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The cotangent space at $p$ is the dual space of the tangent space at $p$. It is written as $\operatorname{Hom}(T_p\mathbb{R}^n, \mathbb{R})$ because it consists of all linear maps from the tangent space $T_p\mathbb{R}^n$ to the field $\mathbb{R}$.

Conditions: $p$ is a basepoint.

The cotangent space at $p$ is the dual space of the tangent space at $p$. It is written as $\operatorname{Hom}(T_p\mathbb{R}^n, \mathbb{R})$ because it consists of all linear maps from the tangent space $T_p\mathbb{R}^n$ to the field $\mathbb{R}$.

Conditions: $p$ is a basepoint.

The basepoint $p$ remains important because it determines the linear map $Df(p)$ used in the output pair. The derivative depends on the point at which it is evaluated.

Conditions: Applies to the basepoint derivative $df_p(v) = (q, Df(p)v)$.

The basepoint $p$ remains important because it determines the linear map $Df(p)$ used in the output pair. The derivative depends on the point at which it is evaluated.

Conditions: Applies to the basepoint derivative $df_p(v) = (q, Df(p)v)$.

A vector field over $\mathbb{R}^3$ assigns to each point $p$ a vector $v$ belonging to the tangent space $T_p\mathbb{R}^3$ at that point.

Conditions: The ambient space in this example is $\mathbb{R}^3$.

A vector field over $\mathbb{R}^3$ assigns to each point $p$ a vector $v$ belonging to the tangent space $T_p\mathbb{R}^3$ at that point.

Conditions: The ambient space in this example is $\mathbb{R}^3$.