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TU B.Sc. First Year Syllabus: mathematics
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TU B.Sc. First Year Mathematics — Complete Syllabus Overview

Mathematics under the Four-Year Bachelor of Science (B.Sc.) programme of Tribhuvan University’s Institute of Science and Technology (IoST), Kirtipur, Kathmandu, consists of two separate theory papers in First Year: MAT 101 (Calculus) and MAT 102 (Analytical Geometry and Vector Analysis), each carrying 75 marks. Together they form the mathematical foundation for all subsequent years of the B.Sc. programme — whether a student majors in Mathematics, Physics, Statistics, Computer Science, or any discipline requiring quantitative reasoning.

The current course of study follows the Revised B.Sc. Mathematics Curriculum 2073 BS (2016), designed by the Mathematics Subject Committee under IoST, TU. Both papers are pure theory courses — there are no separate practical papers for First Year Mathematics. The curriculum was designed to bridge the gap between the NEB Grade 12 Mathematics syllabus and rigorous university-level mathematical analysis, introducing students to formal proof-based reasoning, multi-variable calculus, three-dimensional coordinate geometry, and vector calculus.

Subject Information at a Glance

  • Paper I: MAT 101 — Calculus | Full Marks: 75 | Pass Marks: 26.25 (35%)
  • Paper II: MAT 102 — Analytical Geometry and Vector Analysis | Full Marks: 75 | Pass Marks: 35%
  • Combined Total: 150 marks — both papers are theory only (no practical papers)
  • Year: First Year — Four Year B.Sc. Programme
  • Nature: Theory (both papers)
  • Total Lecture Hours: 80 hours (MAT 101) + 80 hours (MAT 102)
  • Examination: Annual (end-of-year exam conducted by IoST, TU)
  • Conducting Institution: Institute of Science and Technology (IoST), Tribhuvan University
  • Curriculum Authority: Mathematics Subject Committee, IoST, TU / Central Department of Mathematics (CDM), Kirtipur
  • Curriculum Version: Revised Course of Study 2073 BS (2016)
  • Eligibility for Mathematics Major: Must have studied Mathematics with 200 full marks at Grade 12 (or equivalent)

The Four-Year TU B.Sc. Mathematics Programme Structure

First Year Mathematics lays the foundational framework. The full four-year progression is:

  • First Year: MAT 101 Calculus + MAT 102 Analytical Geometry and Vector Analysis
  • Second Year: MAT 201 Linear Algebra + MAT 202 Differential Equations
  • Third Year: MAT 301 Computer Programming + MAT 302 Real Analysis; Elective (one of: Numerical Methods, Discrete Mathematics)
  • Fourth Year: MAT 401 Modern Algebra + MAT 402 Mathematical Analysis + MAT 403 Mechanics + MAT 404 Linear Programming; Elective (one of: Teaching Methodology, Project Work, Bio Mathematics, Mathematical Economics); Compulsory interdisciplinary course MAT 409 Mathematical Modeling

MAT 101: Calculus — Complete Unit-wise Syllabus

MAT 101 covers differential calculus, integral calculus, and an introduction to differential equations — totaling 80 lecture hours across 13 units (Unit 0 through Unit 12).

Unit 0: Review of Elementary Calculus (6 hours)

  • Functions and graphs: domain, range, graph sketching of standard functions
  • Evaluation of limits: direct substitution, factorization, rationalization, standard limits
  • Continuity and discontinuity: definition of continuity at a point and on an interval
  • Test of continuity: left-hand and right-hand limits
  • Properties of continuous functions: intermediate value theorem (concept), extreme value theorem (concept)

Unit 1: Tangents and Normals (6 hours)

  • Tangents and normals to curves: equations using dy/dx at a point
  • Subtangent and subnormal and their lengths: geometric interpretation
  • Derivative of arc length: ds/dx and ds/dy in Cartesian form; ds/dθ in polar form
  • Polar equations of subtangent and subnormal
  • Angle between the radius vector and the tangent in polar coordinates (tan φ = r dθ/dr)
  • Length of perpendicular from the pole on the tangent
  • Pedal equations: derivation from polar equations of curves
  • Angle between two curves (angle of intersection): condition for orthogonality

Unit 2: Higher Order Derivatives and Mean Value Theorems (8 hours)

