Day 2: Introduction to Mathematical Physics

George E. Hrabovsky

MAST

Introduction

This is the first of a series of articles/book chapters/web pages that I intend to write about the nature and processes of mathematics in support of theoretical physics. I plan to author corresponding computational methods writings that will cover the use of Computer Algebra Systems (CAS) in theoretical physics. In this series, I will present physical principles as problems in mathematics without necessarily going into their physical details, beyond what is required to solve them as math problems. This particular writing will serve as both an introduction to mathematics and to support [5], the day one writing on theoretical physics. In each subsequent writing, I will explore in greater depth the mathematical ideas presented in the corresponding theoretical physics writing. These topics may lag behind the methods presented in the corresponding theoretical writings.

In its fundamental essence this will constitute a complete course in mathematical physics from the ground up. I assume that you have little background in physics other than what you are getting from the theoretical physics writings, and that you have had exposure to some algebra and geometry at the high school level. Beyond that you will need to have a certain mental agility to be able to follow arguments where I might throw mathematical methods at problems seemingly by magic.

In what follows, I will begin by giving an overview of logic. Then I will present a large number of arguments by logic that can be used in proofs, I will also give an example of a proof by truth tables and a more traditional proof. I will then make a list of arguments by algebraic manipulation, and I will give an example of a proof for such a manipulation. Following that I will list arguments involving limits and give an example of a proof of one of them. In the next section I will list arguments involving differentiation and give a sample proof of one of them. This will be follwed by a listing of arguments involving integration. I will then make a list of the things to do for day three, and list my conclusions and references.

Logic and proof

To begin with, I will need to present the basic method of mathematics. I will try to make this as basic as possible and still be useful. It is important to realize that in mathematics, until an idea is applied to something concrete, ideas have no meaning. Thus mathematics is the ultimate abstraction from reality; we speak of pure ideas without regard to meaning. It is best to think of mathematics at this level as a kind of structure.

To succeed in mathematics we need to consider several different notions:

Propositional Logic

In the table below you will find definitions and examples of the operations of the logic of propositions. It will be understood that a proposition will be symbolized as p, q, r, s, .... All of these symbols may be used in proofs.

({{Propositional Operation, Symbol, Meaning, Example}, {Negation, ¬, Not, ¬p}, {Conjunction, ∧, This And That, p ∧ q}, {Disjunction, ∨, This Or That, p ∨ q}, {Exclusive Disjunction, ⊻, This Or That But Not Both, p ⊻ q}, {Conditional, ⟹, If p, Then q, p⟹q}, {Converse, ⟹, Converse of If p, Then q
is If q, Then p., q⟹p}, {Contrapositive, ⟹, Contrapositive of If p, 
Then q is If Not-q,
Then Not-p., ¬q⟹¬p}, {Bicondictional, ⇔, p If and Only If q. If and
only if, is sometimes
written iff., p ⇔ q}})

From these symbols we can create logical formulas. The simplest formula is just the statement of a proposition, for example p, or if we are making a statement that a proposition p depends on another idea, say x we would write p(x).

Truth Tables

Every proposition, indeed every logical formula, is either true or false. We can create a table of these values using T for true, and F for false. When we make this array using all possible truth values, we call it a truth table. For example, we can create the truth table for the negation of a proposition p:

({{p, ¬p}, {T, F}, {F, T}})

Here is the truth table for the conjunction between two propositions p and q, where we list all possible truth values of the propositions and apply the definition of the conjunction to determine the resulting truth value.

({{p, q, p ∧ q}, {T, T, T}, {F, T, F}, {T, F, F}, {F, F, F}})

Here is the truth table for a somewhat complicated formula:

({{p, q, r, p ∧ q, ¬r, (p ∧ q) ∨ ¬ r}, {T, T, T, T, F, T}, {F, T, T, F, F, F}, {T, T, F, T, T, T}, {F, T, F, F, T, T}, {T, F, T, F, F, F}, {F, F, T, F, F, F}, {T, F, F, F, T, T}, {F, F, F, F, T, T}})

If two formulas have the same truth table result, then they are said to be logically equivalent. We would write p~q if p and q are logically equivalent. If a formula is always true, then it is called a tautology. If a formula is always false, then it is called a contradiction.

