# Definition:Differential

## Contents

## 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 $\rd f \left({x}\right) : \R \to \R$ defined as:

- $\rd f \left({x}\right) \left({h}\right) = f' \left({x}\right) \cdot h$

where $f' \left({x}\right)$ 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 $x \in U$.

- $\displaystyle \d f \left({x; h}\right) := \sum_{i \mathop = 1}^n \frac {\partial f \left({x}\right)} {\partial x_i} h_i = \frac {\partial f \left({x}\right)} {\partial x_1} h_1 + \frac {\partial f \left({x}\right)} {\partial x_2} h_2 + \cdots + \frac {\partial f \left({x}\right)} {\partial x_n} h_n$

where:

- $h = \left({h_1, h_2, \ldots, h_n}\right) \in \R^n$
- $\dfrac {\partial f} {\partial x_i}$ 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 $\d f \left({x}\right): \R^n \to \R^m$ defined as:

- $\d f \left({x}\right) \left({h}\right) = J_f \left({x}\right) \cdot h$

where:

- $J_f \left({x}\right)$ is the Jacobian matrix of $f$ at $x$.

### Manifolds

Definition:Differential/Manifolds

### 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 $x$:

- $\d f \left({x}\right)$
- $\d f_x$
- $\d_x f$
- $D f \left({x}\right)$
- $D_x f$

Substituting $\d y$ for $\d f \left({x; h}\right)$ and $\d x$ for $h$, the following notation emerges:

- $\d y = f' \left({x}\right) \rd x$

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:

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

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 non-standard analysis is an attempt to address these concerns from a modern perspective.