  • Higher order derivatives: notation, computation of nth derivatives; nth derivative of standard functions (xⁿ, eˣ, sin x, cos x, log x)
  • Leibnitz’s theorem: nth derivative of a product of two functions (statement and application)
  • Rolle’s theorem (without proof): statement, geometric interpretation, verification
  • Lagrange’s Mean Value theorem (without proof): statement, geometric interpretation, applications in proving inequalities
  • Cauchy’s Mean Value theorem (without proof): statement, relationship to Lagrange’s theorem
  • Maclaurin’s theorem (without proof): expansion of functions as power series about x = 0
  • Taylor’s theorem (without proof): expansion of functions about x = a; Taylor and Maclaurin series for eˣ, sin x, cos x, log(1+x), (1+x)ⁿ; applications in approximation and limit evaluation

Unit 3: Application of Derivatives (12 hours — largest MAT 101 unit)

  • Indeterminate forms: 0/0, ∞/∞, 0×∞, ∞−∞, 0⁰, 1^∞, ∞⁰
  • L’Hôpital’s rule (without proof): statement, repeated application, conditions for use
  • Asymptotes: definition, types — horizontal, vertical, oblique; asymptotes of algebraic curves (method of equating highest degree terms); asymptotes parallel to axes
  • Curve tracing techniques: symmetry tests (about x-axis, y-axis, origin), intercepts, asymptotes, regions of non-existence, monotonicity, inflection points
  • Standard curves and their tracing: cissoid, strophoid, folium of Descartes, astroid, cycloid, cardioid, lemniscate of Bernoulli, rose curves (r = a sin nθ, r = a cos nθ)
  • Curvature: concept of curvature, radius of curvature for Cartesian (ρ = [1+(dy/dx)²]^(3/2)/|d²y/dx²|), parametric, and polar forms
  • Chord of curvature: length of chord of curvature through the pole and parallel to axes
  • Curvature at the origin: Newton’s method
  • Centre of curvature and circle of curvature: coordinates of centre of curvature, equation of circle of curvature, evolute as the locus of centre of curvature

Unit 4: Partial Differentiation and Maxima/Minima of Functions of 2 and 3 Variables (8 hours)

  • Basic ideas of limits and continuity of functions of 2 and 3 variables
  • Partial derivatives and their geometrical interpretation: tangent planes, normal lines
  • Higher order partial derivatives: mixed partials, Young’s theorem (equality of mixed partials under continuity)
  • Homogeneous functions: definition, degree; Euler’s theorem (proof for 2 variables), extension to 3 variables
  • Total differentials: total derivative, chain rule for functions of several variables
  • Extreme values of functions of two variables: stationary points, second-order conditions (Hessian determinant criterion for maxima, minima, saddle points)
  • Subsidiary conditions and Lagrange’s method of undetermined multipliers: constrained optimization with one constraint; geometric interpretation; applications in distance, area, and volume optimization problems

Unit 5: Integration and Definite Integrals (6 hours)

  • Integration concepts: antiderivative, indefinite integral, constant of integration
  • Integration techniques: substitution, integration by parts, partial fractions
  • Standard integration formulae: review and extension (including ∫sec x dx, ∫cosec x dx, ∫tan x dx, ∫cot x dx, standard rational forms)
  • Integration of rational functions: proper and improper fractions, partial fraction decomposition
  • Integration of hyperbolic functions: sinh x, cosh x, tanh x and their inverses
  • Integration as the limit of a sum: Riemann sum definition of the definite integral
  • Definite integral and the Fundamental Theorem of Integral Calculus (without proof): first and second fundamental theorems
  • Properties of definite integrals: linearity, additivity, symmetry properties (even/odd function integrals), substitution in definite integrals

Unit 6: Beta and Gamma Functions and Reduction Formulae (4 hours)

  • Improper integrals: types of impropriety (infinite limits, integrand discontinuity), convergence and divergence
  • Gamma function: definition Γ(n) = ∫₀^∞ tⁿ⁻¹e⁻ᵗ dt; properties — Γ(1) = 1, Γ(n+1) = n Γ(n), Γ(1/2) = √π; factorial connection for positive integers
  • Beta function: definition B(m,n) = ∫₀¹ xᵐ⁻¹(1−x)ⁿ⁻¹ dx; symmetry B(m,n) = B(n,m); relation to Gamma function B(m,n) = Γ(m)Γ(n)/Γ(m+n)
  • Properties of Beta and Gamma functions: applications in evaluating definite integrals of the form ∫₀^(π/2) sinᵐx cosⁿx dx
  • Reduction formulae: derivation for ∫sinⁿx dx, ∫cosⁿx dx, ∫tanⁿx dx, ∫secⁿx dx; Wallis formulae

