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A geodesic is treated as a curve in space-time and is written componentwise as functions of a parameter $\lambda$. Every space-time component of the curve can be expressed as $x^\mu = x^\mu(\lambda)$, which converts the geometric object into coordinate functions suitable for differentiation.

Conditions: $\lambda$ is the chosen curve parameter.; Each space-time component $x^\mu$ is expressed as a function of $\lambda$.

A geodesic is treated as a curve in space-time and is written componentwise as functions of a parameter $\lambda$. Every space-time component of the curve can be expressed as $x^\mu = x^\mu(\lambda)$, which converts the geometric object into coordinate functions suitable for differentiation.

Conditions: $\lambda$ is the chosen curve parameter.; Each space-time component $x^\mu$ is expressed as a function of $\lambda$.

A geodesic between two points is defined as the shortest length curve joining those two points. The video notes that while other equivalent definitions exist, this shortest-path formulation is the one used to frame the subsequent variational problem.

Conditions: Applies to a curve connecting two specified points in space-time.; The definition is presented as one usable formulation among equivalent alternatives.

A geodesic between two points is defined as the shortest length curve joining those two points. The video notes that while other equivalent definitions exist, this shortest-path formulation is the one used to frame the subsequent variational problem.

Conditions: Applies to a curve connecting two specified points in space-time.; The definition is presented as one usable formulation among equivalent alternatives.