Differentials

We refer to $dx$ and $df$ as differentials. $df=f^{\prime }(x)dx$

Example

Use differentials to approximate $\sqrt{78}$.

$df=f^{\prime }(x)dx$ $df=\frac{1}{2\sqrt{x}}dx$ $=\frac{1}{2\sqrt{81}}(-3)$ $=-\frac{1}{6}$ That is, when $x$ decreases by 3, $f(x)$ decreases by approximately $\frac{1}{6}$. Thus,

$\sqrt{78}\approx \sqrt{81}+df$ $\sqrt{78}\approx 9+(-\frac{1}{6})$ $\sqrt{78}\approx \frac{53}{6}$

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 $d\vec{l}$) in an arbitrary direction then we will take a small step $dx$ in $\hat{x}$ direction, a small step $dy$ in $\hat{y}$ direction and a small step $dz$ in $\hat{z}$ direction. Thus, a differential length element $d\vec{l}$ will be $\overrightarrow{dl}=dx\hat{x}+dy\hat{y}+dz\hat{z}$

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)