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Preliminaries
Mathematical Physics
Mahfuzur Rahaman Department of Physics, Maulana Azad College, Kolkata-700013, India. Date: January 29, 2026
SI system of units, dimensional analysis. Plotting of functions (both cartesian and po- lar), Limits, Intuitive ideas about continuity and differentiability of a function. Taylor series of one variable and binomial series (statements only); Maxima and minima for functions of one variable. Calculus of functions of more than one variable: Partial derivatives, exact and inexact differentials.
CONTENTS
I. SI system of units 3
II. SI Units of Some Physical Quantities 6
III. Dimensional analysis 9 A. Principle of Dimensional Homogeneity 9 B. Applications of Dimensional Analysis 10
IV. SI Units and Dimensions of Some Physical Quantities 13
V. Plotting of Mathematical Functions: A Physics-Oriented Approach 15
VI. Limits of Functions: Intuitive Definition 23
VII. ϵ – δ Definition of a Limit 28
VIII. Continuity and Differentiability of Functions 31 A. Definition of Continuity 31
- B. Differentiability of a Function: Definition - IX. Intuitive Ideas of Continuity and Differentiability - X. Taylor Series for a Function of a Single Variable - A. Taylor’s Theorem - B. Maclaurin Series - XI. Taylor Series for Functions of Several Variables - A. Taylor Expansion in Two Variables - B. Taylor Expansion about the Origin - C. Taylor Series in Three Variables - D. Limitations
- XII. Binomial series - A. Binomial Theorem for Positive Integers - B. General Binomial Series
- XIII. Maxima and minima - A. Necessary Condition for Extremum - B. Sufficient Condition: Second Derivative Test - C. Geometrical Interpretation - D. Maxima and Minima of Functions of Two Variables - E. Physical Interpretation
- XIV. Partial derivatives - A. First Order Partial Derivative - B. Higher-Order Partial Derivatives - C. Total Differential
- XV. Exact and inexact differentials
Physical Quantity SI Unit Symbol Length metre m Mass kilogram kg Time second s Electric current ampere A Thermodynamic temperature kelvin K Amount of substance mole mol Luminous intensity candela cd
These base units are independent of one another and are defined with high precision. Derived Quantities Derived quantities are those physical quantities that can be expressed in terms of fundamen- tal quantities. Their units, called derived units , are obtained by algebraic combinations of base units. Examples include:
- Velocity: metre per second (m s−^1 )
- Acceleration: metre per second squared (m s−^2 )
- Force: newton (N = kg m s−^2 )
- Energy: joule (J = kg m^2 s−^2 )
The SI system is a coherent system , meaning that derived units involve no numerical factors other than unity. Redefinition of SI Base Units In modern SI (revised in 2019), base units are defined by fixing the numerical values of certain fundamental physical constants. This approach ensures greater precision and long- term stability. Some important definitions are:
- The metre is defined by fixing the speed of light in vacuum, c = 3 × 108 m/s.
- The second is defined using the frequency of radiation corresponding to the transition between two hyperfine levels of the ground state of the cesium-133 atom.
- The kilogram is defined by fixing the value of the Planck constant, h.
This shift from artifact-based definitions to constant-based definitions represents a major advancement in measurement science. SI Prefixes To conveniently express very large or very small quantities, the SI system uses standard prefixes that represent powers of ten.
Prefix Symbol Power of Ten giga G 109 mega M 106 kilo k 103 centi c 10 −^2 milli m 10 −^3 micro μ 10 −^6 nano n 10 −^9 pico p 10 −^12 femto fm 10 −^15
Prefixes simplify numerical expressions and reduce the possibility of errors in calculations. Rules for Writing SI Units To maintain uniformity, the SI system prescribes certain conventions:
- Unit symbols are written in lowercase, except when derived from proper names (e.g., N for newton).
- Unit symbols are never pluralized.
- A space is left between the numerical value and the unit (e.g., 5 m, not 5m).
- Compound units are written using spaces or negative powers (e.g., m/s or m s−^1 ).
