tangenthyperplane

A tangent hyperplane to a surface S in Euclidean space R^n at a point p ∈ S is the affine hyperplane that best approximates S near p, providing a linearized first-order description of the surface.

If S is a smooth hypersurface defined as a level set F(x) = c with F: R^n → R and ∇F(p) ≠ 0, then the tangent hyperplane at p is the set of points x ∈ R^n satisfying ∇F(p) · (x − p) = 0. This expresses the idea that the tangent hyperplane is orthogonal to the gradient and touches the surface at p.

If S is locally the graph of a differentiable function f: R^{n−1} → R, with p = (a, f(a)) and ∂f/∂x_i(p) denoting the partial derivatives, the tangent hyperplane has equation z = f(a) + ∑_{i=1}^{n−1} ∂f/∂x_i(a) (x_i − a_i). This form makes the plane explicit as a first-order linear approximation to the surface near p.

Uniqueness of the tangent hyperplane holds when the gradient ∇F(p) is nonzero; if the gradient vanishes at p, the tangent plane may not be well-defined or unique, signaling a singular point.

In applications, tangent hyperplanes are used for linear approximation in multivariable calculus, differential geometry, optimization, and numerical methods. They generalize to tangent spaces on manifolds and form a key link between local geometry and global structure. See also tangent space, gradient, and level-set methods.