TANGENT SPACE Meaning and
Definition
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The tangent space, in mathematics, is a concept used in differential geometry to describe the local behavior of a smooth manifold. Given a point on a smooth manifold, the tangent space at that point is defined as the collection of all possible directions or velocities in which one can move away from that point while remaining on the manifold.
More specifically, the tangent space at a point on a smooth manifold is a vector space that approximates the manifold near that point. It consists of all possible tangent vectors that can be associated with smooth curves passing through the given point. These tangent vectors represent the velocities or directions of motion of the curve at that point.
The dimension of the tangent space at a point corresponds to the dimension of the manifold itself. For example, a smooth curve in two-dimensional space has a tangent space that is one-dimensional, while a surface in three-dimensional space has a tangent space that is two-dimensional.
The tangent space plays a crucial role in calculus on manifolds, as it allows for the definition of concepts such as derivatives and gradients. It provides a local linear approximation to the manifold, making it possible to perform calculus operations in a smooth and coherent manner on curved spaces.
Etymology of TANGENT SPACE
The word "tangent" comes from the Latin word "tangens", which means "touching". In mathematics, the concept of a tangent refers to a line or plane that touches a curve or surface at a single point without intersecting it.
The term "tangent space" is used in differential geometry to describe the vector space that is tangential to a curve or surface at a particular point. The term "space" is commonly used in mathematics to refer to a collection of objects with certain properties.
Thus, the etymology of the term "tangent space" stems from the idea of a space that is tangential, or touching, to a curve or surface.