Unit 7: Rectification and Quadrature, Volume and Surface Area of Solid of Revolution (8 hours)

  • Rectification (arc length): arc length formulae in Cartesian, parametric, and polar forms; derivation and computation for standard curves
  • Quadrature (area): area formulae in Cartesian (including area between two curves), parametric, and polar forms; area of closed polar curves
  • Volume of solid of revolution: disc method and shell method in Cartesian and parametric forms; Pappus’s centroid theorem for volume
  • Surface area of solid of revolution: formulae in Cartesian and parametric forms; Pappus’s theorem for surface area

Unit 8: Double Integrals (8 hours)

  • Double and iterated integrals in rectangular (Cartesian) coordinates: definition, Fubini’s theorem, evaluation by repeated integration
  • Change of order of integration in double integrals: geometric reasoning, determining new limits
  • Change of variables in double integrals: to polar coordinates (Jacobian = r), to curvilinear coordinates (general Jacobian)
  • Computing area using double integrals: A = ∬ dA
  • Computing volume using double integrals: V = ∬ f(x,y) dA
  • Applications of double integrals in mechanics: mass of a lamina (M = ∬ σ dA), static moments (Mₓ, Mᵧ), centre of gravity (x̄, ȳ), moments of inertia of a lamina (Iₓ, Iᵧ, I₀)

Unit 9: Differential Equations of the First Order and First Degree (6 hours)

  • Introduction to ordinary differential equations: order, degree, general and particular solutions
  • Variables-separable equations: solution by direct integration after separation
  • Homogeneous equations: substitution y = vx (or x = vy); reduction to separable form
  • Equations reducible to homogeneous form: linear fractional equations (ax+by+c)/(dx+ey+f) type
  • Non-homogeneous equations of the first order: linear equations dy/dx + P(x)y = Q(x); integrating factor e^∫P dx
  • Exact differential equations: condition for exactness (∂M/∂y = ∂N/∂x), solution method
  • Condition for exactness and integrating factors: standard integrating factor types — (Mᵧ−Nₓ)/N = function of x alone; (Nₓ−Mᵧ)/M = function of y alone
  • Linear differential equations and equations reducible to linear forms: Bernoulli’s equation

Unit 10: Differential Equations of the First Order but not First Degree (6 hours)

  • Equations solvable for p (= dy/dx): factorization method
  • Equations solvable for y: expressing y as a function of x and p; differentiating and eliminating p
  • Equations solvable for x: expressing x as a function of y and p; differentiating with respect to y
  • Equations homogeneous in x and y: degree of homogeneity, substitution y = vx
  • Clairaut’s equation: y = px + f(p); general solution (family of lines) and singular solution (envelope)

Unit 11: Linear Differential Equations with Constant Coefficients (6 hours)

  • Linear equations with constant coefficients: standard form (aₙDⁿ + … + a₁D + a₀)y = X where D = d/dx
  • Complementary function (CF): auxiliary equation method; cases — distinct real roots, repeated roots, complex conjugate roots
  • Symbolic operator D: shift theorem, operational methods
  • Particular integral (PI): standard forms for X = eˢˣ (method of finding PI using operator (D−a)⁻¹), X = sin(ax) or cos(ax), X = xⁿ, X = eˢˣ·V(x) — each using inverse operator technique
  • Homogeneous linear equations (Euler-Cauchy type): substitution x = eᵗ; reduction to constant coefficient form
  • Equations reducible to homogeneous form: (ax+b)ⁿDⁿ + … type

Unit 12: Applications of First Order and First Degree Differential Equations (6 hours)

  • Formulation of problems into differential equations: translating physical, geometric, and engineering problems into mathematical models
  • Initial conditions and boundary conditions: definition and role in determining particular solutions
  • Solution techniques applied to real problems:
    • Growth and decay problems: population growth, radioactive decay, Newton’s law of cooling
    • Simple interest and compound interest models
    • Mixing problems: concentration of solutions in tanks
    • Geometric applications: orthogonal trajectories of curves — method of finding orthogonal trajectories in Cartesian and polar coordinates

MAT 102: Analytical Geometry and Vector Analysis — Complete Unit-wise Syllabus

MAT 102 covers two-dimensional conics, three-dimensional coordinate geometry (planes, lines, spheres, cones, cylinders, conicoids), and vector analysis — totaling 80 lecture hours across 12 units.

