Agricultural Engineering · Foundations · also Engineering Mathematics I & II, Mathematics (PRC), Higher Mathematics
The algebra, calculus, differential equations and numerical methods used in every numerical question of the degree. GATE AG gives it a fixed block of marks every year.
In the exams: GATE AG Section 1, about 12–13 marks a year (2019–26); ICAR AIEEA-PG Unit I; NCEES FE 8–12 questions; allied subject in PRC ABE
Engineering mathematics is the working language of every other subject in agricultural engineering. A drying curve is the solution of a first-order differential equation; the spacing of field drains comes from the Laplace equation; the frequency at which a tractor seat shakes is an eigenvalue; the volume of a storm hydrograph is a numerical integral. This manual covers the sixteen units of the degree course: linear algebra, the calculus of several variables, series, vector calculus and its integral theorems, ordinary and partial differential equations, Laplace transforms, and the numerical methods used when no closed-form answer exists.
Each chapter defines its terms, states the results you must know, shows how they are derived where the derivation is short, and works at least one example with the numbers checked. The emphasis is on the methods that examiners set as short numerical problems, and on the slips that cost marks: a sign lost in a cofactor, a lower limit forgotten, an integration constant dropped, a step size halved in the wrong place.
Worldwide the subject is examined in much the same way. In India, GATE Agricultural Engineering gives Engineering Mathematics a fixed block of 13 marks, set as one- and two-mark multiple-choice, multiple-select and numerical-answer items, and ICAR AIEEA-PG puts mathematics in its first unit. In the United States, the NCEES Fundamentals of Engineering (Other Disciplines) exam gives mathematics roughly 8 to 12 of its 110 questions, answered with the NCEES reference handbook open on screen. In the Philippines, mathematics is an allied subject in the Agricultural and Biosystems Engineers licensure exam. China's postgraduate entrance exam for agricultural engineering can use a national mathematics paper, which goes deeper into proofs than the others. The content below serves all of them; where an exam differs in emphasis, the Exam Focus section says so.
A matrix is a rectangular array of numbers with m rows and n columns (an m × n matrix). Two matrices of the same size are added entry by entry. The product AB is defined only when the number of columns of A equals the number of rows of B; the entry in row i, column j of AB is the sum of the products of row i of A with column j of B. Matrix multiplication is associative and distributive but not commutative: in general AB ≠ BA, and AB = 0 does not imply A = 0 or B = 0.
The transpose Aᵀ swaps rows and columns, and (AB)ᵀ = BᵀAᵀ. A square matrix is symmetric if Aᵀ = A and skew-symmetric if Aᵀ = −A; every square matrix is the sum of a symmetric part (A + Aᵀ)/2 and a skew-symmetric part (A − Aᵀ)/2. The identity matrix I has ones on the diagonal and zeros elsewhere, so AI = IA = A. A diagonal matrix has non-zero entries only on the diagonal; a triangular matrix has zeros entirely above (lower triangular) or below (upper triangular) the diagonal. An orthogonal matrix satisfies AᵀA = I, so its inverse is its transpose; rotation matrices are the usual example.
In practice, matrices store systems of equations (a network of pipes, a set of mass balances), transformations (rotating coordinates of a survey), and data (yields of several varieties at several sites). Knowing which operations are legal, and in which order, is the first skill.
The determinant is a single number attached to a square matrix. For a 2 × 2 matrix [[a, b], [c, d]] it is ad − bc. For a 3 × 3 matrix it is found by cofactor expansion along any row or column: each entry is multiplied by its cofactor, which is (−1)^(i+j) times the minor (the determinant left after deleting that entry's row and column). The sign pattern starts + − + along the first row. Forgetting the minus sign on the middle term is the single most common slip.
