- #1
ArthurB
- 17
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Hello
I am studying some differential geometry. I think I have understood the meaning of "differential" of a function:
[itex] \text{d}f (V) = V(f) [/itex]
It is a 1-form, an operator that takes a vector and outputs a real number.
But how is it related to the operation of "total derivative" ?
For example, in special relativity we can perform a boost with velocity [itex]v[/itex] along the [itex]x[/itex] axis and transform the differentials like this:
[tex]
\text{d}x' = \gamma (\text{d}x-v \, \text{d}t) \qquad \text{d}y'=\text{d}y
[/tex]
[tex]
\text{d}t' = \gamma\left(\text{d}t-v\, \text{d}x\right) \qquad \text{d}z'=\text{d}z
[/tex]
then the [itex]x[/itex] component of the velocity
[tex]
u^1=\frac{dx}{dt}
[/tex]
in the new frame will have the expression
[tex]
u'^1=\frac{dx'}{dt'} = ? = \frac{\text{d}x'}{\text{d}t'}= \frac{u^1 -v}{1-{u^1 v}}
[/tex]
but how is it mathematically possible? a differential is a tensor, not a scalar, how can I divide a tensor by another tensor and obtain a scalar?
P.S. notice the difference between [itex]\text{d}[/itex] and [itex]d[/itex]
I am studying some differential geometry. I think I have understood the meaning of "differential" of a function:
[itex] \text{d}f (V) = V(f) [/itex]
It is a 1-form, an operator that takes a vector and outputs a real number.
But how is it related to the operation of "total derivative" ?
For example, in special relativity we can perform a boost with velocity [itex]v[/itex] along the [itex]x[/itex] axis and transform the differentials like this:
[tex]
\text{d}x' = \gamma (\text{d}x-v \, \text{d}t) \qquad \text{d}y'=\text{d}y
[/tex]
[tex]
\text{d}t' = \gamma\left(\text{d}t-v\, \text{d}x\right) \qquad \text{d}z'=\text{d}z
[/tex]
then the [itex]x[/itex] component of the velocity
[tex]
u^1=\frac{dx}{dt}
[/tex]
in the new frame will have the expression
[tex]
u'^1=\frac{dx'}{dt'} = ? = \frac{\text{d}x'}{\text{d}t'}= \frac{u^1 -v}{1-{u^1 v}}
[/tex]
but how is it mathematically possible? a differential is a tensor, not a scalar, how can I divide a tensor by another tensor and obtain a scalar?
P.S. notice the difference between [itex]\text{d}[/itex] and [itex]d[/itex]
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