Unit 1: Transformation of Coordinates (6 hours)

  • Introduction to polar, cylindrical, and spherical coordinates: definitions, conversion formulae
  • Transformation of Cartesian coordinates: translation (shift of origin), rotation of axes — transformation formulae, derivation
  • Process involving combination of translation and rotation: removing xy-term by rotation (angle θ = ½ arctan(2B/(A−C)) for Ax²+Bxy+Cy²+… = 0)
  • Invariants in orthogonal transformation: sum of coefficients of x² and y² (A+C), discriminant (B²−4AC), determinant of the matrix of the quadratic form

Unit 2: Conic Sections and Their Properties (14 hours — largest MAT 102 unit)

  • Introduction: conic section as the locus of a point (focus-directrix definition with eccentricity e); conic section as a plane section of a cone (Apollonius’s definition)
  • Central conic sections — Ellipse (e < 1): derivation of standard equation x²/a² + y²/b² = 1 with a > b > 0; relationship b² = a²(1−e²); foci, directrices, major and minor axes, latus rectum length; tracing the ellipse
  • Central conic sections — Hyperbola (e > 1): derivation of standard equation x²/a² − y²/b² = 1; relationship b² = a²(e²−1); foci, directrices, transverse and conjugate axes, latus rectum; asymptotes y = ±(b/a)x
  • Auxiliary circle and eccentric angle: parametric equations x = a cos θ, y = b sin θ (ellipse); eccentric angle and point on auxiliary circle
  • Equations of tangent and normal to ellipse and hyperbola: at point (x₁,y₁) and parametric point (a cos θ, b sin θ); slope form of tangent; condition for a line to be tangent
  • Chord of contact: equation when tangents are drawn from an external point
  • Pole and polar and their properties: definition, equation, properties; harmonic conjugates
  • Diameter: definition, equation of diameter conjugate to direction y = mx for ellipse and hyperbola
  • Conjugate diameters: definition, properties (product of slopes of conjugate diameters = −b²/a² for ellipse); equi-conjugate diameters
  • Asymptotes of hyperbola: equations, properties, angle between asymptotes; the combined equation of asymptotes; relationship between hyperbola, asymptotes, and conjugate hyperbola
  • Hyperbola referred to asymptotes as coordinate axes: equation xy = c²/2; transformation

Unit 3: Polar Equation of a Conic (6 hours)

  • Polar equation of a conic with focus at the pole: l/r = 1 + e cos θ (and variants depending on orientation); identification of conic from equation (e = 0: circle, e < 1: ellipse, e = 1: parabola, e > 1: hyperbola)
  • Equation of a chord of a conic in polar form: chord joining two points (r₁, θ₁) and (r₂, θ₂)
  • Equation of the tangent to a conic in polar form: at a given point
  • Equation of the normal to a conic in polar form
  • Chord of contact in polar form
  • Equation of the polar (polar line) to a conic in polar form
  • Equation of the asymptotes of a hyperbola in polar form

Unit 4: General Equation of the Second Degree (6 hours)

  • General second-degree equation: ax² + 2hxy + by² + 2gx + 2fy + c = 0; conditions for representing a conic (non-degenerate condition: Δ ≠ 0), pair of lines (Δ = 0), point circle
  • Nature of the conic represented: discriminant Δ = abc + 2fgh − af² − bg² − ch²; classification using Δ and the sign of (ab−h²)
  • Centre of conic: method of finding centre by partial differentiation
  • Equation of the tangent at a point (x₁,y₁) and condition of tangency for a line y = mx + c
  • Equation of the pair of tangents from an external point
  • Director circle: locus of intersection of perpendicular tangents to an ellipse/hyperbola
  • Equation of the normal at a point on a conic
  • Pole and polar with respect to a general conic
  • Diameter and conjugate diameters for a general conic
  • Intersection of conics; asymptotes of a general conic

Unit 5: Coordinates in Three-Space and Planes (6 hours)

  • Review of 3D coordinates: rectangular Cartesian coordinates, distance formula, section formula, direction cosines and direction ratios
  • Angle between two lines in 3D: using direction cosines; perpendicularity and parallelism conditions
  • General equation of the first degree representing a plane: ax + by + cz + d = 0; intercept form; normal form
  • Angle between two planes: using direction cosines of normals
  • Plane through three given points: derivation
  • Plane through the intersection of two planes: the family ax+by+cz+d + λ(a’x+b’y+c’z+d’) = 0
  • Condition for the homogeneous second-degree equation to represent a pair of planes: discriminant condition