Properties used constantly: swapping two rows changes the sign; multiplying one row by k multiplies the determinant by k, so for an n × n matrix det(kA) = kⁿ det(A); adding a multiple of one row to another leaves the determinant unchanged; a matrix with two equal or proportional rows has determinant zero; det(AB) = det(A)det(B); det(Aᵀ) = det(A); det(A⁻¹) = 1/det(A). The determinant of a triangular matrix is the product of its diagonal entries, which is why row reduction to triangular form is the efficient way to evaluate large determinants.
Geometrically, |det A| is the factor by which A scales areas (2 × 2) or volumes (3 × 3). A zero determinant means the transformation squashes space into a lower dimension, so it cannot be undone: the matrix is singular and has no inverse.
A square matrix A has an inverse A⁻¹ with AA⁻¹ = A⁻¹A = I exactly when det A ≠ 0. For a 2 × 2 matrix, A⁻¹ = (1/(ad − bc))[[d, −b], [−c, a]]: swap the diagonal entries and change the signs of the off-diagonal ones. In general A⁻¹ = adj(A)/det(A), where the adjoint adj(A) is the transpose of the matrix of cofactors. For anything larger than 3 × 3, Gauss-Jordan elimination on [A | I] is faster. Useful rules: (AB)⁻¹ = B⁻¹A⁻¹ and (Aᵀ)⁻¹ = (A⁻¹)ᵀ.
The rank of a matrix is the largest number of linearly independent rows (equal to the number of independent columns). It is found by row-reducing to echelon form and counting non-zero rows, or as the order of the largest non-zero minor. An n × n matrix has rank n exactly when its determinant is non-zero. A matrix with one row equal to twice another has rank at most n − 1. Rank tells you how many genuinely different equations a system contains.
A system of m equations in n unknowns is written Ax = b. Form the augmented matrix [A | b]. The Rouché-Capelli (rank) test decides the outcome: if rank A ≠ rank [A | b], the system is inconsistent and has no solution; if rank A = rank [A | b] = n, it has a unique solution; if rank A = rank [A | b] = r < n, it has infinitely many solutions with n − r free parameters. A homogeneous system Ax = 0 always has the trivial solution x = 0, and has non-trivial solutions if and only if rank A < n, which for a square matrix means det A = 0.
Gauss elimination reduces [A | b] to upper-triangular form by row operations, then solves by back-substitution. Example: x + y + z = 6, 2x − y + z = 3, x + 2y − z = 2. Subtract 2 × row 1 from row 2 and row 1 from row 3: −3y − z = −9 and y − 2z = −4. From these, y = 2 and z = 3, and then x = 1. Check in the original equations before moving on.
Cramer's rule gives each unknown as a ratio of determinants, xᵢ = Dᵢ/D, where Dᵢ is D with column i replaced by b. It is convenient for 2 × 2 and 3 × 3 systems in an exam and useless for large ones. When a parameter appears in the coefficients, reduce the matrix and look at the last row: a zero coefficient with a zero right-hand side means infinitely many solutions, and a zero coefficient with a non-zero right-hand side means none.
A non-zero vector v is an eigenvector of a square matrix A if Av = λv for some scalar λ, the eigenvalue. The matrix only stretches v, it does not turn it. Rewriting as (A − λI)v = 0 shows that a non-zero v exists only when det(A − λI) = 0. This is the characteristic equation; for an n × n matrix it is a polynomial of degree n in λ, so there are n eigenvalues counted with multiplicity, possibly complex.
For a 2 × 2 matrix the characteristic equation is λ² − (trace A)λ + det A = 0, where the trace is the sum of the diagonal entries. For example A = [[5, 4], [1, 2]] has trace 7 and determinant 6, so λ² − 7λ + 6 = 0 and λ = 1, 6. To find the eigenvector for λ = 6, solve (A − 6I)v = [[−1, 4], [1, −4]]v = 0, which gives v = [4, 1]ᵀ (or any non-zero multiple). For λ = 1, [[4, 4], [1, 1]]v = 0 gives v = [1, −1]ᵀ. Eigenvectors are defined only up to a scale factor; an answer that differs from yours by a constant multiple is still correct.
