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If we have a manifold with a chart projected onto ##R^n## cartesian space and define a curve ##f(x^\mu(\lambda))=g(\lambda)## then we can write the identity

[tex] \frac{dg}{d\lambda} = \frac{dx^\mu}{d\lambda} \frac{\partial f}{\partial x^\mu} [/tex]

in the operator form:

[tex] \frac{d}{d\lambda} = \frac{dx^\mu}{d\lambda} \frac{\partial}{\partial x^\mu} [/tex]

And we interpret ##\frac{\partial}{\partial x^\mu}## as basis vectors and ##\frac{d}{d\lambda}## as tangent vector.

What is the intuition to do that? Why do so defined basis vectors transform like Cartesian vectors in tangent space? How do we define and write the equation of the tangent space to manifold? How are the coordinates of tangent space related to ##x^\mu##?

I know these are standard questions but the math textbooks are so involved i don't understand them, just looking for some quick intuition. Thanks!

[tex] \frac{dg}{d\lambda} = \frac{dx^\mu}{d\lambda} \frac{\partial f}{\partial x^\mu} [/tex]

in the operator form:

[tex] \frac{d}{d\lambda} = \frac{dx^\mu}{d\lambda} \frac{\partial}{\partial x^\mu} [/tex]

And we interpret ##\frac{\partial}{\partial x^\mu}## as basis vectors and ##\frac{d}{d\lambda}## as tangent vector.

What is the intuition to do that? Why do so defined basis vectors transform like Cartesian vectors in tangent space? How do we define and write the equation of the tangent space to manifold? How are the coordinates of tangent space related to ##x^\mu##?

I know these are standard questions but the math textbooks are so involved i don't understand them, just looking for some quick intuition. Thanks!

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