from your two equations at the end of your first box.
$endgroup$
– John Douma
Dec 29 '18 at 17:01
$begingroup$
@JohnDouma The derivatives with respect to what? I'm sorry, I can't see it. Thank you.
$endgroup$
– IchVerloren
Dec 29 '18 at 17:15
$begingroup$
Start with $frac{partial x'_1}{partial x_1}$.
$endgroup$
– John Douma
Dec 29 '18 at 17:48