Two checks catch most arithmetic errors: the sum of the eigenvalues equals the trace, and their product equals the determinant. A matrix is singular exactly when one of its eigenvalues is zero.
If A has eigenvalue λ with eigenvector v, then: kA has eigenvalue kλ; Aᵐ has λᵐ; A⁻¹ (if it exists) has 1/λ; A + cI has λ + c; and any polynomial p(A) has p(λ), all with the same eigenvector. Aᵀ has the same eigenvalues as A (but generally different eigenvectors). The eigenvalues of a triangular or diagonal matrix are its diagonal entries.
A real symmetric matrix has only real eigenvalues, and eigenvectors belonging to different eigenvalues are orthogonal. It can therefore always be diagonalised as A = PDPᵀ with P orthogonal. The eigenvalues of a skew-symmetric real matrix are zero or purely imaginary; those of an orthogonal matrix have absolute value 1.
Eigenvectors for distinct eigenvalues are linearly independent. An n × n matrix with n independent eigenvectors can be diagonalised: P⁻¹AP = D, where the columns of P are the eigenvectors and D holds the eigenvalues. Then Aᵏ = PDᵏP⁻¹, which is how powers of a matrix are computed in practice. A matrix with a repeated eigenvalue may or may not be diagonalisable; count the independent eigenvectors (the geometric multiplicity) to decide.
In engineering, eigenvalues appear as natural frequencies of vibrating systems (a tractor on its tyres, a sprayer boom), as principal stresses in a soil element, and as growth or decay rates of linked populations or storages.
The Cayley-Hamilton theorem states that every square matrix satisfies its own characteristic equation. If det(A − λI) = λⁿ + c₁λⁿ⁻¹ + … + cₙ, then Aⁿ + c₁Aⁿ⁻¹ + … + cₙI = 0. The constant term becomes cₙI, not cₙ — writing a bare number instead of a multiple of I is a common error.
Two uses dominate exams. First, higher powers: for A = [[5, 4], [1, 2]], λ² − 7λ + 6 = 0 gives A² = 7A − 6I, so A³ = 7A² − 6A = 7(7A − 6I) − 6A = 43A − 42I, and so on, without ever multiplying matrices. Second, the inverse: multiply A² − 7A + 6I = 0 by A⁻¹ to get A − 7I + 6A⁻¹ = 0, so A⁻¹ = (7I − A)/6 = (1/6)[[2, −4], [−1, 5]]. Check by multiplying back to I.
The theorem also shows that any polynomial in an n × n matrix can be reduced to a polynomial of degree at most n − 1, which is the basis of several methods for computing matrix functions such as e^(At) in control and vibration problems.
lim(x→a) f(x) = L means f(x) can be made as close to L as desired by taking x close enough to a (but not equal to it). The limit exists only if the left-hand and right-hand limits are equal. Standard limits worth memorising: sin x/x → 1 as x → 0; (1 − cos x)/x² → 1/2; (eˣ − 1)/x → 1; ln(1 + x)/x → 1; (1 + a/x)ˣ → eᵃ as x → ∞; and (xⁿ − aⁿ)/(x − a) → naⁿ⁻¹.
Indeterminate forms 0/0 and ∞/∞ are resolved by L'Hôpital's rule (differentiate numerator and denominator separately and take the limit again) or, often faster, by Maclaurin series. Forms 0·∞, ∞ − ∞, 1^∞, 0⁰ and ∞⁰ are first rearranged into 0/0 or ∞/∞, usually by taking logarithms. A 1^∞ form is not equal to 1: that is the trap in (1 + 2/x)ˣ.
A function is continuous at a if f(a) is defined, the limit exists, and the two are equal. A removable discontinuity, such as (x² − 4)/(x − 2) at x = 2, is fixed by defining f(2) as the limit, here 4. A function differentiable at a point is continuous there, but not conversely: |x| is continuous at 0 and not differentiable.
