Tangent plane to a graph

Definition

Let $U \subseteq \mathbb{R}^2$ be open, let $f \colon U \to \mathbb{R}$, and let $(a, b) \in U$ be a point at which $f$ is differentiable. The tangent plane to the graph $z = f(x, y)$ at the point $(a, b, f(a, b))$ is the plane in $\mathbb{R}^3$ with the equation $$z = f(a, b) + \frac{\partial f}{\partial x}(a, b)\,(x - a) + \frac{\partial f}{\partial y}(a, b)\,(y - b).$$ More generally, let $U \subseteq \mathbb{R}^n$ be open and let $f$ be differentiable at $a \in U$. The tangent hyperplane to the graph at $(a, f(a))$ is the set of points $(x, z) \in \mathbb{R}^n \times \mathbb{R}$ with $z = f(a) + \nabla f(a) \cdot (x - a)$.