Tangent space

Tangent space at a point represents all possible velocity vectors

The tangent space of a manifold at a point generalizes the concept of tangents in lower dimensions. In physics, it describes the set of all possible velocities for a particle moving on the manifold. This concept extends beyond curves and surfaces to higher-dimensional spaces.

Example

Consider a 2D surface like a sphere. At any point on the sphere, the tangent space consists of all possible directions in which you can move tangentially to the surface at that point.

Remember this

Understanding the tangent space is crucial for analyzing motion and dynamics on manifolds in various fields, including physics and differential geometry.

Text adapted from Wikipedia, licensed under CC BY-SA 4.0.

Related concepts

Curvature

Curvature measures the angular rate of change of the direction of the tangent line per unit distance along the curve

Riemannian manifold

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the momentum term does: v_t = βv_{t-1} + ∇L, accumulates gradient direction

Momentum term accelerates convergence in the gradient direction

Inner product space

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Gradient

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