For functions of two variables, a limit at (a, b) must be the same along every path of approach. If two paths give different values (for xy/(x² + y²), along y = mx the value is m/(1 + m²)), the limit does not exist.
For f(x, y), the partial derivative ∂f/∂x is the derivative with respect to x with y held constant. Higher derivatives follow: fₓₓ, f_yy and the mixed derivatives fₓy and f_yx. If the mixed derivatives are continuous they are equal (Schwarz or Clairaut theorem), which is a useful check.
The total differential df = fₓ dx + f_y dy estimates the change in f for small changes in x and y. It is the basis of error analysis: for a quantity Q = k xᵃ yᵇ, the relative error is approximately dQ/Q = a dx/x + b dy/y, so an error of 1% in a radius becomes about 2% in an area and about 3% in a volume. Engineers use this to decide which measurement most needs to be accurate — for instance the depth reading in a weir formula that uses depth to the power 1.5.
Chain rule: if z = f(x, y) with x = x(t) and y = y(t), then dz/dt = fₓ dx/dt + f_y dy/dt. If x and y depend on two variables (u, v), then ∂z/∂u = fₓ ∂x/∂u + f_y ∂y/∂u, and similarly for v. For an implicit relation f(x, y) = 0, dy/dx = −fₓ/f_y. The Jacobian ∂(u, v)/∂(x, y) = uₓv_y − u_yvₓ measures how a change of variables scales area; it reappears when changing variables in double integrals.
A function f(x, y) is homogeneous of degree n if f(tx, ty) = tⁿ f(x, y) for all t > 0. Polynomials in which every term has the same total degree are homogeneous: x³ + y³ + 3x²y has degree 3. Ratios are homogeneous if numerator and denominator are: (x³ + y³)/(x + y) has degree 3 − 1 = 2, and (x^(1/3) + y^(1/3))/(x^(1/2) + y^(1/2)) has degree 1/3 − 1/2 = −1/6. The degree can be negative, fractional or zero.
Euler's theorem: if f is homogeneous of degree n with continuous partial derivatives, then x fₓ + y f_y = n f. Differentiating once more gives the second-order form x² fₓₓ + 2xy fₓy + y² f_yy = n(n − 1) f.
The standard exam twist is a function that is not homogeneous itself but becomes so after applying a known function. If u = sin⁻¹(g) with g homogeneous of degree n, then z = sin u = g satisfies x zₓ + y z_y = n z, which gives cos u (x uₓ + y u_y) = n sin u, so x uₓ + y u_y = n tan u. Similarly, if u = ln g with g of degree n, x uₓ + y u_y = n; if u = tan⁻¹(g), the result is n sin u cos u = (n/2) sin 2u. Always ask first: which function of u is homogeneous?
Euler's theorem is used beyond the classroom. A production function Q = A Lᵃ Kᵇ is homogeneous of degree a + b; Euler's theorem then says L ∂Q/∂L + K ∂Q/∂K = (a + b)Q, so a + b = 1 means constant returns to scale. The same idea underlies dimensional analysis in fluid mechanics.
A point where all first partial derivatives vanish is a stationary (critical) point. For f(x, y), solve fₓ = 0 and f_y = 0 together. To classify each point, compute r = fₓₓ, s = fₓy and t = f_yy there and form D = rt − s²:
If D > 0 and r > 0, the point is a local minimum. If D > 0 and r < 0, it is a local maximum. If D < 0, it is a saddle point — a maximum along one direction and a minimum along another. If D = 0 the test is inconclusive and higher-order terms or a direct study of f near the point are needed.
Worked example: f = x² + xy + y² − 3x. fₓ = 2x + y − 3 = 0 and f_y = x + 2y = 0 give x = 2, y = −1. Here r = 2, s = 1, t = 2, so D = 4 − 1 = 3 > 0 with r > 0: a minimum, with value f(2, −1) = 4 − 2 + 1 − 6 = −3.
