Mathematical physics fiziks notes for BSc students. PDF
| Title | Mathematical physics fiziks notes for BSc students. |
|---|---|
| Course | Statistical Physics |
| Institution | Savitribai Phule Pune University |
| Pages | 249 |
| File Size | 22.7 MB |
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| Total Downloads | 116 |
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Summary
formula sheet for mathematical physics for BSc students for exam preparations and revision.All formulas in detail....
Description
fiziks InstituteforNET/JRF,GATE,IIT‐JAM,JEST,TIFRandGREinPHYSICALSCIENCES
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) Headoffice fiziks,H.No.23,G.F,JiaSarai, NearIIT,HauzKhas,NewDelhi‐16 Phone:011‐26865455/+91‐9871145498 Website: www.physicsbyfiziks.com
Branchoffice AnandInstituteofMathematics, 28‐B/6,JiaSarai,NearIIT HauzKhas,NewDelhi‐16
fiziks
InstituteforNET/JRF,GATE,IIT‐JAM,JEST,TIFRandGREinPHYSICALSCIENCES 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 rˆ / 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) Headoffice Branchoffice fiziks,H.No.23,G.F,JiaSarai, AnandInstituteofMathematics, NearIIT,HauzKhas,NewDelhi‐16 28‐B/6,JiaSarai,NearIIT Phone:011‐26865455/+91‐9871145498 HauzKhas,NewDelhi‐16 Website: www.physicsbyfiziks.com
fiziks
InstituteforNET/JRF,GATE,IIT‐JAM,JEST,TIFRandGREinPHYSICALSCIENCES 1. Linear Algebra and Matrices…………………………………………………(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…………………………………………………………….(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………………………………………………………………..(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 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 Headoffice fiziks,H.No.23,G.F,JiaSarai, NearIIT,HauzKhas,NewDelhi‐16 Phone:011‐26865455/+91‐9871145498 Website: www.physicsbyfiziks.com
Branchoffice AnandInstituteofMathematics, 28‐B/6,JiaSarai,NearIIT HauzKhas,NewDelhi‐16
fiziks
InstituteforNET/JRF,GATE,IIT‐JAM,JEST,TIFRandGREinPHYSICALSCIENCES 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
Headoffice fiziks,H.No.23,G.F,JiaSarai, NearIIT,HauzKhas,NewDelhi‐16 Phone:011‐26865455/+91‐9871145498 Website: www.physicsbyfiziks.com
Branchoffice AnandInstituteofMathematics, 28‐B/6,JiaSarai,NearIIT HauzKhas,NewDelhi‐16
fiziks
InstituteforNET/JRF,GATE,IIT‐JAM,JEST,TIFRandGREinPHYSICALSCIENCES 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
(Α + Β)
−Β
(Β + Α)
Α
(Α − Β)
Α
Β
Headoffice fiziks,H.No.23,G.F,JiaSarai, NearIIT,HauzKhas,NewDelhi‐16 Phone:011‐26865455/+91‐9871145498 Website: www.physicsbyfiziks.com
Branchoffice AnandInstituteofMathematics, 28‐B/6,JiaSarai,NearIIT HauzKhas,NewDelhi‐16
fiziks
InstituteforNET/JRF,GATE,IIT‐JAM,JEST,TIFRandGREinPHYSICALSCIENCES (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 A.B = AB cos θ
Α
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
(
)(
)
Α
C
θ Β
C 2 = A2 + B2 − 2 AB cos θ
Headoffice fiziks,H.No.23,G.F,JiaSarai, NearIIT,HauzKhas,NewDelhi‐16 Phone:011‐26865455/+91‐9871145498 Website: www.physicsbyfiziks.com
