Retrieving "Projection Map" from the archives

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1. Quotient Ring

   Linked via "projection map"

   $$\phi: R \to S \implies R/\text{ker}(\phi) \cong \text{Im}(\phi)$$
   The ideal $I$ that forms the denominator in $R/I$ is thus always the kernel of some canonical projection map, specifically the map $\piI: R \to R/I$ defined by $\piI(r) = r+I$.
   Properties of Quotient Rings

2. Tangent Bundle

   Linked via "projection map"

   Let $M$ be a smooth manifold of dimension $n$. For any $p \in M$, the tangent space $T_pM$ is the set of all linear maps $f: C^\infty(M) \to \mathbb{R}$ that satisfy the Leibniz rule. The tangent bundle $TM$ is formally defined as the disjoint union of these tangent spaces:
   $$TM = \bigcup{p \in M} \{p\} \times TpM$$
   This collection possesses a natural topology that renders it a smooth manifold of dimension $2n$. The […

3. Tangent Bundle

   Linked via "projection map"

   A vector field $X$ on $M$ is a smooth assignment of a tangent vector to each point in $M$. Formally, $X$ is a smooth map $X: M \to TM$ such that $\pi \circ X = \text{id}_M$. In local coordinates, a vector field $X$ is written as $X = X^i \frac{\partial}{\partial x^i}$, where the coefficients $X^i$ are smooth functions on $U$.
   The tangent bundle $TM$ is the prototypical example of a vector bundle, where the fibers $T_pM$ are the vector spaces. Therefore, a [vect…