Definition:Differential of Mapping

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Definition

Real Function

Let $U \subset \R$ be an open set.

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

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


The differential of $f$ at $x$ is the linear transformation $\map {\d f} x : \R \to \R$ defined as:

$\map {\map {\d f} x} h = \map {f'} x \cdot h$

where $\map {f'} x$ is the derivative of $f$ at $x$.


Real-Valued Function

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 $x \in U$.

\(\ds \map {\d f} {x; h}\) \(=\) \(\ds \sum_{i \mathop = 1}^n \frac {\map {\partial f} x} {\partial x_i} h_i\)
\(\ds \) \(=\) \(\ds \frac {\map {\partial f} x} {\partial x_1} h_1 + \frac {\map {\partial f} x} {\partial x_2} h_2 + \cdots + \frac {\map {\partial f} x} {\partial x_n} h_n\)

where:

$h = \tuple {h_1, h_2, \ldots, h_n} \in \R^n$
$\dfrac {\partial f} {\partial x_i}$ is the partial derivative of $f$ with respect to $x_i$.


Vector-Valued Function

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

Let $f: U \to \R^m$ be a vector-valued function.

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


The differential of $f$ at $x$ is the linear transformation $\d \map f x: \R^n \to \R^m$ defined as:

$\map {\d \map f x} h = \map {J_f} x \cdot h$

where:

$\map {J_f} x$ is the Jacobian matrix of $f$ at $x$.


Manifolds

Let $M$ and $N$ be smooth manifolds with or without boundary.

Let $F : M \to N$ be a smooth mapping.

Let $T_p M$ be the tangent space at $p \in M$.


Then the differential of $F$ at $p$ is defined and denoted as the smooth mapping $\d F_p : T_p M \to T_{\map F p} N$ such that:

$\forall f \in \map {C^\infty} N : \forall p \in M : \forall v \in T_p M : \map {\map {\d F_p} v} f = \map v {f \circ F}$


Functional

Let $J \sqbrk y$ be a differentiable functional.

Let $h$ be an increment of the independent variable $y$.


Then the term linear with respect to $h$ is called the differential of the functional $J$, and is denoted by $\delta J \sqbrk {y; h}$.


Notation

There are various notations for the differential of a function $f$ at $x$:

  • $\map {\d f} x$
  • $\d f_x$
  • $\d_x f$
  • $\map {D f} x$
  • $D_x f$


Substituting $\d y$ for $\map {\d f} {x; h}$ and $\d x$ for $h$, the following notation emerges:

$\d y = \map {f'} x \rd x$

hence:

$\d y = \dfrac {\d y} {\d x} \rd x$


Warning

It is generally considered to be incorrect to consider $\d y$ as:

a small change in $y$ caused by a small change $\d x$ in $x$.

This is nearly true for small values of $\d x$, but will only ever be exactly true when $f$ has a graph which is a straight line.

If it is necessary to talk about small changes then the notation $\delta x$ and $\delta y$ are to be used instead.

Thus:

$\ds \lim_{\delta x \mathop \to 0} \delta y = \frac {\d y} {\d x} \delta x$


Received wisdom tells us that an even worse misconception is the idea that $\d y$ and $\d x$ are infinitesimal quantities which are obtained by letting $\delta x$ and $\delta y$ tend to zero.

Then $\dfrac {\d y} {\d x}$ could be regarded as the quotient of these quantities, and the whole concept of a limit could be disposed of. This was the original idea that Isaac Newton based his Theory of Fluxions on. However, useful as this approach is, it is generally considered that does not have any logical basis.

However, the field of nonstandard analysis is an attempt to address these concerns from a modern perspective.


Also see

  • Results about differentials can be found here.


Sources

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