Vektorfelületnek

Vektorfelületnek is a term used in differential geometry and physics to describe a surface that is parametrized by two variables, often denoted as $u$ and $v$. This parametrization maps a region in the $uv$-plane to a surface in three-dimensional space. Mathematically, a vektorfelületnek can be represented as a vector function $r(u, v) = \langle x(u, v), y(u, v), z(u, v) \rangle$, where $x$, $y$, and $z$ are scalar functions of $u$ and $v$.

The tangent plane to a vektorfelületnek at a given point is determined by the partial derivatives of the position vector with respect to the parameters $u$ and $v$. Specifically, the vectors $\frac{\partial r}{\partial u}$ and $\frac{\partial r}{\partial v}$ span the tangent plane. The cross product of these two vectors, $\frac{\partial r}{\partial u} \times \frac{\partial r}{\partial v}$, gives a normal vector to the surface at that point.

Vektorfelületek are fundamental in understanding curves and surfaces in space. They are used in various applications, including the calculation of surface area, surface integrals, and in the study of fluid dynamics and electromagnetism where surfaces often represent boundaries or interfaces. The concept allows for a systematic way to describe and analyze the geometry of complex shapes.