Unit 6: Straight Lines in 3D (8 hours)

  • Representation of a line as the intersection of two planes: solving the system of two linear equations simultaneously
  • Line in symmetric form: (x−x₁)/l = (y−y₁)/m = (z−z₁)/n; direction ratios from the symmetric form
  • Line through two given points: deriving direction ratios and equation
  • Reduction of the general form (two planes) to the symmetrical form
  • Perpendicular distance of a point from a line in 3D
  • Condition for a line to lie in a plane: two conditions (point on plane, direction perpendicular to normal)
  • General equation of a plane containing a given line
  • Coplanar lines: condition for two lines to be coplanar (using determinant condition)
  • Skew lines: definition, shortest distance between two skew lines — magnitude (formula using cross product) and equation of the line of shortest distance
  • Intersection of three planes: consistent, inconsistent, and dependent systems

Unit 7: Sphere (4 hours)

  • Sphere and equation of a sphere: standard form (x−a)²+(y−b)²+(z−c)² = r²; general form x²+y²+z²+2ux+2vy+2wz+d = 0; centre (−u,−v,−w) and radius √(u²+v²+w²−d)
  • Representation by the general second-degree equation: conditions for the general equation to represent a sphere (coefficients of x², y², z² equal; no xy, yz, xz terms)
  • Sphere through four given points: four equations in four unknowns (u, v, w, d)
  • Plane section of a sphere: the section is a circle; finding centre and radius of the circular section
  • Intersection of two spheres: the intersection lies on a plane (the radical plane); coaxial spheres
  • Sphere with a given diameter: equation using endpoints (x₁,y₁,z₁) and (x₂,y₂,z₂): (x−x₁)(x−x₂)+(y−y₁)(y−y₂)+(z−z₁)(z−z₂) = 0
  • Tangent plane to a sphere at a point: equation; condition of tangency for a plane

Unit 8: Cone and Cylinder (8 hours)

  • Cone: definition (locus of a line through a fixed point generating a fixed curve); equation of a cone with vertex at origin: homogeneous equation f(x/z, y/z) = 0; general equation of a cone
  • Condition that the general second-degree equation represents a cone
  • Condition that a cone has three mutually perpendicular generators: trace on a coordinate plane conditions
  • Tangent lines and tangent plane to a cone at a point: equation of tangent plane
  • Condition of tangency for a plane and a cone
  • Reciprocal cone: definition and equation
  • Enveloping cone of a sphere (or conicoid): definition and equation
  • Right circular cone: axis, semi-vertical angle; equation given vertex, axis direction, and semi-vertical angle
  • Cylinder: definition (surface generated by a line moving parallel to a fixed direction and intersecting a fixed curve)
  • Equation of a cylinder with given generators parallel to z-axis
  • Enveloping cylinder of a sphere (or conicoid)
  • Right circular cylinder: definition, equation given axis and radius

Unit 9: Central Conicoids (10 hours)

  • Conicoids and central conicoids: general definition; standard equation ax²+by²+cz² = 1 (ellipsoid, hyperboloid of one sheet, hyperboloid of two sheets depending on signs of a, b, c)
  • Intersection of a line with a conicoid: quadratic in the parameter t; tangency condition (discriminant = 0)
  • Tangent at a point: equation of the tangent plane at (x₁,y₁,z₁) to ax²+by²+cz² = 1
  • Condition of tangency for a plane with a conicoid
  • Director sphere: locus of the intersection of three mutually perpendicular tangent planes
  • Equation of the normal at a point on a conicoid
  • Cubic curve through the feet of six normals from an external point
  • Polar plane and plane of contact
  • Enveloping cone of the central conicoid from an external point
  • Enveloping cylinder to a conicoid in a given direction
  • Section of a conicoid: plane sections are conics; diametrical plane definition
  • Conjugate diameters and diametrical planes of an ellipsoid: properties, relation between conjugate semi-diameters (sum of squares of conjugate semi-diameters = a²+b²+c²)

Unit 10: Product of Three or More Vectors (6 hours)

  • Scalar triple product: definition a·(b×c); geometric meaning (volume of parallelepiped); properties — cyclic symmetry, sign change on interchange; determinant representation
  • Condition of coplanarity of three vectors: scalar triple product = 0; application to coplanarity of four points
  • Vector triple product: a×(b×c) = (a·c)b − (a·b)c (expansion formula); proof and applications
  • Scalar product of four vectors: (a×b)·(c×d) = determinant form |a·c a·d; b·c b·d|
  • Vector product of four vectors: (a×b)×(c×d) — expansion using vector triple product formula
  • Reciprocal system of vectors: definition of reciprocal vectors a‘, b‘, c‘ satisfying a·a‘ = 1, a·b‘ = 0, etc.; application in crystallography and oblique coordinate systems