Basic Set Theory

The language of modern mathematics is a combination of logic and set theory. We understand a set to be a collection of objects of some kind. Here is a table of basic ideas from set theory.

({{Idea, Symbol, Meaning}, {Element of a Set, x∈X, x is an element of the set X.  }, {Subset of a Set, X⊆Y, The set Xis a subset of the
set Y if every element of X
is also an element of Y.                  }, {Equal Sets, X=Y, The set Xis equal to the set Y 
if every element of X is also 
an element of Yand every 
element of Y is also an 
element of X.}, {Unequal Sets, X≠Y, X and Yare not equal.           }, {Proper Subset, X⊂Y, X⊆Yand X≠Y.                          }})

Predicate Logic

Not all mathematical statements are propositions. Indeed, (Sqrt[(x/2)])=0, is neither true nor false as it is presented. It becomes a proposition only if we define x in some way. We need to develop a couple of additional ideas.

Proof Methods

In what follows, we will identify the starting proposition, the given, as the hypothesis and symbolize it by p. The conjecture to be proved, the conclusion, will be symbolized by q.

Proof by Truth Table

This is the most rudimentary style of proof. The primary limitation is the amount of work it requires, and the ever-expanding size of the resulting truth table. You begin by producing the truth table for the hypothesis, and then the conclusion; if they are the same, then they are logically equivalent, thus the hypothesis iff the conclusion.

Direct proof

This is at once the most effective proof and the most difficult. Here are the steps:

Often this process is ended by writing Q.E.D. standing for qoud erat demonstratum, meaning roughly, "Which was to be demonstrated."

Proof by contrapositive

The contrapositive and the conditional are logically equivalent, thus if we can prove the contrapositive, we have proven the conditional. We begin this method of proof by stating the conclusion.

Reductio ad absurdum (RAA)

I gave Galileo's example of this type of proof in the Day One Theoretical Physics Article.

Mathematical induction

This requires knowing that the natural numbers are 1, 2, 3, and so on.

Proof by cases - divide and conquer

The final style of proof is given in the next two sections:

Proof by cases - Bootstrap

We continue with the second method for case analysis:

Counterexamples

Up to now we have considered how to construct a mathematical proof. We can also disprove a conjecture by showing a single case where the conclusion is not true. Such an instance is called a counterexample of the conjecture.

Arguments by logic

The following are arguments of logic. It is a useful exercise to prove each of these, either by writing their truth tables, or by other methods.

({{Argument, Name, Formula, Explanation}, {1, Contradiction, (p ∧ ¬ p)⟹F, A proposition and its negation cannot 
both  be true.                                                              }, {2, Double
Negative, ¬(¬p)⟹p, The negation of a negation of a 
proposition  is  the proposition.                      }, {3, Law of the
Excluded
Middle, (p ∨ ¬ p), Either something is true or it is not. This 
is similar to argument 1. }, {4, Commutation, (p * q)⟺(q*p), This is true when you replace * with 
either  ∧ or ∨.                                                        }, {5, Associativity, (p*q)*r⟺
p*(q*r), This is true when you replace * with 
either  ∧ or ∨.                                                          }, {6, Law of the
Contrapositive, (p⟹q)⟺
(¬ q⟹¬ p), This is the basis for proof by 
contrapositive.                                                          }, {7, DeMorgan's
Laws, ¬ (p * q)⟺
  (¬p ∘ ¬q), This is true when you replace * with 
either ∧ or ∨and ∘ with either ∨ or ∧,
respectively.}, {8, Distribution, p*(q∘r)⟺
   (p*q)∘(p*r), This is true when you replace * with 
either  ∧ or ∨and ∘ with either ∨ or ∧, 
respectively.}})

Proof: Here is an example of a proof by truth table, we will prove Argument 1, Contradiction.

thus (p ∧ ¬ p)~Contradiction, which proves argument 1. QED.

