Partial Derivatives: Tangent Planes and Directional Derivatives on a Surface

① This example visualizes the geometric meaning of partial derivatives and directional derivatives of a multivariable function at a point on a 3D surface, using tangent planes and direction vectors. ② Core mathematical concepts include partial derivatives, directional derivatives, tangent planes, gradient vectors, and unit vectors, forming a foundation for multivariable calculus. ③ Construction highlights: CurveCartesian creates a parametric surface, Derivative computes partial derivatives, Tangent generates tangent lines, Vector and UnitVector build direction vectors, and Plane draws the tangent plane; sliders a and b control the point of interest, with objects yc, f, p, a, xc, A, g, b, fx, h, fy, B, u, v, n, r, q. ④ Suitable for university-level calculus or multivariable calculus courses, ideal for classroom demonstrations or student exploration to understand local linear approximation and the role of the gradient direction.