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Advanced General Physics I
Fall 2019
Lecture 2
Differential and integral calculus and exponential
and logarithmic functions
Mathematics is the language of physics. We would not be able to seriously discuss physics
without knowing some basic ideas of mathematics. Before we begin our study of classical
mechanics, we’ll review most of the mathematics that we will be need. The math subjects
that we’ll cover are: differential calculus, integral calculus, complex variables, trigonometry,
differential equations, vectors and vector calculus. For some of you, this will be a review
while others might be seeing the topics for the first time. Despite the impression that you
might have from high school or college math courses, these are not difficult subjects. There
are only a few basics things that you need to know and we will cover them as simply as
possible. In this lecture, we’ll cover differential and integral calculus and exponential and
logarithmic functions
Mathematics is a collection of topics: calculus, number theory, geometry, etc. Each
topic is based on a set of axioms and postulates on which is built a full structure through
theorems derived through logical proofs using the axioms and previously derived theorems.
Math would seem to have nothing to do with “reality” but rather exists in its own Platonic
world based solely on axioms and theorems.
Oneofthegreat, perhaps the greatest, mysteries is why mathematics provides a description
andexplanation of the real world. We don’t know why this relationship between mathemat-
ics and physics exists and perhaps we never will. In the meanwhile, it is a continuous source
of wonderment that abstract mathematics is somehow fundamentally related to reality and
constrains the way nature behaves.
The roles of mathematicians and physicists are very different. Mathematicians build a
Platonic world with no concern for reality. They only care that this world is consistent and
based on rigorous proofs. Physicists, on the other hand, simply want to use mathematics as
a tool to describe nature and are generally not concerned with the rigor of the mathematics.
For them, the correctness of the mathematics comes from its ability to provide a correct
description of nature. If the physics that they develop from the mathematics gives a good
description of nature, then it is assumed that the math is right and that it could be shown
to be rigorous if desired. This is very troubling to mathematicians who don’t have the real
world as a test but can rely only on the proven rigor of the mathematics.
Throughout history, there has been a close symbiotic relationship between physics and
mathematics. Sometimes physicists develop a new field in mathematics to help them solve
some problem in physics. A prime example is the creation of calculus by Newton that
allowed him to develop a theory of mechanics. Other times physicists will find that some
abstract field in mathematics plays a fundamental role in physics. An example of this that
we will soon discuss is vectors. Vector spaces are a purely abstract concept developed by
mathematicians that was later found to have many important applications in physics. One
of the most remarkable examples of what would seem to be purely Platonic mathematics
is the fundamental and essential role that complex numbers play in the theory of quantum
mechanics.
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1. Differential Calculus
1.1 The basic idea of differential calculus
The first math topic that we will discuss is calculus. Newton invented calculus as part of
his effort to develop a theory of mechanics. A working knowledge of calculus is essential
for doing physics. There are two types of calculus: differential and integral. We will first
discuss differential calculus.
Differential calculus is a procedure or set of rules for determining the slope of a line tangent
to the curve of a function at a point. For example, if we have a function, f(x), the derivative
of f(x) with resect to x evaluated at x gives the slope of the tangent line at x , as shown
0 0
in the figure below. We express the derivative as:
df(x) ∆f(x) f(x )−f(x )
f′(x) = = lim = lim 2 1
dx ∆x→0 ∆x x ,x →x x −x
1 2 0 2 1
where I’ve used here the common notation of writing the derivative of f(x) as f′x.
exp(x)
f(x)
tangent line
x
0 x
That’s it. That’s all that differential calculus is about, finding slopes of tangents to curves
of functions.
1.2 Rules of differential calculus
The following are a few rules for determining derivatives that are important to know.
1) The differentiation is a linear operation. That means that:
d[af(x)] = a df(x) and d [f(x)+g(x)] = df(x) + dg(x)
dx dx dx dx dx
2) The derivative of a power of x is given by:
n
dx = nxn−1
dx
3) The derivative of the product of two functions f(x) and g(x) is given by the Leibniz
or product rule:
2
d [f(x)g(x)] = df(x) g(x) + f(x) dg(x)
dx dx dx
4) The derivative of a function of of f(x), g(f(x)), with respect to x is given by the
so-called chain rule:
dg(f(x)) = dg(f(x)) df(x)
dx df(x) dx
Some of you might know about the quotient rule. This is not a separate rule but is just
a combination of the product rule with the chain rule. For those of you who might be
interested, here is how to show that:
d f(x) = df(x) 1 + f(x) d 1
dx g(x) dx g(x) dx g(x)
d f(x) = df(x) 1 + f(x)d[g−1(x)]
dx g(x) dx g(x) dx
= df(x) 1 + f(x)d[g−1(x)] d[g(x)]
dx g(x) dg(x) dx
= df(x) 1 − f(x) 1 dg(x)
dx g(x) g2 dx
= df(x)/dx − f(x)dg(x)/dx
g(x) g2(x)
= g(x)df(x)/dx − f(x)dg(x)/dx
g2(x) g2(x)
= g(x)f′(x)−f(x)g′(x)
g2(x)
1.3 Second derivatives
The second derivative of a function is the derivative of the derivative. In other words, it
2 2
gives the slope of the slope of the function. The illogical notation of d /dx is commonly
used to denote the second derivative. The second derivative is the derivative operator acting
twice.
2 ′
d f(x) = d df(x) = df (x)
dx2 dx dx dx
Note that the second derivative is also a linear operation:
2 2 2 2 2
d [af(x)] = a d f(x) and d [f(x)+g(x)] = d f(x) + d g(x)
2 2 2 2 2
dx dx dx dx dx
3
Keep in mind that the second derivative is simply the derivative operator operating on a
function twice. It is definitely not the square of the first derivative.
d2f(x) 6= df(x)2
dx2 dx
1.4 Partial derivatives
We will often need the derivative of a multivariable function with respect to one of the
variables. This is called the partial derivative. We use the symbol ∂/∂x rather than d/dx
to indicate that we are taking the derivative with respect to one variable while the other
variables are held fixed. For example:
∂f(x,y)
∂x
means to take the derivative of f(x,y) with respect to x while holding the value of y fixed.
1.5 Finding the minimum and maximum of a function
An important use of differential calculus is in finding the extrema, the local maxima and
minima, of a function. As shown in the figure below, at a local maximum or minimum
the slope of the function is zero, that is, the derivative of the function is zero. The first
derivative alone, however, does not tell us if the extremum is a local maximum or a local
minimum. For that, we need to look at the second derivative of the function. If the second
derivative is negative then it is a local maximum. If the second derivative is positive then
it is a local minimum, as shown in the figure.
f(x) df d2f
dx =0 dx2 < 0
x**2-x**4+1. maxima
1.2
1.1
1
minima
0.9 df d2f
=0 2 > 0
0.8 dx dx
−0.4 −0.2 0 0.2 0.4 0.6 0.8 1
x
1.6 Role of differential calculus in physics
Differential calculus plays a prominent role in physics. One of the most common is in
determining the velocity of a particle by taking the derivative of the position of the particle
with respect to time:
v = dx(t)
dt
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