The same test in matrix form uses the Hessian, the matrix of second derivatives [[r, s], [s, t]]. A positive definite Hessian (all eigenvalues positive) means a minimum, negative definite means a maximum, and eigenvalues of mixed sign mean a saddle. This form extends directly to three or more variables. The absolute (global) extrema of a function on a closed, bounded region can also lie on the boundary, so in applied problems check the edges as well as the interior.
Engineering problems usually have constraints: a fixed length of fence, a fixed tank volume, a fixed budget. For a single constraint the simplest route is often to eliminate a variable: a paddock fenced on three sides with 400 m of fence has x + 2y = 400, so area A = (400 − 2y)y, and dA/dy = 0 gives y = 100 m, x = 200 m.
When elimination is awkward, use Lagrange multipliers. To find extrema of f(x, y, z) subject to g(x, y, z) = c, form L = f − λ(g − c) and solve ∂L/∂x = ∂L/∂y = ∂L/∂z = 0 together with the constraint. Geometrically, at the optimum the gradient of f is parallel to the gradient of g: ∇f = λ∇g. The multiplier λ has a meaning of its own — it is the rate at which the optimal value of f changes as the constraint level c changes (a shadow price in economics).
Example: maximise xyz with x + y + z = 12 (x, y, z > 0). The conditions yz = xz = xy = λ give x = y = z = 4, so the maximum product is 64. Example: the shortest distance from the origin to the plane ax + by + cz = d lies along the normal and equals |d|/√(a² + b² + c²).
A classic design result: an open-top tank with a square base and fixed volume V uses the least sheet metal when the depth is half the base side. For 32 m³ that is a 4 m × 4 m base, 2 m deep, with 48 m² of sheet. A closed box of fixed volume uses least material as a cube.
A sequence a₁, a₂, a₃, … converges to L if aₙ gets and stays arbitrarily close to L as n grows. Rational expressions are handled by dividing by the highest power of n: n/(2n + 1) = 1/(2 + 1/n) → 1/2. Useful facts: rⁿ → 0 when |r| < 1; n^(1/n) → 1; (1 + 1/n)ⁿ → e; and any power of n grows more slowly than any exponential aⁿ with a > 1, which in turn grows more slowly than n!.
A monotonic sequence that is bounded converges. This is how many iterative numerical methods are shown to converge: each iterate moves the same way and cannot pass a known bound. A sequence defined by a recurrence, xₙ₊₁ = g(xₙ), converges to a fixed point x* = g(x*) when |g'(x*)| < 1 near that point; this is the condition behind fixed-point iteration, and the smaller |g'(x*)|, the faster the approach. Divergent sequences are of two kinds: those that grow without bound, such as 2ⁿ, and those that oscillate, such as (−1)ⁿ.
A series Σaₙ converges if its partial sums Sₙ = a₁ + … + aₙ approach a finite limit. The first check is the nth-term test: if aₙ does not tend to zero, the series diverges. The converse is false. The harmonic series Σ1/n has terms tending to zero yet diverges; its partial sums grow like ln n.
The geometric series Σarⁿ (n from 0) converges to a/(1 − r) when |r| < 1 and diverges otherwise. Check the starting index: Σ(2/3)ⁿ from n = 0 is 3, but from n = 1 it is 2. The p-series Σ1/nᵖ converges for p > 1 and diverges for p ≤ 1. Telescoping series collapse: Σ1/[n(n + 1)] = Σ[1/n − 1/(n + 1)] = 1.
For series of positive terms: the comparison test (smaller than a convergent series converges, larger than a divergent one diverges); the limit comparison test (if aₙ/bₙ → a finite non-zero limit, both behave alike); the ratio test (if aₙ₊₁/aₙ → ρ, the series converges for ρ < 1, diverges for ρ > 1, and the test fails at ρ = 1); the root test (the same with (aₙ)^(1/n)); and the integral test (Σf(n) and ∫f(x)dx converge together when f is positive and decreasing). The ratio test is the one to reach for when factorials or exponentials appear; it fails on every p-series.