Here is an example of how to discover a proof. We will prove Argument 2, Double Negative. We need to show that the double negative is equivalent to the initial proposition. We start by stating that the negation of a proposition always has the opposite truth value of a proposition, thus we can write

q=¬p.

The negation of q will then have the opposite truth value from q, we can write,

r=¬q.

Since a proposition is either true or false, when a negation is false the starting proposition is true. When r is false, then q must be true, this also means that p is false. Similarly when r is true q is false, and thus p is true. Therefore we see that r and p are the same. Since r is the double negative of p, then we can say that the double negative of any proposition is the same as the proposition. This has been a proof by cases. QED.

Arguments by algebraic manipulation

In this section I will assume that you are familiar with the basic operations of arithmetic: addition, subtraction, multiplication, division, exponentiation, and root-taking. Once again, it is a useful exercise to prove these arguments for yourself.

Algebra Table 1

({{Argument, Name, Formula, Meaning}, {9, Fraction Multiplication, (a/b)·(c/d)=((a·c)/(b·d)), Multiplying fractions involves 
first multiplying the 
denominators  and then the 
numerators.}, {10, Fraction Division, (a/b)÷(c/d)=((a·d)/(b·c)), Dividing fractions is just 
mutiplying the dividend by 
the inversion of the divisor.       }, {11, Cancellation, (a/a)=1, Common factors in the 
denominator and numerator 
can  be  replaced by 1, 
term-by-term.  Thus
 ((a+1)/a)=1+(1/a) and not 1.                  }, {12, Multplying a Term by 1, a·(b/b)=a·1=a, We can always multiply a term 
by 1. }, {13, Adding and Subtraction
Fractions, (a/b)±(c/d)=(a/b) (d/d)±(c/d)(b/b)
                =((a·d ± b·c)/(b·d)), Here we use argument 12 to
get the results we need.}, {14, Adding Terms, n · a + m · a + b =
 (n + m) · a + b, We can add like terms be 
adding their  coefficients, 
unlike  terms  cannot  be added.}, {15, Adding Opposite Sign
Integers, a + (-b) = a - b, Subtraction and adding 
opposite sign integers are equal.}, {16, Adding Same Sign
Integers, (-a) + (-b) = 
 -(a + b),  }, {17, Multiplying Opposite
Sign Integers, a · (-b) = -(a · b),  }, {18, Multiplying Same Sign
Integers, a · b = (-a)·(-b),  }, {19, Double Negative, -(-a) = a,  }, {20, Multiplying Exponents, (a^m)·(a^n)=(a^(m+n)),  }})

Algebra Table 2

({{Argument, Name, Formula, Meaning}, {21, Dividing Exponents, ((a^m)/(a^n))=(a^(m-n)),  }, {22, Reciprocal, (a^(-1))=(1/a),  }, {23, Negative Power, (a^(-n))=(1/(a^n)),  }, {24, Power of a Product, (((a·b))^n)=(a^n)·(b^n),  }, {25, Power of a Quotient, ((a^n)/(b^n))=((((a/b)))^n),  }, {26, Rewriting Division, (a/b)=a·(b^(-1)),  }, {27, Multiplying Same Sign
Integers, a · b = (-a)·(-b),  }, {28, Product of Roots, ((a·b)^(1/n))=(a^(1/n))·(b^(1/n)),  }, {29, Quotient of Roots, ((a/b)^(1/n)) =((a^(1/n))/(b^(1/n))),  }, {30, Power of a Root, ((a^m)^(1/n))=((((a^(1/n))))^m),  This also means that if m=n
then we have ((a^n)^(1/n))=|a|.}, {31, Root as an Reciprocal 
Power, (a^(1/n))=(a^(1/n)),  }, {32, Root as a Fractional 
Power, ((a^m)^(1/n))=(a^(m/n)),  }, {33, Commutative Property, a * b = b * a, Here we replace * by either 
+ or ·. }, {34, Associative Property, (a*b) * c=a* (b * c),  Here we replace * by either 
+ or ·. }, {35, Distributive Property, a · (b + c)=
(a · b )+( a · c), This is also the basis for 
factoring, in which case we 
reverse the  process. }, {36, Multiplying Binomials, (a + b)·(c + d) =
(a · c) + (a · d) +
(b · c) + (b · d),  }})

Proof of argument 9, Fraction Multiplication.

