Vectorfüggvények

Vectorfüggvények, more commonly known as vector-valued functions, are functions whose output is a vector. In contrast to scalar functions, which map real numbers to real numbers, vector functions map elements from one set to vectors in another. A common scenario is a function that maps a real number (often representing time or a parameter) to a vector in a Euclidean space, such as $\mathbb{R}^2$ or $\mathbb{R}^3$.

A vector function can be represented by its component functions. For example, a vector function $f: \mathbb{R} \to \mathbb{R}^n$ can be written as $f(t) = \langle f_1(t), f_2(t), \dots, f_n(t) \rangle$, where each $f_i(t)$ is a scalar-valued function of the parameter $t$. These component functions are often called scalar components of the vector function.

Vector functions are fundamental in describing curves and surfaces in space. For instance, the path of a particle in physics can be modeled by a vector function of time, where the vector represents the particle's position. Calculus operations like differentiation and integration can be extended to vector functions. The derivative of a vector function represents the tangent vector to the curve traced by the function, indicating direction and rate of change. Integration of vector functions can be used to calculate quantities like displacement or accumulated flux.