fiziks Institute for NET/JRF, GATE, IIT‐JAM, JEST, TIFR and GRE in PHYSICAL SCIENCES
fiziks Forum for CSIR-UGC JRF/NET, GATE, IIT-JAM/IISc, JEST, TIFR and GRE in PHYSICS & PHYSICAL SCIENCES
Mathematical Physics
(IIT-JAM/JEST/TIFR/M.Sc Entrance)
Head office fiziks, H.No. 23, G.F, Jia Sarai, Near IIT, Hauz Khas, New Delhi‐16 Phone: 011‐26865455/+91‐9871145498 Website: www.physicsbyfiziks.com
Branch office Anand Institute of Mathematics, of Mathematics, 28‐B/6, Jia Sarai, Near IIT Hauz Khas, New Delhi‐16
fiziks Institute for NET/JRF, GATE, IIT‐JAM, JEST, TIFR and GRE in PHYSICAL SCIENCES MATHEMATICAL METHODS 1(A). Vector Analysis 1.1(A) Vector Algebra..................................................................................................(1-7) 1.1.1 Vector Operations
1.1.2 Vector Algebra: Component Form 1.1.3 Triple Products 1.1.4 Position, Displacement, and Separation Vectors 1.2(A) Differential Calculus......................................................... ............................(8-16) 1.2.1 “Ordinary” Derivatives 1.2.2 Gradient 1.2.3 The Operator ∇ 1.2.4 The Divergence 1.2.5 The Curl 1.2.6 Product Rules 1.2.5 Second Derivatives 1.3(A) Integral Calculus............................................................ .............................(16-27) 1.3.1 Line, Surface, and Volume Integrals 1.3.2 The Fundamental Theorem of Calculus 1.3.3 The Fundamental Theorem for Gradients 1.3.4 The Fundamental Theorem for Divergences 1.3.5 The Fundamental Theorem for Curls 1.4(A) Curvilinear Coordinates................................................... ..........................(28-39) 1.4.1 Spherical Polar Coordinates
1.4.2 Cylindrical Polar Coordinates 1.5(A) The Dirac Delta Function............................................................................(39-41) 1.5.1 The Divergence of r ˆ / r 2
1.5.2 The One- Dimensional Dirac Delta Function 1.5.3 The Three-Dimensional Delta Function 1.6(A) The Theory of Vector Fields............................................................................(42)
1.6.1 The Helmholtz Theorem 1.6.2 Potentials Questions and Solutions..........................................................................................(43-57) Head office fiziks, H.No. 23, G.F, Jia Sarai, Near IIT, Hauz Khas, New Delhi‐16 Phone: 011‐26865455/+91‐9871145498 Website: www.physicsbyfiziks.com
Branch office Anand Institute of Mathematics, of Mathematics, 28‐B/6, Jia Sarai, Near IIT Hauz Khas, New Delhi‐16
fiziks Institute for NET/JRF, GATE, IIT‐JAM, JEST, TIFR and GRE in PHYSICAL SCIENCES 1. Linear Algebra and Matrices…………………… Matrices………………………………………………… ……………………………(58-100) (58-100)
1.1 Linear Dependence and Dimensionality of a Vector Space 1.2 Properties of Matrices 1.3 Eigen value problem 1.4 Different Types of Matrices and their properties 1.5 Cayley–Hamilton Theorem 1.6 Diagonalisation of Matrix 1.7 Function of Matrix 2. Complex Number……………………………… Number………………………………………………………… …………………………….(101-147) ….(101-147)
2.1 Definition 2.2 Geometric Representation of Complex Numbers 2.3 De Moivre’s Theorem 2.4 Complex Function 2.4.1 Exponential Function of a Complex Variable 2.4.2 Circular Functions of a Complex Variable 2.4.3 Hyperbolic Functions 2.4.4 Inverse Hyperbolic Functions 2.4.5 Logarithmic Function of a Complex Variable 2.5 Summation of Series
C + iS Method
3. Fourier Series………………………… Series…………………………………………………… ……………………………………..(148-184) …………..(148-184)
3.1 Half-Range Fourier Series 3.2 Functions defined in two or more sub-ranges 3.3 Complex Notation for Fourier series 4 Calculus of Single S ingle and Multiple Variables…………………………………(185-220)