This has been a direct proof.

Arguments involving limits

In [5] we explored the idea of a limit. We begin with the formal definition of the limit:

Definition 1 The Limit: The limit of some function f(x) as x approaches some specific value a is symbolized by

(lim_(x->a)) f(x) = L.

While this definition is adequate, it will eventually be replaced by a more accurate one.

Limit Table 1

({{Argument, Name, Formula, Meaning}, {37, Constant Multiple 
Rule for Limits, (lim_(x->a)) [c f(x)]=
c (lim_(x->a)) f(x), The limit of a constant 
multiple  of a function 
is the constant 
multiple of the limit. }, {38, Sum/Difference Rule
for Limits, (lim_(x->a)) [f(x)±g(x)]= 
(lim_(x->a)) f(x)±(lim_(x->a)) g(x),  The limit of a sum is 
the sum of the limits.}, {39, Product Rule for
Limits, (lim_(x->a)) [f(x)·g(x)]= 
(lim_(x->a)) f(x)·(lim_(x->a)) g(x), The limit of a product 
is the product of the 
limits. }, {40, Quotient Rule for
Limits, (lim_(x->a)) [((f(x))/(g(x)))]=(((lim_(x->a)) f(x))/((lim_(x->a)) g(x))), The limit of a quotient 
is the  quotient of the 
limits.}, {41, Power Rule for Limits, (lim_(x->a)) (([f(x)])^n)=
(([(lim_(x->a)) f(x)])^n),  The limit of a power is 
the power of the limit.}, {42, Root Rule for Limits, (lim_(x->a)) ((f(x))^(1/n))=
(((lim_(x->a)) f(x))^(1/n)), The limit of an nth root 
is the nth root of the 
limit. }, {43, Constant Limit Rule, (lim_(x->a)) (c)=c, The limit of a constant 
is that constant. }, {44, Limiting Value, (lim_(x->a)) (x)=a,  }, {45, Power of a Limiting
Value, (lim_(x->a)) ((x^n))=(a^n),  }, {46, Limit of a Polynomial
p(x), (lim_(x->a)) p(x)=p(a),   }, {47, Limit Theorem 1, If f(x)≤g(x) for all x on
an interval [a,b], and
a≤c≤b, then
(lim_(x->c)) f(x)≤(lim_(x->c)) g(x),   }, {48, Squeeze Theorem, If f(x)≤h(x)≤g(x) and
(lim_(x->c)) f(x)=
(lim_(x->c)) g(x)=L
and a ≤ c ≤ b, then
(lim_(x->c)) h(x)=L,  }})

Limit Table 2

({{Argument, Name, Formula, Meaning}, {49, Infinite Limit, (lim_(x->a)) f(x)= ∞, A limit is considered to
be infinite  if and only 
if we can make f(x) 
arbitrarily large for all
values of x sufficiently 
close to aso long as
 x ≠ a. It is important to
remember that infinity is
not a number, it is a place
holder for a quantity that
grows without limit.}, {50, Negative Infinite
Limit, (lim_(x->a)) f(x)= -∞, A limit is considered to
be negatively infinite if 
and only if we can make
 f(x)arbitrarily large and 
negative for all values of 
x sufficiently close to a 
so long as x ≠ a. }, {51, Limits at Infinity, (lim_(x->±∞)) (c/(x^r))= 0, Infinity and zero are 
inverses.}, {52, Infinite Polynomial
Limit, (lim_(x->±∞)) p(x)= 
(lim_(x->±∞)) (Subscript[a, n])(x^n), The infinite limit of a 
polynomial
p(x)=(Subscript[a, n])(x^n)+(Subscript[a, n-1])(x^(n-1))
+...+(Subscript[a, 0])
is the same as the infinite 
limit  of the highest order 
term of the polynomial.}, {53, Continuous Functions, (lim_(x->a)) f(x)= f(a), A function is continuous, 
or smooth, if at any point
 a the limit of the function 
is the limit at that point. }})

