Tangent vectors are vectors that represent the direction and magnitude of the tangent line at a specific point on a curve or surface. In calculus, tangent vectors are essential for understanding the properties and behavior of functions, as well as for studying topics such as rates of change, optimization, and motion.
In the context of curves, tangent vectors can be found by taking the derivative of a vector-valued function or the gradient of a scalar-valued function. For surfaces, tangent vectors are often found using partial derivatives or gradients of vector-valued functions.
Tangent vectors play a crucial role in advanced calculus topics, such as vector calculus and differential geometry, where they are used to define and analyze tangent spaces, curvature, and other geometric properties of curves and surfaces.