# Definition:Differential of Mapping

## 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

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

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

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

Then the **differential of $F$**, denoted by $d F$, is the mapping $d F_p : T_p M \to T_{\map F p} N$ such that:

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

### 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

- Definition:Jacobian Matrix
- Definition:Derivative
- Definition:Differentiable Mapping
- Definition:Differential Form
- Definition:Exterior Derivative

- Results about
**differentials**can be found**here**.

## Sources

- 1998: David Nelson:
*The Penguin Dictionary of Mathematics*(2nd ed.) ... (previous) ... (next):**differential** - 2008: David Nelson:
*The Penguin Dictionary of Mathematics*(4th ed.) ... (previous) ... (next):**differential**