Gradient
Differential or (exterior) derivative

The best linear approximation to a differentiable function f: R n → R {\displaystyle f:\mathbb {R} ^{n}\to \mathbb {R} } at a point x {\displaystyle x} in R n {\displaystyle \mathbb {R} ^{n}} is a linear map from R n {\displaystyle \mathbb {R} ^{n}} to R {\displaystyle \mathbb {R} } which is often denoted by d f x {\displaystyle df_{x}} or D f (x) {\displaystyle Df(x)} and called the differential or total derivative of f {\displaystyle f} at x {\displaystyle x}. The function d f {\displaystyle df}, which maps x {\displaystyle x} to d f x {\displaystyle df_{x}}, is called the total differential or exterior derivative of f {\displaystyle f} 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 the function, 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 (∇ f) x ⋅ v = d f x (v) {\displaystyle (\nabla f)_{x}\cdot v=df_{x}(v)} for any v ∈ R n {\displaystyle v\in \mathbb {R} ^{n}}, where ⋅ {\displaystyle \cdot } 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 R n {\displaystyle \mathbb {R} ^{n}} is viewed as the space of (dimension n {\displaystyle n}) column vectors (of real numbers), then one can regard d f {\displaystyle df} as the row vector with components (∂ f ∂ x 1, …, ∂ f ∂ x n), {\displaystyle \left({\frac {\partial f}{\partial x_{1}}},\dots,{\frac {\partial f}{\partial x_{n}}}\right),} so that d f x (v) {\displaystyle df_{x}(v)} is given by matrix multiplication. Assuming the standard Euclidean metric on R n {\displaystyle \mathbb {R} ^{n}}, the gradient is then the corresponding column vector, that is, (∇ f) i = d f i T. {\displaystyle (\nabla f)_{i}=df_{i}^{\mathsf {T}}.}
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