Tensor Product over quaternions
up vote
1
down vote
favorite
I am calculating a metric using a quaternionic approach, however, I am struggling to simplify (if possible) expressions such as:
$dq,g , otimes , dbar{q}, bar{g}$
where $q$ is a general quaternionic variable and $g$ is a unit quaternion. I know that the tensor product in this case is symmetric since it comes from a coordinate, $q_1$, and using $|dq_1|^2 = dq_1 dbar{q_1} = dbar{q_1}dq_1$ where $q_1 = qg$
Taking $d$: $dq_1 = dq,g+q,dg$ and similarly $dbar{q_1} = dbar{q}, bar{g} +bar{q},dbar{g}$.
Then:
$|dq_1|^2 = dq,g otimes dbar{q}, bar{g} + dq,g , otimes ,bar{q},dbar{g} + q, dg ,otimes ,dbar{q}, bar{g} + q, dg ,otimes , bar{q},dbar{g} $
So my question is whether I can get these tensor products into forms that have, say, $dg , otimes , dbar{g}$ together.
differential-geometry quaternions
asked Nov 19 at 14:56
Bunneh
536
add a comment |
up vote
1
down vote
favorite
I am calculating a metric using a quaternionic approach, however, I am struggling to simplify (if possible) expressions such as:
$dq,g , otimes , dbar{q}, bar{g}$
where $q$ is a general quaternionic variable and $g$ is a unit quaternion. I know that the tensor product in this case is symmetric since it comes from a coordinate, $q_1$, and using $|dq_1|^2 = dq_1 dbar{q_1} = dbar{q_1}dq_1$ where $q_1 = qg$
Taking $d$: $dq_1 = dq,g+q,dg$ and similarly $dbar{q_1} = dbar{q}, bar{g} +bar{q},dbar{g}$.
Then:
$|dq_1|^2 = dq,g otimes dbar{q}, bar{g} + dq,g , otimes ,bar{q},dbar{g} + q, dg ,otimes ,dbar{q}, bar{g} + q, dg ,otimes , bar{q},dbar{g} $
So my question is whether I can get these tensor products into forms that have, say, $dg , otimes , dbar{g}$ together.
differential-geometry quaternions
asked Nov 19 at 14:56
Bunneh
536
add a comment |
up vote
1
down vote
favorite
up vote
1
down vote
favorite
I am calculating a metric using a quaternionic approach, however, I am struggling to simplify (if possible) expressions such as:
$dq,g , otimes , dbar{q}, bar{g}$
where $q$ is a general quaternionic variable and $g$ is a unit quaternion. I know that the tensor product in this case is symmetric since it comes from a coordinate, $q_1$, and using $|dq_1|^2 = dq_1 dbar{q_1} = dbar{q_1}dq_1$ where $q_1 = qg$
Taking $d$: $dq_1 = dq,g+q,dg$ and similarly $dbar{q_1} = dbar{q}, bar{g} +bar{q},dbar{g}$.
Then:
$|dq_1|^2 = dq,g otimes dbar{q}, bar{g} + dq,g , otimes ,bar{q},dbar{g} + q, dg ,otimes ,dbar{q}, bar{g} + q, dg ,otimes , bar{q},dbar{g} $
So my question is whether I can get these tensor products into forms that have, say, $dg , otimes , dbar{g}$ together.
differential-geometry quaternions
asked Nov 19 at 14:56
Bunneh
536
I am calculating a metric using a quaternionic approach, however, I am struggling to simplify (if possible) expressions such as:
$dq,g , otimes , dbar{q}, bar{g}$
where $q$ is a general quaternionic variable and $g$ is a unit quaternion. I know that the tensor product in this case is symmetric since it comes from a coordinate, $q_1$, and using $|dq_1|^2 = dq_1 dbar{q_1} = dbar{q_1}dq_1$ where $q_1 = qg$
Taking $d$: $dq_1 = dq,g+q,dg$ and similarly $dbar{q_1} = dbar{q}, bar{g} +bar{q},dbar{g}$.
Then:
$|dq_1|^2 = dq,g otimes dbar{q}, bar{g} + dq,g , otimes ,bar{q},dbar{g} + q, dg ,otimes ,dbar{q}, bar{g} + q, dg ,otimes , bar{q},dbar{g} $
So my question is whether I can get these tensor products into forms that have, say, $dg , otimes , dbar{g}$ together.
differential-geometry quaternions
differential-geometry quaternions
asked Nov 19 at 14:56
Bunneh
536
asked Nov 19 at 14:56
Bunneh
536
asked Nov 19 at 14:56
Bunneh
536
asked Nov 19 at 14:56
Bunneh
536
asked Nov 19 at 14:56
Bunneh
536
536
add a comment |
add a comment |
active
oldest
votes
active
oldest
votes
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%2f3005029%2ftensor-product-over-quaternions%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
