Differential and Applications of Calculus Physics Notes
Differential Calculus:
Differentiation is an operation that makes us able to find out a function that outputs the rate of change of one variable with respect to another variable.
Let y be a physical quantity that depends upon another quantity, say x. So, when there is a change in the value of x, then the value of y also changes. For example, the derivative of the position of a moving object with respect to time is the velocity of the object, which measures how quickly the position of the object changes when time is changed.
Let, y = f(x)
It means that y is a function of x, and the value of y changes with the change in the value of x. Here, x is said to be an independent variable because it does not depend upon any other variable, whereas y is a dependent variable as it varies with the value of x.
Let there be a small change (increase) in the value of x, so that its value changes to x + δx. So, the value of y will also change to y + δy.
When δx approaches zero, then δy also approaches zero.
In mathematical form, the rate of change of y w.r.t. the change in x is written as \(\frac{d y}{d x}\).
A derivative is the limit of the ratio of the change in a function to the corresponding change in its independent variable as the t change in the latter approaches zero.
It can also be expressed as:
Here, we should note that differentiation is a mathematical process and d/dx is an operator which acts on y to give dy/ dx.
Applications of Differential Calculus in Physics:
There are many physical quantities that’1 on other quantities. For example distance defends upon time, velocity depends upon distance and Vme, pressure of a gas depends upon volume &nd temperature, etc. .
Let us assume that the displacement \(\vec{r}\) is function of time t, i.e.,
This quantity is the velocity of the object. So, the derivative of \(\vec{r}\) w.r.t. t is the velocity.
\(\vec{v}=\frac{d \vec{r}}{d t}=\frac{d}{d t}(\vec{r})\)
Similarly, the velocity \(\vec{v}\) depends upon time t. So, the rate of change of velocity is the acceleration:
Instantaneous acceleration,
\(\vec{a}=\frac{d \vec{v}}{d t}=\frac{d}{d t}\left(\frac{d \vec{r}}{d t}\right)=\frac{d^{2} r}{d t^{2}}\)
\(\frac{d^{2} r}{d t^{2}}\) is the second derivative of displacement vector \(\vec{r}\) w.r.t time (t).
Another example : Work done (W) by a force is a function of t and power P is the rate of change of work done.
∴ P = \(\frac{d W}{d t}\)
or, P = \(\frac{d E}{d t}\) (E = Energy)
dt
The number of active atoms (N) present in radioactive substance is a function of time and the rate of decay A is
A = – \(\frac{d N}{d t}\)