Definition:Differential of Mapping/Real-Valued Function/Point

Definition
Let $U\subset\R^n$ be an open set.

Let $f : U \to \R$ be a real-valued function.

Let $f$ be differentiable at a point $x \in U$.

The differential of $f$ at $x$ is the linear transformation $df(x) : \R^n \to \R$ defined as:
 * $df(x)(h_1,\ldots,h_n) = \dfrac{\partial f}{\partial x_1}(x) \cdot h_1 + \cdots + \dfrac{\partial f}{\partial x_n}(x) \cdot h_n$

where:
 * $\dfrac {\partial f}{\partial x_i}(x)$ is the $i$th partial derivative of $f$ at $x$.


 * $\displaystyle \mathrm d f \left({x; h}\right) := \sum_{i \mathop = 1}^n \frac{\partial f}{\partial x_i} h_i = \frac{\partial f}{\partial x_1} h_1 + \frac{\partial f}{\partial x_2} h_2 + \cdots + \frac{\partial f}{\partial x_n} h_n$

where:
 * $h = \left({h_1, h_2, \ldots, h_n}\right) \in \R^n$
 * $\dfrac {\partial f}{\partial x_i}$ is the partial derivative of $f$ with respect to $x_i$.