Branchoffice AnandInstituteofMathematics, 28‐B/6,JiaSarai,NearIIT HauzKhas,NewDelhi‐16
fiziks
InstituteforNET/JRF,GATE,IIT‐JAM,JEST,TIFRandGREinPHYSICALSCIENCES (iv) Cross Product of Two Vectors
The cross product of two vectors is define by A × B = AB sin θ nˆ
Α
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 ˆn . (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
Headoffice fiziks,H.No.23,G.F,JiaSarai, NearIIT,HauzKhas,NewDelhi‐16 Phone:011‐26865455/+91‐9871145498 Website: www.physicsbyfiziks.com
Branchoffice AnandInstituteofMathematics, 28‐B/6,JiaSarai,NearIIT HauzKhas,NewDelhi‐16
fiziks
InstituteforNET/JRF,GATE,IIT‐JAM,JEST,TIFRandGREinPHYSICALSCIENCES 1.1.2 Vector Algebra: Component Form Let xˆ, yˆ and ˆz 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 = Ax xˆ + Ay yˆ + Az zˆ z
z
zˆ xˆ
A
yˆ
Ax xˆ
y
x
x
Az zˆ y
Ay yˆ
The numbers Ax , A y , and Az 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 = ( Ax 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 + ( aA y ) 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 = ( Ax xˆ + Ay yˆ + Az ˆz ). ( 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 = Ax2 + A2y + Az2 ⇒ A = Ax2 + A2y + Az2
Similarly,
xˆ × xˆ = yˆ × yˆ = zˆ × zˆ = 0, xˆ × ˆy = − ˆy × xˆ = ˆz yˆ × zˆ =− zˆ × yˆ = xˆ zˆ × xˆ = − xˆ × zˆ = yˆ
Headoffice fiziks,H.No.23,G.F,JiaSarai, NearIIT,HauzKhas,NewDelhi‐16 Phone:011‐26865455/+91‐9871145498 Website: www.physicsbyfiziks.com
Branchoffice AnandInstituteofMathematics, 28‐B/6,JiaSarai,NearIIT HauzKhas,NewDelhi‐16
fiziks
InstituteforNET/JRF,GATE,IIT‐JAM,JEST,TIFRandGREinPHYSICALSCIENCES (iv) Rule: To calculate the cross product, form the determinant whose first row is ˆx , ˆy , zˆ , whose second row is A (in component form), and whose third row is B . xˆ yˆ zˆ A × B = Ax A y A z = ( Ay Bz − Az B y ) xˆ + ( Az Bx − Ax Bz ) yˆ + ( Ax B y − A yB x ) zˆ Bx B y B z z
Example: Find the angle between the face diagonals of a cube. Solution: The face diagonals A and B are A = 1xˆ + 0 yˆ + 1zˆ ; B = 0 xˆ + 1yˆ + 1zˆ So, ⇒ A.B = 1 Also, ⇒ A.B = AB cosθ =
2 2 cosθ ⇒ cosθ =
( 0,0,1) Α θ
1 ⇒ θ = 600 2
x
(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; B = xˆ + yˆ + zˆ So, ⇒ A. B = 1 + 1 − 1 = 1
Β
θ
(0,1,0) y
Α
1 ⎛ 1⎞ Also,⇒ A.B = AB cos θ = 3 3 cos θ ⇒ cos θ = ⇒ θ = cos−1 ⎜ ⎟ 3 ⎝ 3⎠
x
(1,0,0 ) z
Example: Find the components of the unit vector nˆ perpendicular
3
to the plane shown in the figure. Solution: The vectors A and B can be defined as A × B 6 xˆ + 3 yˆ + 2 zˆ ˆ = = A = − xˆ + 2 yˆ; B = − xˆ+ 3zˆ ⇒ n 7 A× B
nˆ
B 1
2 y
A
x
Headoffice fiziks,H.No.23,G.F,JiaSarai, NearIIT,HauzKhas,NewDelhi‐16 Phone:011‐26865455/+91‐9871145498 Website: www.physicsbyfiziks.com
Branchoffice AnandInstituteofMathematics, 28‐B/6,JiaSarai,NearIIT HauzKhas,NewDelhi‐16
fiziks
InstituteforNET/JRF,GATE,IIT‐JAM,JEST,TIFRandGREinPHYSICALSCIENCES 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
and A cosθ is the altitude. Evidently,
B
A. B × C = B. C × A = C . A × B
(
)
(
)
(
)