Unit 11: Differentiation of Vectors (8 hours)

  • Vector function of a single variable: r(t) = f(t)i + g(t)j + h(t)k; representation in terms of unit vectors
  • Limit and continuity of vector functions: definitions analogous to scalar functions
  • Differentiation of a vector function with respect to a scalar: dr/dt — component-wise differentiation; geometric meaning (tangent vector to a space curve)
  • Higher derivatives: d²r/dt², d³r/dt³; velocity and acceleration interpretation in kinematics
  • Partial derivatives of vectors: ∂r/∂u, ∂r/∂v for surfaces r(u,v)
  • Differentiation of the product of a scalar and a vector: d(φa)/dt = (dφ/dt)a + φ(da/dt)
  • Differentiation of the scalar product of two vectors: d(a·b)/dt = (da/dt)·b + a·(db/dt)
  • Differentiation of the vector product of two vectors: d(a×b)/dt = (da/dt)×b + a×(db/dt) (order matters)
  • Differentiation of the scalar triple product and vector triple product of three vectors

Unit 12: Gradient, Divergence and Curl, and Expansion Formulae (8 hours)

  • Scalar point function and vector point function: definitions; scalar field and vector field
  • Vector differential operator ∇ (del/nabla): ∇ = i ∂/∂x + j ∂/∂y + k ∂/∂z
  • Gradient of a scalar field: grad φ = ∇φ; directional derivative as ∇φ · n̂; gradient in polar coordinates
  • Geometric meaning of gradient: normal to level surfaces; direction of maximum rate of increase
  • Condition for a scalar point function to be constant: ∇φ = 0
  • Total differential: dφ = ∇φ · dr = (∂φ/∂x)dx + (∂φ/∂y)dy + (∂φ/∂z)dz
  • Directional derivative: Dᵤφ = ∇φ · û; maximum directional derivative = |∇φ|
  • Divergence of a vector field: div F = ∇·F = ∂F₁/∂x + ∂F₂/∂y + ∂F₃/∂z; physical meaning (flux per unit volume); solenoidal vector field (∇·F = 0)
  • Curl of a vector field: curl F = ∇×F (3×3 determinant); physical meaning (circulation per unit area); irrotational field (∇×F = 0)
  • Expansion formulae (first order): ∇(φ+ψ), ∇(φψ), ∇(a·b), ∇×(a×b), ∇·(φa), ∇×(φa), ∇·(a×b)
  • Second order differential operators: ∇²φ (Laplacian = ∂²φ/∂x² + ∂²φ/∂y² + ∂²φ/∂z²); curl(grad φ) = 0; div(curl F) = 0; expansion formulae involving second order operators

Recommended Textbooks and References

For MAT 101 (Calculus) — as listed in the official CDM TU syllabus

  • Singh, M.B. and Bajracharya, B.C. Differential Calculus. Sukunda Pustak Bhandar, Kathmandu, 1995 (primary Nepali reference — widely available)
  • Pant, G.D. and Shrestha, G.S. Integral Calculus and Differential Equations. Sunita Prakashan, Kathmandu, 1994 (primary Nepali reference)
  • Murray, D.A. Introductory Course in Differential Equations. Oriental Longman
  • Apostol, T.M. Calculus, Vols I & II. Wiley Eastern Ltd., New Delhi, 1986 (standard international reference)
  • Das and Mukherjee. Differential Calculus. U.N. Dhar and Sons, Calcutta
  • Das and Mukherjee. Integral Calculus. U.N. Dhar and Sons, Calcutta
  • Maskey, S.M. Calculus. Ratna Pustak Bhandar, Kathmandu, 2008

For MAT 102 (Analytical Geometry and Vector Analysis) — as listed in the official CDM TU syllabus

