# Differentials

We refer to $dx$ and $df$ as differentials. $df=f_{′}(x)dx$

### Example

Use differentials to approximate $78 $.

$df=f_{′}(x)dx$ $df=2x 1 dx$ $=281 1 (−3)$ $=−61 $ That is, when $x$ decreases by 3, $f(x)$ decreases by approximately $61 $. Thus,

$78 ≈81 +df$ $78 ≈9+(−61 )$ $78 ≈653 $

### Multivariate Differentials

Same thing, but use $df=f_{x}dx+f_{y}dy$

## Physics Stuff

### Differential Length Element

we want to make a small differential length (let us call it $dl$) in an arbitrary direction then we will take a small step $dx$ in $x$ direction, a small step $dy$ in $y $ direction and a small step $dz$ in $z$ direction. Thus, a differential length element $dl$ will be $dl=dxx+dyy +dzz$

### Differential Elements

To define arbitrary shapes, we use differential elements.

Thus, there are three differential elements in length i.e. $dx$, $dy$, and $dz$. $dx$ is a small step we take in the 𝑥̂ direction, $dy$ is a small step we take in the 𝑦̂ direction and dz is a small step we take in 𝑧̂ direction

Differential surface elements ($dS$) are very-very small surfaces defined by two differential elements)