Limit Table 3

({{Argument, Name, Formula, Meaning}, {54, Intermediate Value
Theorem (IVT), If f (x) is continuous on 
an interval [a, b] , and if n
 is a number such that 
f (a) ≤ n ≤ f (b), then
there exists some number
c such that, a < c < b, 
and f (c) = n., This is a special way of 
saying that every 
continuous function 
will take on all values 
between f (a) and f (b).   }})

Here is a proof of Argument 37, The Constant Multiple Rule for Limits. For this we immediately require a more precise definition of the limit than the one we have above. We need to define the absolute value,

|x|={({{x, x≥0}, {-x, x<0}}).

So, we redefine the limit,

Definition 1a: The Limit. For some function f(x) then

(lim_(x->a)) f(x) = L.

|f(x)-L|<ϵ

0<|x-a|<δ.

Proof of Argument 37: To accomplish this proof we need to show that the limit of the constant multiple of an arbitrary function is the same as the constant multiple of the limit of the function. It seems the most straightforward way to do this is to compute both expressions and show they are equivalent. Let us begin with the hypothesis, (lim_(x->a)) [c f(x)]. By the definition we have some any value ϵ>0 such that

| c f(x) - L | < ϵ

whenever there is a value δ>0 such that,

0<|x-a|<δ.

If we think about this for a while we realize that in this situation the limit L is the product of a different limit M and the constant c, by the definition of the limit. Thius gives us a nice clue as to how to complete the proof; we need to show that (1) is equivalent to |c| |f(x) - L | < ϵ. We begin by rewriting (1)

| c f(x) - c M | < ϵ.

By argument 35, (the distributive property) we can rewrite this,

|c| | f(x) - M | < ϵ.

So, by (1) we have (lim_(x->a)) [c f(x)]=M, and by (2) we have c (lim_(x->a)) f(x)=M, thus we have (lim_(x->a)) [c f(x)]=c (lim_(x->a)) f(x), QED.

Arguments involving differentiation

In [5] I introduced the idea of differential calculus. We will present the following definition of the derivative of a function:

Definition 2 The Derivative: The derivative of a function, f(t) is given as,

((dx)/(dt))=((d x)/(d t))=f'(t)=(Subscript[D, t])x=(lim_(Δ t->0))((Δ f(t))/(Δ t))=(lim_(Δ t->0))((f(t + Δ t) - f(t))/(Δ t)).

Differentiation Table 1

({{Argument, Name, Formula, Meaning}, {55, Differentiability,  , A function is differentiable at
some point a if f' (a) exists.      }, {56, Differentiability on 
an Interval,  , A function is differentiable on
an interval [a,b] if f' (t) 
exists for every point a≤t≤b.}, {57, Continuity,  , If f (t) is differentiable at 
t = a,then f (t) is
continuous at  t = a.}, {58, Slope of a Tangent
Line,  , The slope of a line tangent to a 
point a on f (t) is f' (a).                }, {59, Constant 
Derivative
Rule, (d/(dt))c=0, The derivative of a constant is
 0.                                                                   }, {60, Constant
Multiple
Rule, (d/(dt))c f(t)=c (d/(dt)) f(t), The derivative of a constant
multiple is the constant 
multiple of the derivative.           }, {61, Sum Rule, [f(t) ± g(t)]'=
f'(t) ± g'(t), The derivative of a sum is the
sum of the derivatives.                    }, {62, Power Rule, (d/(dt))(t^n)=n (t^(n-1)),  }, {63, Product Rule, [f(t) · g(t)]'=
f'(t) g(t)+ 
g'(t) f(t),  }, {64, Quotient Rule, (d/(dt))[((f(t) )/(g(t)))]=
((f'(t) g(t)- g'(t) f(t))/((g(t))^2)),  }, {65, Chain Rule, ((dx)/(dy))=((dx)/(dt))((dt)/(dy)), This allows you to change 
variables in differentiation. It is 
a direct application of 
argument 11, above.  }})

I will now prove Argument 60, The Constant Multiple Rule. We begin with the hypothesis.