  • Sthapit, Y.R. and Bajracharya, B.C. A Text Book of Three Dimensional Geometry. Sukunda Pustak Bhandar, Kathmandu (primary Nepali reference)
  • Loney, S.L. Elements of Coordinate Geometry. MacMillan Books Co., NY, 1984
  • Bell, J.T. An Elementary Treatise of Coordinate Geometry of Three Dimensions. MacMillan Book Co., NY
  • Singh, M.B. and Bajracharya, B.C. A Text Book of Vector Analysis. Sukunda Pustak Bhandar, Kathmandu (primary Nepali reference)
  • Narayan, S. Analytical Solid Geometry. S. Chand and Co.
  • Prasad, Lalji. Vector Analysis. Paramoung Publication, 1986
  • Joshi, M.R. Analytical Geometry. Sukunda Pustak Bhandar, Kathmandu

Difference Between NEB Grade 12 and TU B.Sc. First Year Mathematics

Students entering TU B.Sc. First Year Mathematics from NEB Grade 12 encounter a significant step-up in both abstraction and rigor. NEB Grade 12 Mathematics (Basic Mathematics Grade XII) covers calculus through applications of derivatives, definite integrals, and an introduction to differential equations — all at a relatively procedural level. TU MAT 101 revisits and extends this in Unit 0, then immediately advances into topics with no NEB equivalent: formal curve tracing using curvature and asymptotes, the Beta and Gamma functions, double integrals with change of variables and applications in mechanics, and differential equations of the first order but not first degree (Clairaut’s equation, p-discriminant, c-discriminant methods) and higher-order linear differential equations with symbolic operator methods.

MAT 102 similarly extends NEB content significantly. NEB Grade 12 covers conic sections (standard equations of parabola, ellipse, hyperbola, focus-directrix properties) and dot and cross products of vectors — but only in two dimensions and at an introductory level. TU MAT 102 Unit 2 covers ellipse and hyperbola in exhaustive detail (conjugate diameters, asymptotes, pole-polar theory), Unit 3 introduces polar equations of conics, Units 5–9 develop three-dimensional geometry from planes and lines through spheres, cones, cylinders, and conicoids (quadric surfaces) — content entirely absent from the NEB curriculum — and Units 10–12 introduce vector triple products, vector differentiation, and the full machinery of vector calculus (gradient, divergence, curl, and the ∇ operator).

Tips to Succeed in TU B.Sc. First Year Mathematics

  • Read both official syllabuses from the CDM TU website: The official PDFs for Math 101 and Math 102 (downloadable from cdmathtu.edu.np) list every topic and teaching hour. Use these as your master checklist — if a topic is on the syllabus, it can appear on the exam.
  • Master curve tracing early in MAT 101: Unit 3 (Applications of Derivatives, 12 hours) is the largest in MAT 101 and requires skills — finding asymptotes, tracing standard curves like the cissoid, folium of Descartes, cardioid, and lemniscate — that are entirely new to most students. Practice curve tracing weekly from the first month rather than leaving it to revision.
  • For double integrals (Unit 8), always sketch the region: The most common source of marks lost in double integral problems is setting wrong limits of integration. Always draw the region D, identify the boundary curves, and determine correct limits before computing. Changing the order of integration requires re-sketching — do this geometrically, not algebraically.
  • MAT 102 Unit 2 (Conic Sections, 14 hours) requires deep mastery: This is the most extensive unit in MAT 102. Move systematically through ellipse (standard form, focal properties, tangent/normal, conjugate diameters) before moving to hyperbola. Do not attempt the general second-degree equation (Unit 4) until Unit 2 is fully understood.
  • In 3D geometry (Units 5–9), always write the equation of a plane/line in standard form first: Many 3D geometry problems — finding the distance from a point to a plane, the shortest distance between skew lines, or the tangent plane to a sphere — require identifying the standard form of the geometric object first. Students who rush to plug into formulas without first expressing the equation in standard form make consistent errors.
  • Build a systematic derivation notebook for Unit 12 (Gradient, Divergence, Curl): The expansion formulae in Unit 12 (∇(φa), ∇·(a×b), ∇×(a×b)) are tested in long-answer questions and require proof. Derive each formula at least twice in your own notation, paying attention to when the order of factors matters (especially for cross products).
  • Use Nepali-authored textbooks as your primary study material: Singh & Bajracharya’s Differential Calculus, Pant & Shrestha’s Integral Calculus, Sthapit & Bajracharya’s Three Dimensional Geometry, and Singh & Bajracharya’s Vector Analysis are all widely available across Nepal, written in a style aligned with the TU syllabus, and structured around the types of problems that appear in TU examinations. These should be your first resource, supplemented by Apostol for deeper theoretical understanding.
  • Solve past TU B.Sc. Mathematics First Year question papers: The 2082 Regular examination question papers for both Calculus and Analytical Geometry & Vector Analysis are available through EducateNepal and similar platforms. These reveal exactly which units carry the most weight in the annual paper and the typical format and length of answers expected.

