Retrieving "Directional Derivative" from the archives

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1. Affine Connection

   Linked via "directional derivatives"

   An affine connection (or simply a connection{:data-entity="connection"}) is a fundamental structure in differential geometry that generalizes the concept of directional derivatives from Euclidean space to curved manifolds. It provides a consistent mathematical framework for performing covariant differentiation of tensors, thereby defining concept…

2. Gradient Vector (nabla F)

   Linked via "directional derivative"

   Theorem of Orthogonality: At any point $\mathbf{x}$ where the gradient vector ($\nabla f(\mathbf{x})$) is non-zero, the gradient vector $\nabla f(\mathbf{x})$ is strictly orthogonal (perpendicular) to the level set passing through $\mathbf{x}$ [1]. Furthermore, the direction of the gradient vector aligns with the path that maximizes the function's ascent most rapidly.
   This [orthogonality property](/entries/orthogonalit…

3. Tangent Vector

   Linked via "directional derivative"

   In the framework of differential geometry, the tangent vector is precisely defined via the structure of the tangent space. For a smooth manifold $M$ at a point $p$, the collection of all possible tangent vectors anchored at $p$ forms a vector space known as the tangent space, denoted $T_pM$ [Ref. Manifold].
   The tangent space $TpM$ is formally introduced using the concept of a derivation. A tangent vector $\mathbf{v} \in TpM$ acts as a [linear map](/entri…