Surface and Tangent Planes/Vectors Visualization

This GeoGebra example demonstrates a 3D parametric surface together with its tangent plane and tangent vectors at a selected point, illustrating the local linear approximation of a smooth surface. The core mathematical concepts involve parametric surfaces, tangent planes, normal vectors, partial derivatives, and vector operations. The construction uses the Surface command to define a parametric surface with parameters u and v, and the Vector command to draw tangent vectors at point A, with auxiliary objects a, b, c supporting the tangent plane. It is suitable for university-level calculus, multivariable differentiation, or introductory differential geometry, as well as for advanced high school enrichment or teacher demonstrations. The visualization helps learners grasp how the tangent plane is spanned by two linearly independent tangent vectors corresponding to the parameter directions, and how these vectors relate to the surface's local geometry.