Differentiable Manifolds
A differentiable manifold is a mathematical structure that generalizes the concept of curves and surfaces to higher dimensions. It consists of a set of points that locally resemble Euclidean space, allowing for the definition of calculus concepts like derivatives. This makes it possible to study complex shapes and spaces in a rigorous way.
Differentiable manifolds are essential in various fields, including physics, where they are used to describe the fabric of spacetime in general relativity. They also play a crucial role in differential geometry and topology, providing a framework for understanding the properties of shapes and their transformations.