Arguments involving integration

In [5] I introduced the idea of an integral. Given any function f(t), its antiderivative is the function F(t) such that,

F'(t) = f(t).

The most general antiderivative is called the indefinite integral and is written,

∫f(t)dt = F(t) + c.

Here are some arguments involving integration.

({{Argument, Name, Formula, Meaning}, {66, Constant Multiple
Rule, ∫k f(t)dt=k∫f(t)dt , This allows us to factor any
multiplicative constants out 
of the integrand.}, {67, Sum Rule for
Integrals, ∫f(t)±g(t)dt=
∫f(t)dt±∫g(t)dt, The integral of the sum is the 
sum of the integrals.}, {68, Power Rule for
Integrals, ∫(t^n)dt=((t^(n+1))/(n+1))+c, Here n≠-1.                                   }, {69, Constant Rule for
Integrals, ∫kdt=k t + c,  }, {70, Substitution Rule, ∫f( g(t)) g'(t)dt=
∫f(v)dv, Here we understand that
v = g(t). This is an 
application of the Chain Rule
(Argument 65) above.}, {71, Fundamental
Theorem of
Calculus, (∫_a^b)f(t)dt=
∫f(b)dt-∫f(a)dt, In essence this defines an
integral over a range from a
to b. Such an integral is called
a definite integral. The values a
and b are called the limits of
integration.}, {72, Interchanging the
Limits of Integration, (∫_a^b)f(t)dt=-(∫_b^a)f(t)dt, We can interchange the limits 
of integration by changing 
the sign.}, {73, The Same Limits, (∫_a^a)f(t)dt=0,  }, {74, Splitting the Limits
of Integration, (∫_a^b)f(t)dt=
(∫_a^c)f(t)dt+(∫_c^b)f(t)dt, We can split the limits of 
integration so long as
 a≤c≤b.                 }, {75, Equivalent Integrals, (∫_a^b)f(t)dt=(∫_a^b)f(x)dx,  }, {76, Constant Rule for
Definite Integrals, (∫_a^b)c dt=c·(b-a),   }})

I will prove Argument 65, the Constant Multiple Rule for Integration. Here we begin with the definition of the integral.

Things to do for Day Three

Begin building your library of references. Definitely begin with reference [4]. Reference [6] is a gold mine, and I use it now, even though I have the immensely powerful computer algebra system (CAS) Mathematica. Check out the web sites listed in [7]. Work some problems in logic, proofs, and calculus. Do not be concerned with how you did on them, just get some experience. Even getting a wrong answer is good as long as you try. If you do get a wrong answer, figure out what you did wrong.

Conclusions

I have presented a fairly good reference for beginning to explore the mathematics used in physics. This is a good beginning.

References

[1] Stan Gibilisco, (2005), Math Proof Demystified, McGraw-Hill Companies, Inc. This  book has good parts and parts that I do not like so much, and it is inexpensive, too.

[2] Steven Galovich, (1989), Introduction to Mathematical Structures, Harcourt Brace Jovanovich. This truly wonderful book is out of print, but you can still find it here and there. It is my favorite book on logic, proofs, and set theory. I picked up my copy at a used book store for only $8.

[3] Rhonda Huettenmueller, (2003),  Algebra Demystified, McGraw-Hill Companies, Inc. This is a pretty good book if you need a review of algebra.

[4] Paul Dawkins, (2007), Calculus I, available at http://tutorial.math.lamar.edu/pdf/CalcI/CalcI_Complete.pdf as a free download. Also available from this web site are Calculus II, Calculus III, Algebra, Linear Algebra, and Differential Equations. These books are all very nice.

[5] George E. Hrabovsky, (2009), Day 1: Introduction to Theoretical Physics. MASTers Notes, Issue 1, (http://www.madscitech.org/notes.html).

[6] Murray R. Speigel, (1995), Mathematical Handbook, 34th printing, McGraw-Hill, this is one of the Schaum's Outline series.