Editing Gradient (section)

====Differential or (exterior) derivative====
The best linear approximation to a differentiable function
<math display="block">f : \R^n \to \R</math>
at a point <math>x</math> in <math>\R^n</math> is a linear map from <math>\R^n</math> to <math>\R</math> which is often denoted by <math>df_x</math> or <math>Df(x)</math> and called the [[differential (calculus)|differential]] or [[total derivative]] of <math>f</math> at <math>x</math>. The function <math>df</math>, which maps <math>x</math> to <math>df_x</math>, is called the [[total differential]] or [[exterior derivative]] of <math>f</math> and is an example of a [[differential 1-form]].

Much as the derivative of a function of a single variable represents the [[slope]] of the [[tangent]] to the [[graph of a function|graph]] of the function,<ref>{{harvtxt|Protter|Morrey|1970|pp=21,88}}</ref> the directional derivative of a function in several variables represents the slope of the tangent [[hyperplane]] in the direction of the vector.

The gradient is related to the differential by the formula
<math display="block">(\nabla f)_x\cdot v = df_x(v)</math>
for any <math>v\in\R^n</math>, where <math>\cdot</math> is the [[dot product]]: taking the dot product of a vector with the gradient is the same as taking the directional derivative along the vector.

If <math>\R^n</math> is viewed as the space of (dimension <math>n</math>) column vectors (of real numbers), then one can regard <math>df</math> as the row vector with components
<math display="block">\left( \frac{\partial f}{\partial x_1}, \dots, \frac{\partial f}{\partial x_n}\right),</math>
so that <math>df_x(v)</math> is given by [[matrix multiplication]]. Assuming the standard Euclidean metric on <math>\R^n</math>, the gradient is then the corresponding column vector, that is,
<math display="block">(\nabla f)_i = df^\mathsf{T}_i.</math>