Advantages of the SI System The SI system offers several advantages:
2. Mechanics Quantity Symbol SI Unit Defining Equation Displacement x m — Velocity v m s−^1 v = dxdt Acceleration a m s−^2 a = dvdt Force F newton (N) F = ma Momentum p kg m s−^1 p = mv Work W joule (J) W = −→ F · −→ s Energy E joule (J) E = W Power P watt (W) P = dWdt Pressure p pascal (Pa) p = FA Density ρ kg m−^3 ρ = mV 3. Oscillations and Waves Quantity Symbol SI Unit Defining Equation Time period T s T =^1 f Frequency f hertz (Hz) f = T^1
Angular frequency ω rad s−^1 ω = 2 πf
Wave speed v m s−^1 v = f λ Wavelength λ m —
4. Thermodynamics Quantity Symbol SI Unit Defining Equation Heat Q joule (J) Q = mc ∆ T Specific heat c J kg−^1 K−^1 c = (^) mQ ∆ T Latent heat L J kg−^1 Q = mL Entropy S J K−^1 dS = dQT
5. Electricity Quantity Symbol SI Unit Defining Equation Charge Q coulomb (C) Q = It Electric field −→ E N C−^1 −→ E =
−→ F
q Potential V volt (V) V = Wq Capacitance C farad (F) C = QV Resistance R ohm (Ω) R = VI Resistivity ρ Ωm R = ρ Al
6. Magnetism Quantity Symbol SI Unit Defining Equation Magnetic field B tesla (T) F = qvB Magnetic flux Φ B weber (Wb) Φ B = −→ B · −→ A Induced emf ε volt (V) ε = − d Φ dtB Inductance L henry (H) ε = − LdIdt 7. Optics Quantity Symbol SI Unit Defining Equation Refractive index μ dimensionless μ = (^) vc Focal length f m (^) f^1 =^1 v +^1 u Power of lens P dioptre (D) P =^1 f 8. Modern Physics Quantity Symbol SI Unit Defining Equation Photon energy E joule (J) E = hν
de Broglie wavelength λ m λ = hp
Activity A becquerel (Bq) A = dNdt
Every physical equation must be dimensionally homogeneous; that is, the di- mensions of all terms on both sides of an equation must be the same.
For example, consider the equation for distance in uniformly accelerated motion:
s = ut +^12 at^2
Each term has the dimension of length:
[ ut ] = [ at^2 ] = [ L ]
Thus, the equation is dimensionally consistent.
B. Applications of Dimensional Analysis
Dimensional analysis is widely used to verify whether a given physical equation is di- mensionally correct. If the dimensions of the two sides of an equation do not match, the equation is certainly incorrect. For example, consider the incorrect relation:
v = u + at^2
Here, [ u ] = [ LT −^1 ] , [ at^2 ] = [ L ]
Since the dimensions are different, the equation is dimensionally incorrect. Derivation of Relations Between Physical Quantities Dimensional analysis can be used to derive the form of a physical relationship between quantities. Example: Suppose the time period T of a simple pendulum depends on its length l and acceleration due to gravity g. Assume
T ∝ lagb
Then, [ T ] = [ La ( LT −^2 ) b ] = [ La + bT −^2 b ]
Equating dimensions, (^)
a + b = 0 − 2 b = 1
⇒ a =^12 , b = −^12
Hence,
T ∝
√ l g The numerical factor 2 π cannot be obtained by dimensional analysis. Conversion of Units Dimensional analysis helps in converting a physical quantity from one system of units to another. Example: Convert force from CGS to SI units.
1 dyne = 1 g cm s−^2 1 N = 10^5 dyne
Estimation and Order of Magnitude Dimensional analysis is useful in estimating physical quantities and determining their order of magnitude, especially when exact solutions are difficult. Example: Estimating the speed of waves on a stretched string or the energy released in an explosion. Dimensionless Quantities Some physical quantities have no dimensions and are called dimensionless quantities. Examples include:
strain , refractive index , coefficient of friction , Mach number
Although dimensionless, these quantities may still have physical units (e.g., angle in radians). Limitations of Dimensional Analysis Despite its usefulness, dimensional analysis has certain limitations:
- It cannot determine numerical constants such as 2, π , or 12.
- It cannot distinguish between different physical quantities having the same dimensions (e.g., work and torque).