Career Pathways for TU B.Sc. Mathematics Graduates

B.Sc. Mathematics equips graduates with rigorous analytical skills applicable across a wide range of fields. Career pathways include:

  • Academic and research: M.Sc. and Ph.D. in Mathematics, Statistics, Computer Science, or related fields at TU and international universities; university and college teaching (requiring M.Sc. and TSC clearance)
  • Data science and analytics: the mathematical foundation in calculus, linear algebra, and statistics is directly relevant to careers in data science, machine learning, and quantitative finance — growing sectors in Nepal’s digital economy
  • Banking and finance: actuarial science, risk analysis, quantitative modelling in commercial and development banks; NRB and government financial institutions
  • Government services: Lok Sewa Aayog (Public Service Commission) examinations for positions requiring quantitative/analytical skills in survey departments, statistics departments, and planning commissions
  • Education: secondary and higher secondary level mathematics teaching (TU B.Sc. Mathematics with BEd provides the strongest qualification)
  • Software and engineering: B.Sc. Mathematics graduates frequently pursue M.Sc. Computer Science or M.Sc. Information Technology, leveraging their mathematical background for roles in algorithm design, cryptography, and scientific computing

Frequently Asked Questions (FAQs)

What are the two papers in TU B.Sc. First Year Mathematics?

The two First Year Mathematics papers under the Revised B.Sc. Curriculum 2073 are MAT 101 (Calculus), carrying 75 full marks (pass mark 26.25), and MAT 102 (Analytical Geometry and Vector Analysis), carrying 75 full marks (pass mark 35%). Both are pure theory papers — there are no practical papers for First Year Mathematics.

How many units are in MAT 101 and MAT 102?

MAT 101 (Calculus) has 13 units: Unit 0 (Review of Elementary Calculus) through Unit 12 (Applications of First Order Differential Equations), totaling 80 lecture hours. MAT 102 (Analytical Geometry and Vector Analysis) has 12 units: Unit 1 (Transformation of Coordinates) through Unit 12 (Gradient, Divergence and Curl), totaling 80 lecture hours.

What is the difference between the old and the new (Revised 2073) B.Sc. Mathematics course structure?

Under the old three-year B.Sc. curriculum, the course codes were Math(101) Calculus and Math(102) Analytical Geometry & Vector Analysis for First Year. The Revised 2073 four-year B.Sc. curriculum uses the same two papers — MAT 101 Calculus and MAT 102 Analytical Geometry & Vector Analysis — but with updated content and teaching hours distributed across 80 hours per paper. The content of both papers is closely aligned between old and new syllabuses; the primary change is the extension to a four-year programme, the addition of new subjects in later years (Linear Algebra, Real Analysis, Modern Algebra, Mathematical Analysis), and some content reorganization.

Does TU B.Sc. First Year Mathematics have any practical papers?

No. Unlike Chemistry, Botany, Zoology, Microbiology, Physics, and Geology, First Year Mathematics has no separate practical papers. Both MAT 101 and MAT 102 are pure theory papers, each with a written annual examination of 3 hours.

Which textbooks are officially recommended for TU B.Sc. First Year Mathematics?

For MAT 101 (Calculus), the official CDM TU syllabus recommends: Singh & Bajracharya (Differential Calculus, Sukunda Pustak Bhandar), Pant & Shrestha (Integral Calculus and Differential Equations, Sunita Prakashan), Apostol (Calculus, Vols I & II), and Maskey (Calculus, Ratna Pustak Bhandar). For MAT 102, the official recommendations include: Sthapit & Bajracharya (Three Dimensional Geometry, Sukunda Pustak Bhandar), Loney (Elements of Coordinate Geometry), Singh & Bajracharya (Vector Analysis, Sukunda Pustak Bhandar), and Narayan (Analytical Solid Geometry, S. Chand).

Where can I find TU B.Sc. First Year Mathematics past papers and notes?

The 2082 Regular examination papers for both Calculus and Analytical Geometry & Vector Analysis are available through EducateNepal (educatenepal.com). The official syllabuses for Math 101 and Math 102 are downloadable as PDFs directly from the Central Department of Mathematics website (cdmathtu.edu.np). Notes and solved problems are available through MeroSpark (old.merospark.com) and various Nepali educational platforms. The official syllabus PDF is also embedded at the top of this page.

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