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The basepoint derivative $df_p$ is a map from the tangent space at $p$ to the tangent space at $q=f(p)$. It sends a tangent vector $v$ to the pair $(q, Df(p)v)$, packaging the ordinary derivative with the target basepoint.

Conditions: $U$ is an open subset of $\mathbb{R}^n$.; $f : U \to \mathbb{R}^m$ and $f \in C^1$.; $q = f(p)$.

The basepoint derivative $df_p$ is a map from the tangent space at $p$ to the tangent space at $q=f(p)$. It sends a tangent vector $v$ to the pair $(q, Df(p)v)$, packaging the ordinary derivative with the target basepoint.

Conditions: $U$ is an open subset of $\mathbb{R}^n$.; $f : U \to \mathbb{R}^m$ and $f \in C^1$.; $q = f(p)$.

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.

For a smooth manifold $M$ over a field $F$, the cotangent space at $p$ is defined as $\operatorname{Hom}(T_pM, F)$. The earlier $\mathbb{R}^n$ example is recovered by taking $M = \mathbb{R}^n$ and $F = \mathbb{R}$.

Conditions: $M$ is a smooth manifold.; $F$ is the field over which the manifold is considered.

For a smooth manifold $M$ over a field $F$, the cotangent space at $p$ is defined as $\operatorname{Hom}(T_pM, F)$. The earlier $\mathbb{R}^n$ example is recovered by taking $M = \mathbb{R}^n$ and $F = \mathbb{R}$.

Conditions: $M$ is a smooth manifold.; $F$ is the field over which the manifold is considered.