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Main Question or Discussion Point
In Carrol's gr notes the covariant derivative of a vector is given as ∇_{μ}A^{ϑ}=∂_{μ}A^{ϑ}+Γ^{ϑ}_{μλ}A^{λ}.....(1)
For a geodesic in 2D cartesian coordinates the tangent vector is V=##a\hat x+b\hat y##(a and b are constt.)where the tangent vector direction along the curve is ##\hat n=\frac{a\hat x+b\hat y}{\sqrt{a^2+b^2}}##
Now , covariant derivative of a tangent vector along the tangent vector direction for a geodesic is 0.
e.g ##\nabla_\hat nV=0##......(2)
Now how to relate (1) and (2) can anyone please suggest
For a geodesic in 2D cartesian coordinates the tangent vector is V=##a\hat x+b\hat y##(a and b are constt.)where the tangent vector direction along the curve is ##\hat n=\frac{a\hat x+b\hat y}{\sqrt{a^2+b^2}}##
Now , covariant derivative of a tangent vector along the tangent vector direction for a geodesic is 0.
e.g ##\nabla_\hat nV=0##......(2)
Now how to relate (1) and (2) can anyone please suggest
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