4.1 Limits 4.1.1 Right Hand and Left hand Limits 4.1.2 Theorem of Limits 4.1.3 L’Hospital’s Rule 4.1.4 Continuity Head office fiziks, H.No. 23, G.F, Jia Sarai, Near IIT, Hauz Khas, New Delhi‐16 Phone: 011‐26865455/+91‐9871145498 Website: www.physicsbyfiziks.com
Branch office Anand Institute of Mathematics, of Mathematics, 28‐B/6, Jia Sarai, Near IIT Hauz Khas, New Delhi‐16
fiziks Institute for NET/JRF, GATE, IIT‐JAM, JEST, TIFR and GRE in PHYSICAL SCIENCES
4.2 Differentiability 4.2.1 Tangents and Normal 4.2.2 Condition for tangent to be parallel or perpendicular to x-axis 4.2.3 Maxima and Minima 4.3 Partial Differentiation 4.3.1 Euler theorem of Homogeneous function 4.3.2 Maxima and Minima (of function of two independent variable) 4.4 Jacobian 4.4.1 Properties of Jacobian 4.5 Taylor’s series and Maclaurine series expansion 4.5.1 Maclaurine’s Development 5. Differential Equations of the first Order and first Degree………………(221-244)
5.1 Linear Differential Equations of First Order 5.1.1 Separation of the variables 5.1.2 Homogeneous Equation 5.1.3 Equations Reducible to homogeneous form 5.1.4 Linear Differential Equations 5.1.5 Equation Reducible to Linear Form 5.1.6 Exact Differential Equation 5.1.7 Equations Reducible to the Exact Form 5.2 Linear Differential Equations of Second Order with constant Coefficients
Head office fiziks, H.No. 23, G.F, Jia Sarai, Near IIT, Hauz Khas, New Delhi‐16 Phone: 011‐26865455/+91‐9871145498 Website: www.physicsbyfiziks.com
Branch office Anand Institute of Mathematics, 28‐B/6, Jia Sarai, Near IIT Hauz Khas, New Delhi‐16
fiziks Institute for NET/JRF, GATE, IIT‐JAM, JEST, TIFR and GRE in PHYSICAL SCIENCES 1(A).Vector Analysis 1.1
Vector Algebra
Vector quantities have both direction as well as magnitude such as velocity, acceleration, force
and momentum etc. We will use A for any general vector and its magnitude by A . In diagrams vectors are denoted by arrows: the length of the arrow is proportional to the magnitude of the
vector, and the arrowhead indicates its direction. Minus A ( − A ) is a vector with the same
magnitude as A but of opposite direction.
Α −Α
1.1.1 Vector Operations
We define four vector operations: addition and three kinds of multiplication. (i) Addition of two vectors
Place the tail of B at the head of A ; the sum, A + B , is the vector from the tail of A to the head
of B .
Addition is commutative: A + B = B + A
(
)
(
)
Addition is associative: A + B + C = A + B + C
( )
To subtract a vector, add its opposite: A − B = A + − B
Β A
−Β Β+Α
Α+Β
)
)
Α
Α−Β
Α
Β
Head office fiziks, H.No. 23, G.F, Jia Sarai, Near IIT, Hauz Khas, New Delhi‐16 Phone: 011‐26865455/+91‐9871145498 Website: www.physicsbyfiziks.com
Branch office Anand Institute of Mathematics, 28‐B/6, Jia Sarai, Near IIT Hauz Khas, New Delhi‐16
fiziks Institute for NET/JRF, GATE, IIT‐JAM, JEST, TIFR and GRE in PHYSICAL SCIENCES (ii) Multiplication by scalar
Multiplication of a vector by a positive scalar a, multiplies the magnitude but leaves the direction unchanged. (If a is negative, the direction is reversed.) Scalar multiplication is distributive:
(
)
a A + B = a A + aB
2Α
Α (iii) Dot product of two vectors
The dot product of two vectors is define by
Α
A.B = AB cos θ
θ
Β
where θ is the angle they form when placed tail to tail. Note that A.B is itself a scalar. The dot product is commutative,
A.B = B. A
(
)
A. B + C = A.B + A.C .
and distributive,
Geometrically A.B is the product of A times the projection of B along A (or the product of B
times the projection of A along B ).
If the two vectors are parallel, A.B = AB
If two vectors are perpendicular, then A.B = 0
Law of cosines
Let C = A − B and then calculate dot product of C with itself.
