quality:corruption, structural:expansion
Multivariable differentiability

A function f (x) is differentiable in a neighborhood of a point a if there is an n -tuple of numbers dependent on a in general, A (a) = (A (a), A (a), …, A (a)), so that:

where α (x) → 0 {\displaystyle \alpha ({\boldsymbol {x}})\to 0} as | x − a | → 0 {\displaystyle |{\boldsymbol {x}}-{\boldsymbol {a}}|\to 0}. This means that if f is differentiable at a point a, then f is continuous at x = a, although the converse is not true - continuity in the domain does not imply differentiability in the domain. If f is differentiable at a then the first order partial derivatives exist at a and:

for i = 1, 2, …, n, which can be found from the definitions of the individual partial derivatives, so the partial derivatives of f exist.

Assuming an n -dimensional analogue of a rectangular Cartesian coordinate system, these partial derivatives can be used to form a vectorial linear differential operator, called the gradient (also known as " nabla " or " del ") in this coordinate system:

used extensively in vector calculus, because it is useful for constructing other differential operators and compactly formulating theorems in vector calculus.

Then substituting the gradient ∇ f (evaluated at x = a) with a slight rearrangement gives:

where · denotes the dot product. This equation represents the best linear approximation of the function f at all points x within a neighborhood of a. For infinitesimal changes in f and x as x → a:

which is defined as the total differential, or simply differential, of f, at a. This expression corresponds to the total infinitesimal change of f, by adding all the infinitesimal changes of f in all the x directions. Also, df can be construed as a covector with basis vectors as the infinitesimals dx in each direction and partial derivatives of f as the components.

Geometrically ∇ f is perpendicular to the level sets of f, given by f (x) = c which for some constant c describes an (n − 1) -dimensional hypersurface. The differential of a constant is zero:

in which d x is an infinitesimal change in x in the hypersurface f (x) = c, and since the dot product of ∇ f and d x is zero, this means ∇ f is perpendicular to d x.

In arbitrary curvilinear coordinate systems in n dimensions, the explicit expression for the gradient would not be so simple - there would be scale factors in terms of the metric tensor for that coordinate system. For the above case used throughout this article, the metric is just the Kronecker delta and the scale factors are all 1.