Ax Ay Az In component form A . B ×C = B x B y B z
(
)
Cx Cy Cz 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
rˆ r x
z
(x, y , z )
R
r′
y
field point
x
r
y
Headoffice fiziks,H.No.23,G.F,JiaSarai, NearIIT,HauzKhas,NewDelhi‐16 Phone:011‐26865455/+91‐9871145498 Website: www.physicsbyfiziks.com
Branchoffice AnandInstituteofMathematics, 28‐B/6,JiaSarai,NearIIT HauzKhas,NewDelhi‐16
fiziks
InstituteforNET/JRF,GATE,IIT‐JAM,JEST,TIFRandGREinPHYSICALSCIENCES 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 = x 2 + y 2 + z 2 is the distance from the origin, r xxˆ + y ˆy + z ˆz is a unit vector pointing radially outward. and rˆ = = r x2 + y2 + z2 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 and a unit vector in the direction from r ′ to r is Rˆ = = R r
− r′ . − r′
In Cartesian coordinates, R = ( x − x′ ) xˆ + ( y − y′ ) yˆ + ( z − z′ ) ˆz R=
( x − x ′) + ( y − y ′) + (z − z ′) 2
2
2
( x − x′ ) xˆ + ( y − y′ ) yˆ + (z − z′) zˆ Rˆ = ( x − x′ )2 + ( y − y′ )2 + ( z − z′ )2
Headoffice fiziks,H.No.23,G.F,JiaSarai, NearIIT,HauzKhas,NewDelhi‐16 Phone:011‐26865455/+91‐9871145498 Website: www.physicsbyfiziks.com
Branchoffice AnandInstituteofMathematics, 28‐B/6,JiaSarai,NearIIT HauzKhas,NewDelhi‐16
fiziks
1.2
InstituteforNET/JRF,GATE,IIT‐JAM,JEST,TIFRandGREinPHYSICALSCIENCES Differential Calculus
1.2.1 “Ordinary” Derivatives
Suppose we have a function of one variable: f(x) then the derivative, df /dx tells us how rapidly the function f(x) varies when we change the argument x by a tiny amount, dx: ⎛ df ⎞ df = ⎜ ⎟dx ⎝ dx ⎠ In words: If we change x by an amount dx, then f changes by an amount df; the derivative is the proportionality factor. For example in figure (a), the function varies slowly with x, and the derivative is correspondingly small. In figure (b), f increases rapidly with x, and the derivative is large, as we move away from x = 0. Geometrical Interpretation: The derivative df / dx is the slope of the graph of f versus x. f
f
(a )
x
(b )
x
1.2.2 Gradient
Suppose that we have a function of three variables-say, V (x, y, z) in a ⎛ ∂V ⎛ ∂V ⎞ dV = ⎜ ⎟ dx + ⎜⎜ ⎝ ∂x ⎠ ⎝ ∂y
⎞ ⎛ ∂V ⎞ ⎟⎟ dy + ⎜ ⎟ dz. ⎝ ∂z ⎠ ⎠
This tells us how V changes when we alter all three variables by the infinitesimal amounts dx, dy, dz. Notice that we do not require an infinite number of derivatives-three will suffice: the partial derivatives along each of the three coordinate directions. Thus
⎛ ∂V ∂V ∂V dV = ⎜ ˆx + ˆy+ ∂ ∂ ∂z x y ⎝
where ∇V =
( )( )
⎞ zˆ ⎟ ⋅ ( dxxˆ + dyy ˆ + dzzˆ) = ∇V ⋅ d l , ⎠
∂V ∂V ∂V xˆ + ˆy + zˆ is the gradient of V . ∂x ∂y ∂z
∇V is a vector quantity, with three components. Headoffice fiziks,H.No.23,G.F,JiaSarai, NearIIT,HauzKhas,NewDelhi‐16 Phone:011‐26865455/+91‐9871145498 Website: www.physicsbyfiziks.com
Branchoffice AnandInstituteofMathematics, 28‐B/6,JiaSarai,NearIIT HauzKhas,NewDelhi‐16
fiziks
InstituteforNET/JRF,GATE,IIT‐JAM,JEST,TIFRandGREinPHYSICALSCIENCES Geometrical Interpretation of the Gradient
Like any vector, the gr...
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