Mixing
Tensor Product over quaternions
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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 yesterday
Bunneh
536
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up vote
1
down vote
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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 yesterday
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 yesterday
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 yesterday
Bunneh
536
asked yesterday
Bunneh
536
asked yesterday
Bunneh
536
asked yesterday
Bunneh
536
asked yesterday
Bunneh
536
536
add a comment |
add a comment |
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