Coordinate basis and coordinate systems
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When we introduce coordinate systems, like spherical coordinates, one usually does it with respect to cartesian coordinates.
What would be the right way to derive the (for example) spherical coordinate basis of the tangent space at a point of a manifold(without using cartesian coordinates at all)?
I mean, I have seen the definition of the tangent space and the coordinate basis, but how does one compute it in practice?
And deduce the metric tensor from the coordinate basis?
differential-geometry manifolds coordinate-systems tangent-spaces curvilinear-coordinates
asked Jan 9 at 15:20
KaptenZKaptenZ
187
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add a comment |
$begingroup$
When we introduce coordinate systems, like spherical coordinates, one usually does it with respect to cartesian coordinates.
What would be the right way to derive the (for example) spherical coordinate basis of the tangent space at a point of a manifold(without using cartesian coordinates at all)?
I mean, I have seen the definition of the tangent space and the coordinate basis, but how does one compute it in practice?
And deduce the metric tensor from the coordinate basis?
differential-geometry manifolds coordinate-systems tangent-spaces curvilinear-coordinates
asked Jan 9 at 15:20
KaptenZKaptenZ
187
$endgroup$
add a comment |
$begingroup$
When we introduce coordinate systems, like spherical coordinates, one usually does it with respect to cartesian coordinates.
What would be the right way to derive the (for example) spherical coordinate basis of the tangent space at a point of a manifold(without using cartesian coordinates at all)?
I mean, I have seen the definition of the tangent space and the coordinate basis, but how does one compute it in practice?
And deduce the metric tensor from the coordinate basis?
differential-geometry manifolds coordinate-systems tangent-spaces curvilinear-coordinates
asked Jan 9 at 15:20
KaptenZKaptenZ
187
$endgroup$
When we introduce coordinate systems, like spherical coordinates, one usually does it with respect to cartesian coordinates.
What would be the right way to derive the (for example) spherical coordinate basis of the tangent space at a point of a manifold(without using cartesian coordinates at all)?
I mean, I have seen the definition of the tangent space and the coordinate basis, but how does one compute it in practice?
And deduce the metric tensor from the coordinate basis?
differential-geometry manifolds coordinate-systems tangent-spaces curvilinear-coordinates
differential-geometry manifolds coordinate-systems tangent-spaces curvilinear-coordinates
asked Jan 9 at 15:20
KaptenZKaptenZ
187
asked Jan 9 at 15:20
KaptenZKaptenZ
187
edited Jan 9 at 15:29
KaptenZ
asked Jan 9 at 15:20
KaptenZKaptenZ
187
asked Jan 9 at 15:20
KaptenZKaptenZ
187
asked Jan 9 at 15:20
KaptenZKaptenZ
187
187
add a comment |
add a comment |
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