(
)(
)
C.C = A − B . A − B = A. A − A.B − B. A + B.B C 2 = A2 + B 2 − 2 AB cos θ
Head office fiziks, H.No. 23, G.F, Jia Sarai, Near IIT, Hauz Khas, New Delhi‐16 Phone: 011‐26865455/+91‐9871145498 Website: www.physicsbyfiziks.com
C
Α θ
Β
Branch office Anand Institute of Mathematics, 28‐B/6, Jia Sarai, Near IIT Hauz Khas, New Delhi‐16
fiziks Institute for NET/JRF, GATE, IIT‐JAM, JEST, TIFR and GRE in PHYSICAL SCIENCES (iv) Cross Product of Two Vectors
The cross product of two vectors is define by
Α
A × B = AB sin θ nˆ
θ
Β
where nˆ is a unit vector( vector of length 1) pointing perpendicular to the plane of A and B .Of course there are two directions perpendicular to any plane “in” and “out.” The ambiguity is resolved by the right-hand rule: let your fingers point in the direction of first vector and curl around (via the smaller angle)
toward the second; then your thumb indicates the direction of nˆ . (In figure A × B points into the
page; B × A points out of the page) The cross product is distributive,
(
) (
) (
)
A × B + C = A × B + A × C
but not commutative.
(
)
(
)
In fact B × A = − A × B .
Geometrically, A × B is the area of the parallelogram generated by A and B . If two vectors are parallel, their cross product is zero.
In particular A × A = 0 for any vector A
Head office fiziks, H.No. 23, G.F, Jia Sarai, Near IIT, Hauz Khas, New Delhi‐16 Phone: 011‐26865455/+91‐9871145498 Website: www.physicsbyfiziks.com
Branch office Anand Institute of Mathematics, 28‐B/6, Jia Sarai, Near IIT Hauz Khas, New Delhi‐16
fiziks Institute for NET/JRF, GATE, IIT‐JAM, JEST, TIFR and GRE in PHYSICAL SCIENCES 1.1.2 Vector Algebra: Component Form
Let xˆ, yˆ and zˆ be unit vectors parallel to the x, y and z axis, respectively. An arbitrary vector A can be expanded in terms of these basis vectors
A = A x xˆ + Ay yˆ + Az zˆ z
z zˆ
A
xˆ
A x xˆ
y
yˆ
A z zˆ y
x
A y yˆ
x
The numbers A x , A y , and A z are called component of A ; geometrically, they are the projections
of A along the three coordinate axes. (i) Rule: To add vectors, add like components.
(
) (
)
(
)
A + B = A x xˆ + Ay yˆ + Az zˆ + Bx xˆ + By yˆ + Bz zˆ = ( Ax + Bx ) xˆ + Ay + By yˆ + ( Az + Bz ) zˆ
(ii) Rule: To multiply by a scalar, multiply each component.
(
)
A = ( aA x ) xˆ + aAy yˆ + ( aAz ) zˆ
Because xˆ , yˆ and zˆ are mutually perpendicular unit vectors xˆ.xˆ = yˆ .yˆ = zˆ.zˆ = 1; xˆ.yˆ = xˆ.zˆ = yˆ.zˆ = 0
Accordingly, A.B = ( A x xˆ + Ay yˆ + Az zˆ ) . ( Bx xˆ + By yˆ + Bz zˆ ) = Ax Bx + Ay By + Az Bz (iii) Rule: To calculate the dot product, multiply like components, and add.
In particular, A. A = A x2 + Ay2 + Az2 ⇒ A = Similarly,
Ax2 + Ay2 + Az2
xˆ × xˆ = ˆy × yˆ = zˆ × zˆ = 0, xˆ × yˆ = − yˆ × xˆ = zˆ yˆ × zˆ =− zˆ × yˆ = xˆ zˆ × xˆ = − xˆ × zˆ = yˆ
Head office fiziks, H.No. 23, G.F, Jia Sarai, Near IIT, Hauz Khas, New Delhi‐16 Phone: 011‐26865455/+91‐9871145498 Website: www.physicsbyfiziks.com
Branch office Anand Institute of Mathematics, 28‐B/6, Jia Sarai, Near IIT Hauz Khas, New Delhi‐16
fiziks Institute for NET/JRF, GATE, IIT‐JAM, JEST, TIFR and GRE in PHYSICAL SCIENCES (iv) Rule: To calculate the cross product, form the determinant whose first row is xˆ, yˆ , zˆ , whose
second row is A (in component form), and whose third row is B .
xˆ
yˆ
zˆ Az = ( A y Bz − Az B y ) xˆ + ( Az Bx − Ax Bz ) yˆ + ( Ax By − Ay Bx ) zˆ
A × B = A x Ay
B x By Bz z
Example: Find the angle between the face diagonals of a cube.
(0,0,1)
Solution: The face diagonals A and B are
A = 1xˆ + 0 yˆ + 1zˆ;
B = 0 xˆ + 1 yˆ + 1 zˆ
Α
So, ⇒ A.B = 1
Also, ⇒ A. B = AB cos θ = 2 2 cos θ ⇒ cosθ =
1 2
⇒ θ = 60 0
x
θ
(0,1,0)
Β
y
(1,0,0) z
(0,0,1)
Example: Find the angle between the body diagonals of a cube.
