Table Of ContentPhysics 501
Math Models of Physics Problems
Luis Anchordoqui
Wednesday, September 4, 13 1
Bibliography
L. A. Anchordoqui and T. C. Paul,
``Mathematical Models of Physics Problems’‘
(Nova Publishers, 2013)
G. F. D. Duff and D. Naylor,
``Differential Equations of Applied Mathematics’‘
(John Wiley & Sons, 1966)
G. B. Arfken, H. J. Weber, and F. E. Harris
``Mathematical Methods for Physicists’’ (7th Edition)
(Academic Press, 2012)
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Provisional Course Outline
(Please note this may be revised during the course
to match coverage of material during lectures, etc.)
1st week - Elements of Linear Algebra
2nd week - Analytic Functions
3rd week - Integration in the Complex Plane
4th week - Isolated Singularities and Residues
5th week - Initial Value Problem (Picard’s Theorem)
6th week - Initial Value Problem (Green Matrix)
7th week - Boundary Value Problem (Sturm-Liouville Operator)
8th week - Midterm-exam (October 23)
9th week - Boundary Value Problem (Special Functions)
10th week - Fourier Series and Fourier Transform
11th week - Hyperbolic Partial Differential Equation (Wave equation)
12th week - Parabolic Partial Differential Equation (Diffusion equation)
13th week - Elliptic Partial Differential Equation (Laplace equation)
14th week - Midterm-exam (December 11)
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Elements of Linear Algebra
1.1 Linear Spaces
1.2 Matrices and Linear Transformations
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Linear Spaces
Definition 1.1.1.
F +
field
A is a set together with two operations and
·
�, µ, ⌫ F :
for which all axioms below hold
8 2
(i) closure sum � + µ and product � µ again belong to F
� � ·
(ii) associative law � + (µ + ⌫) = (� + µ) + ⌫ & � (µ ⌫) = (� µ) ⌫
� � · · · ·
(iii) commutative law � + ⌫ = ⌫ + � & � µ = µ �
� � · ·
(iv) distributive laws � (µ + ⌫) = � µ + � ⌫
� � · · ·
(� + µ) ⌫ = � ⌫ + µ ⌫
and
· · ·
(v) existence of an additive identity there exists an element
� �
0 F for which � + 0 = �
2
(vi) existence of a multiplicative identity
there exists an element
� �
1 F with 1 = 0 for which 1 � = �
2 6 ·
(vii) existence of additive inverse
to every � F there corresponds
� �
2
an additive inverse � such that � + � = 0
� �
(viii) existence of multiplicative inverse to every � F
� � 2
1 1
� � � = 1
there corresponds a multiplicative inverse � such that �
·
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Example 1.1.1.
F
Underlying every linear space is a field
examples are R and C
Definition 1.1.2.
A linear space V is a collection of objects
with a (vector) addition and scalar multiplication defined
which is closed under both operations
Such a vector space satisfies following axioms:
➢ commutative law of vector addition
x + y = y + x, x, y V
8 2
➢ associative law of vector addition
x + (y + w) = (x + y) + w, x, y, w V
8 2
0 x + 0 = x, x V
➢ There exists a zero vector such that
8 2
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x V
➢ To every element
2
x
there corresponds an inverse element
�
x + ( x) = 0
such that
�
➢ associative law of scalar multiplication
(�µ ) x = � (µ x), x V and �, µ F
8 2 2
➢ distributive laws of scalar multiplication
(� + µ) x = � x + µ x, x V and �, µ F
8 2 2
� (x + y) = � x + � y, x, y V and � F
8 2 2
1 x = x, x V
➢
· 8 2
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Example 1.1.2.
n
Cartesian space R is prototypical example
n
of real -dimensional vector space
x = (x , . . . , x ) n x
Let 1 n be an ordered tuple of real numbers i
x
to which there corresponds a point with these Cartesian
x
coordinates and a vector with these components
We define addition of vectors by component addition
x + y = (x + y , . . . , x + y )
1 1 n n (1.1.1.)
and scalar multiplication by component multiplication
� x = (�x , . . . ,�x ) (1.1.2.)
1 n
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Definition 1.1.3.
Given a vector space V over a field F
a subset W of V is called subspace
if W is vector space over F under operations already defined on V
Corollary 1.1.1
W V V
A subset of a vector space is a subspace of
,
(i)W
is nonempty (ii) if x, y W ☛ then x + y W
2 2
(iii) x W and � F ☛ then � x W
2 2 · 2
After defining notions of vector spaces and subspaces
next step is to identify functions that can be used
to relate one vector space to another
Functions should respect algebraic structure of vector spaces
so we require they preserve addition and scalar multiplication
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Definition 1.1.4.
V W F
Let and be vector spaces over field
A linear transformation from V to W is a function T : V W
!
T (�x + µy) = �T (x) + µT (y)
such that (1.1.3.)
for all vectors x, y V �, µ F
and all scalars
2 2
If a linear transformation is one-to-one and onto
it is called vector space isomorphism ☛ or simply isomorphism
Definition 1.1.5.
S = x , , x V F
Let be a set of vectors in vector space over field
1 n
· · ·
n
� F
y = � x
Any vector of form i i for i
2
i=1
X S
is called linear combination of vectors in
S V V
Set is said to span if each element of
S
can be expressed a s linear combination of vectors in
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Description:G. B. Arfken, H. J. Weber, and F. E. Harris ``Mathematical Models of Physics
Problems'' . We define addition of vectors by component addition and scalar
