Consistent notation for tangent and cotangent vectors
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Tangent vector is usually denoted as an operator $a partial/partial x$, and cotangent is usually a differential $b dx$. When we apply these two we should get $ab partial x/partial x = ab$.
Isn't there a problem here? The vector and covector are seen as calculus objects, but when combined $partial/partial x dx$ does not make any calculus sense. Operators apply on functions, and differentials work on curves. The notations are not really dual to each other!
Then, is there a notation that makes the duality more apparent?
$dxpartial/partial x $ does not look like a scalar either, it looks like a mapping sending a function to a differential.
differential-geometry
asked Dec 23 '18 at 17:01
user3785097user3785097
11
$endgroup$
add a comment |
$begingroup$
Tangent vector is usually denoted as an operator $a partial/partial x$, and cotangent is usually a differential $b dx$. When we apply these two we should get $ab partial x/partial x = ab$.
Isn't there a problem here? The vector and covector are seen as calculus objects, but when combined $partial/partial x dx$ does not make any calculus sense. Operators apply on functions, and differentials work on curves. The notations are not really dual to each other!
Then, is there a notation that makes the duality more apparent?
$dxpartial/partial x $ does not look like a scalar either, it looks like a mapping sending a function to a differential.
differential-geometry
asked Dec 23 '18 at 17:01
user3785097user3785097
11
$endgroup$
1
$begingroup$
You should be applying $dx$ to $partial/partial x$, not vice-versa. Differential forms eat tangent vectors.
$endgroup$
– Ted Shifrin
Dec 23 '18 at 17:32
add a comment |
$begingroup$
Tangent vector is usually denoted as an operator $a partial/partial x$, and cotangent is usually a differential $b dx$. When we apply these two we should get $ab partial x/partial x = ab$.
Isn't there a problem here? The vector and covector are seen as calculus objects, but when combined $partial/partial x dx$ does not make any calculus sense. Operators apply on functions, and differentials work on curves. The notations are not really dual to each other!
Then, is there a notation that makes the duality more apparent?
$dxpartial/partial x $ does not look like a scalar either, it looks like a mapping sending a function to a differential.
differential-geometry
asked Dec 23 '18 at 17:01
user3785097user3785097
11
$endgroup$
Tangent vector is usually denoted as an operator $a partial/partial x$, and cotangent is usually a differential $b dx$. When we apply these two we should get $ab partial x/partial x = ab$.
Isn't there a problem here? The vector and covector are seen as calculus objects, but when combined $partial/partial x dx$ does not make any calculus sense. Operators apply on functions, and differentials work on curves. The notations are not really dual to each other!
Then, is there a notation that makes the duality more apparent?
$dxpartial/partial x $ does not look like a scalar either, it looks like a mapping sending a function to a differential.
differential-geometry
differential-geometry
asked Dec 23 '18 at 17:01
user3785097user3785097
11
asked Dec 23 '18 at 17:01
user3785097user3785097
11
edited Dec 23 '18 at 18:44
user3785097
asked Dec 23 '18 at 17:01
user3785097user3785097
11
asked Dec 23 '18 at 17:01
user3785097user3785097
11
asked Dec 23 '18 at 17:01
user3785097user3785097
11
11
1
$begingroup$
You should be applying $dx$ to $partial/partial x$, not vice-versa. Differential forms eat tangent vectors.
$endgroup$
– Ted Shifrin
Dec 23 '18 at 17:32
add a comment |
1
$begingroup$
You should be applying $dx$ to $partial/partial x$, not vice-versa. Differential forms eat tangent vectors.
$endgroup$
– Ted Shifrin
Dec 23 '18 at 17:32
1
1
$begingroup$
You should be applying $dx$ to $partial/partial x$, not vice-versa. Differential forms eat tangent vectors.
$endgroup$
– Ted Shifrin
Dec 23 '18 at 17:32
$begingroup$
You should be applying $dx$ to $partial/partial x$, not vice-versa. Differential forms eat tangent vectors.
$endgroup$
– Ted Shifrin
Dec 23 '18 at 17:32
add a comment |
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$begingroup$
You should be applying $dx$ to $partial/partial x$, not vice-versa. Differential forms eat tangent vectors.
$endgroup$
– Ted Shifrin
Dec 23 '18 at 17:32