# How does calculus relate to physics?

Mar 28, 2018

Many laws of physics involve differentiation and integration, so it is important to understand what these mean.

#### Explanation:

The first example most students meet is the idea that velocity is the rate of change of position. In one dimension $v = \dot{x}$ (where, throughout this answer, $\dot{x}$ is short for $\frac{\mathrm{dx}}{\mathrm{dt}}$.) Similarly acceleration $a = \dot{v} = \ddot{x}$. Conversely, velocity is the integral of acceleration and position is the integral of velocity: hence all the distance-time graphs and velocity-time graphs you were inflicted with.

Moving to three dimensions, all the ideas of calculus in one dimension carry over to three dimensions as vector calculus, so you get acceleration $\vec{a} = \dot{\vec{v}} = \ddot{\vec{r}}$, leading to Newton's Second Law $\vec{F} = m \vec{a} = m \frac{{d}^{2} \vec{r}}{{\mathrm{dt}}^{2}}$.

Basic vector calculus is straightforward. Just as $\frac{\mathrm{dx}}{\mathrm{dt}} = {\lim}_{h \to 0} \frac{f \left(t + h\right) - f \left(t\right)}{h}$ is the definition of scalar differentiation, vector differentiation is defined as $\frac{\mathrm{dv} e c r}{\mathrm{dt}} = {\lim}_{h \to 0} \frac{\vec{r} \left(t + h\right) - \vec{r} \left(t\right)}{h}$, that is, velocity is rate of change of position.

But calculus really comes into its own in exploring fields, which mean that you need to understand partial differentiation. This crops up when some scalar or vector quantity is a function not of one variable but two, three or four. For example, the potential energy V (a scalar quantity) of a body in a gravitational field will be described as $V \left(x , y , z\right)$, or the electric field $\vec{E}$ will be described as $\vec{E} \left(x , y , x\right)$ .

Partial differentiation is then the operation of finding the rate of change of $V$ or $E$ when one of the parameters changes while the others stay fixed.

Partial differential leads to the three very important vector operators: grad, div and curl which save an enormous amount of writing of individual components of vectors.

If you study electromagnetic fields, you will need Maxwell's Equations which express how electric and magnetic phenomena can be described as wave equations involving vector differential equations in three spatial dimensions and one time:

$c u r l \vec{E} = - {\mu}_{0} \frac{d \vec{B}}{\mathrm{dt}}$
$c u r l \vec{H} = {\epsilon}_{0} \frac{d \vec{E}}{\mathrm{dt}}$

Eventually you get onto tensors and Einstein's approach, which unifies time with the three spatial co-ordinates and makes the mathematical models consistent with reality.

IMHO, the ideas above are far more important than knowing how to differentiate ${x}^{3} {e}^{\sqrt{x}} \sin x$ which never comes up in physics!
I hope this road-map of how calculus relates to physics is useful.