Solution: The body diagonals A and B are
A = xˆ + yˆ − zˆ;
Β
B = xˆ + yˆ + zˆ
θ
Also, ⇒ A.B = AB cos θ = 3 3 cosθ ⇒ cosθ =
1
−1 ⎛ 1 ⎞
x
⇒ θ = cos ⎜ ⎟ 3 ⎝3⎠
3
to the plane shown in the figure.
nˆ
Solution: The vectors A and B can be defined as A = − xˆ + 2 yˆ ;
(1,0,0) z
Example: Find the components of the unit vector nˆ perpendicular
y
Α
So, ⇒ A. B = 1 + 1 − 1 = 1
(0,1,0)
B = − xˆ + 3 zˆ ⇒ nˆ =
A × B A × B
=
B
2
6 xˆ + 3 yˆ + 2 zˆ 1
7
y
A
x
Head office fiziks, H.No. 23, G.F, Jia Sarai, Near IIT, Hauz Khas, New Delhi‐16 Phone: 011‐26865455/+91‐9871145498 Website: www.physicsbyfiziks.com
Branch office Anand Institute of Mathematics, 28‐B/6, Jia Sarai, Near IIT Hauz Khas, New Delhi‐16
fiziks Institute for NET/JRF, GATE, IIT‐JAM, JEST, TIFR and GRE in PHYSICAL SCIENCES 1.1.3 Triple Products
Since the cross product of two vectors is itself a vector, it can be dotted or crossed with a third vector to form a triple product.
(
)
(i) Scalar triple product: A. B × C
(
)
Geometrically A. B × C is the volume of the parallelepiped
A nˆ θ
generated by A, B and C , since B × C is the area of the base,
C
B
and A cos θ is the altitude. Evidently,
(
)
(
)
(
A. B × C = B. C × A = C. A × B
(
A x Ay
)
Az
)
In component form A. B × C = B x By Bz C x C y C z
(
) (
)
Note that the dot and cross can be interchanged: A. B × C = A × B .C
(
)
(ii) Vector triple product: A × B × C
The vector triple product can be simplified by the so-called BAC-CAB rule:
(
)
( )
(
A × B × C = B A.C − C A.B
)
1.1.4 Position, Displacement, and Separation Vectors z
source point
ˆ r r x
z
( x, y, z )
R
r ′
y
field point
x r
y
Head office fiziks, H.No. 23, G.F, Jia Sarai, Near IIT, Hauz Khas, New Delhi‐16 Phone: 011‐26865455/+91‐9871145498 Website: www.physicsbyfiziks.com
Branch office Anand Institute of Mathematics, 28‐B/6, Jia Sarai, Near IIT Hauz Khas, New Delhi‐16
fiziks Institute for NET/JRF, GATE, IIT‐JAM, JEST, TIFR and GRE in PHYSICAL SCIENCES
The location of a point in three dimensions can be described by listing its Cartesian coordinates ( x, y, z ) . The vector to that point from the origin is called the position vector:
r = xxˆ + y yˆ + z zˆ .
Its magnitude, r =
2 2 2 x + y + z is the distance from the origin,
r xxˆ + y yˆ + z zˆ ˆ= = and r is a unit vector pointing radially outward. r x2 + y 2 + z 2
The infinitesimal displacement vector, from ( x, y, z ) to ( x + dx, y + dy, z + dz ) , is
d l = dxxˆ + dy yˆ + dz zˆ .
Note: In electrodynamics one frequently encounters problems involving two points-typically, a
source point , r ′ , where an electric charge is located, and a field point, r , at which we are
calculating the electric or magnetic field. We can define separation vector from the source
point to the field point by R ;
R = r − r ′ .
R = r − r ′ ,
Its magnitude is
R r − r ′ and a unit vector in the direction from r ′ to r is Rˆ = = . R r − r ′
In Cartesian coordinates,
R = ( x − x′ ) xˆ + ( y − y′ ) yˆ + ( z − z′ ) zˆ
R = Rˆ =
2
2
( x − x′ ) + ( y − y′) + ( z − z′)
2
( x − x′ ) xˆ + ( y − y′ ) yˆ + ( z − z′ ) zˆ 2
2
2
( x − x′ ) + ( y − y′ ) + ( z − z′ )
Head office fiziks, H.No. 23, G.F, Jia Sarai, Near IIT, Hauz Khas, New Delhi‐16 Phone: 011‐26865455/+91‐9871145498 Website: www.physicsbyfiziks.com
Branch office Anand Institute of Mathematics, 28‐B/6, Jia Sarai, Near IIT Hauz Khas, New Delhi‐16