IV. SI UNITS AND DIMENSIONS OF SOME PHYSICAL QUANTITIES
Physical Quantity Symbol SI Unit Dimensions Length l metre (m) [ L ] Mass m kilogram (kg) [ M ] Time t second (s) [ T ] Electric current I ampere (A) [ A ] Temperature T kelvin (K) [Θ] Amount of substance n mole (mol) [ N ] Luminous intensity Iv candela (cd) [ J ] Displacement x m [ L ] Velocity v m s−^1 [ LT −^1 ] Acceleration a m s−^2 [ LT −^2 ] Force F newton (N) [ M LT −^2 ] Momentum p kg m s−^1 [ M LT −^1 ] Work / Energy W, E joule (J) [ M L^2 T −^2 ] Power P watt (W) [ M L^2 T −^3 ] Pressure p pascal (Pa) [ M L −^1 T −^2 ] Density ρ kg m−^3 [ M L −^3 ] Time period T s [ T ] Frequency f hertz (Hz) [ T −^1 ] Angular frequency ω rad s−^1 [ T −^1 ] Wavelength λ m [ L ] Wave speed v m s−^1 [ LT −^1 ]
Physical Quantity Symbol SI Unit Dimensions
Heat Q joule (J) [ M L^2 T −^2 ]
Specific heat c J kg−^1 K−^1 [ L^2 T −^2 Θ−^1 ]
Latent heat L J kg−^1 [ L^2 T −^2 ]
Entropy S J K−^1 [ M L^2 T −^2 Θ−^1 ]
Electric charge Q coulomb (C) [ AT ]
Electric field E N C−^1 [ M LT −^3 A −^1 ]
Electric potential V volt (V) [ M L^2 T −^3 A −^1 ]
Capacitance C farad (F) [ M −^1 L −^2 T^4 A^2 ]
Resistance R ohm (Ω) [ M L^2 T −^3 A −^2 ]
Resistivity ρ Ωm [ M L^3 T −^3 A −^2 ]
Magnetic field B tesla (T) [ M T −^2 A −^1 ]
Magnetic flux Φ B weber (Wb) [ M L^2 T −^2 A −^1 ]
Induced emf ε volt (V) [ M L^2 T −^3 A −^1 ]
Inductance L henry (H) [ M L^2 T −^2 A −^2 ]
Refractive index μ dimensionless [1]
Focal length f m [ L ]
Power of lens P dioptre (D) [ L −^1 ]
Photon energy E joule (J) [ M L^2 T −^2 ]
de Broglie wavelength λ m [ L ]
Radioactive activity A becquerel (Bq) [ T −^1 ]
This is a straight line in the x – t plane.
- Slope = v (velocity)
- Intercept = x 0 (initial position)
A steeper line corresponds to a higher speed. Quadratic Functions y = ax^2 + bx + c
The graph is a parabola. Physics Example: Motion with Constant Acceleration For motion under constant acceleration a ,
x ( t ) = ut +^12 at^2 Power-Law Functions
2
4
y
Power Law vs Exponential Growth
x^2 x^3 x^4 x x
FIG. 1: Plot of different power of x and exp x
y = xn
Physics Examples
- Inverse-square law: F ( r ) ∝ (^) r^12 used in gravitation and electrostatics.
- Kinetic energy: K =^12 mv^2
Such plots often show rapid variation near the origin and slow variation at large values.
3.0^ y
Exponential Growth and Decay
x
x^
2
-^ x
-^ x^
2
FIG. 2: Plot of different powers of e
Exponential Functions y = eax
Physics Example: Radioactive Decay
N ( t ) = N 0 e − λt
- 2 π - 32 π - π - π 2 π 2 π 32 π 2 π
x
Function Value
Sine and Cosine Functions
sin x
cos x
FIG. 4: Plot of sin( x ) and cos( x )
Trigonometric Functions
y = A sin( ωx + ϕ ) , y = A cos( ωx + ϕ )
Physics Example: Simple Harmonic Motion
x ( t ) = A cos( ωt )
Key features:
- Periodicity with period T = (^2) ωπ.
- Amplitude represents maximum displacement.
- Phase determines initial conditions.
Discontinuous and Piecewise Functions
- 2 π - 3 2 π - π - π 2 π 2 π 32 π 2 π
x
5
Function Value
Sec and Cosec Functions
sec x cosec x
FIG. 5: Plot of sec( x ) and csc( x )
Step Function
f ( x ) =
0 , x < 0 1 , x ≥ 0
Physics Example: Switching Fields Used to model sudden application of force or electric field. Rational Functions and Asymptotes Mathematical Form
y =^1 x
Properties:
- Vertical asymptote at x = 0
- Horizontal asymptote at y = 0
