 # Tangent plane

Consider the surface, $LaTeX: \textstyle S = \{x \mbox{ in } R^n : h(x)=0\},$ where $LaTeX: \textstyle h \in C^1.$ A differentiable curve passing thru $LaTeX: \textstyle x^* \in S$ is $LaTeX: \textstyle \{x(t): x(0)=x* \mbox{ and } h(x(t))=0 \mbox{ for all } t \in (-e,e)\},$ for which the derivative, $LaTeX: x'(t),$ exists, where $LaTeX: e > 0.$ The tangent plane at $LaTeX: x^*$ is the set of all initial derivatives: $LaTeX: \{x'(0)\}.$ (This is a misnomer, except in the special case of one function and two variables at a non-stationary point.) An important fact that underlies the classical Lagrange multiplier theorem when the rank of $LaTeX: \textstyle \nabla h(x^*)$ is full row ( $LaTeX: x^*$ is then called a regular point): the tangent plane is $LaTeX: \textstyle \{d: \nabla h(x^*) d = 0\}.$
Extending this to allow inequalities, the equivalent of the tangent plane for a regular point $LaTeX: (x^*)$ is the set of directions that satisfy first-order conditions to be feasible: $LaTeX: \{d: \nabla h(x*) d = 0 \mbox{ and } \nabla g_{i(x*)} d \le 0 \mbox{ for all } i: g_{i(x*)} = 0 \}.$