An alternating series Σ(−1)ⁿ⁺¹bₙ converges if bₙ decreases to zero (the Leibniz test), and the error after stopping is at most the first term left out. A series is absolutely convergent if Σ|aₙ| converges, and conditionally convergent if it converges only because of the sign changes. 1 − 1/2 + 1/3 − … converges to ln 2 but only conditionally.
A power series Σcₙ(x − a)ⁿ converges for |x − a| < R and diverges for |x − a| > R, where the radius of convergence R is found from the ratio test: R = lim |cₙ/cₙ₊₁| when that limit exists. At the two end points x = a ± R, the series must be tested separately. For Σxⁿ/3ⁿ, R = 3; multiplying the coefficients by n or by 1/n² does not change R. Series for eˣ, sin x and cos x have R = ∞; the series for ln(1 + x) and 1/(1 − x) have R = 1.
Inside its interval of convergence a power series can be differentiated and integrated term by term, which is how many standard series are generated: integrating 1/(1 + x) = 1 − x + x² − … gives ln(1 + x) = x − x²/2 + x³/3 − …, and differentiating Σxⁿ = 1/(1 − x) gives Σnxⁿ⁻¹ = 1/(1 − x)². Sums such as Σn/2ⁿ = 2 and Σn²/2ⁿ = 6 come from these manipulations.
If f has derivatives of all orders at x = a, its Taylor series about a is f(a) + f'(a)(x − a) + f''(a)(x − a)²/2! + f'''(a)(x − a)³/3! + …. With a = 0 it is called the Maclaurin series. Truncating after the term in (x − a)ⁿ leaves a remainder that, in Lagrange's form, equals f⁽ⁿ⁺¹⁾(ξ)(x − a)ⁿ⁺¹/(n + 1)! for some ξ between a and x; this bounds the error of the approximation.
Maclaurin series to know by heart: eˣ = 1 + x + x²/2! + x³/3! + …; sin x = x − x³/3! + x⁵/5! − …; cos x = 1 − x²/2! + x⁴/4! − …; ln(1 + x) = x − x²/2 + x³/3 − … (−1 < x ≤ 1); (1 + x)ⁿ = 1 + nx + n(n − 1)x²/2! + … (|x| < 1 for non-integer n); and tan x = x + x³/3 + 2x⁵/15 + …. New series come from substitution: e^(2x) has x³ coefficient 2³/3! = 4/3, not 8 and not 1/6.
Example: √1.1 = (1 + 0.1)^(1/2) ≈ 1 + 0.05 − 0.00125 = 1.04875, against the true 1.048809. The first neglected term, +x³/16 = 0.0000625, accounts for almost all the difference. Angles must be in radians in every trigonometric series.
Taylor series drive much of engineering practice: linearising a non-linear equation about an operating point, deriving finite-difference formulas, evaluating limits, and building the Newton-Raphson and Runge-Kutta methods of Chapter 11. A function of two variables has the expansion f(a + h, b + k) = f + (hfₓ + kf_y) + (h²fₓₓ + 2hk fₓy + k²f_yy)/2! + …, which is where the second-derivative test of Chapter 4 comes from.
A periodic function of period 2π that is piecewise smooth can be written f(x) = a₀/2 + Σ[aₙ cos nx + bₙ sin nx], with aₙ = (1/π)∫f(x) cos nx dx and bₙ = (1/π)∫f(x) sin nx dx, both integrals over one period (−π to π). With the a₀/2 convention the constant term is the mean value of f; some books write a₀ for the mean and change the formula for a₀ accordingly, so check which convention a question uses. For period 2L replace nx by nπx/L and 1/π by 1/L.
Symmetry halves the work. An even function (f(−x) = f(x)) has only the constant and cosine terms; an odd function (f(−x) = −f(x)) has only sine terms. For f(x) = x on (−π, π), bₙ = 2(−1)ⁿ⁺¹/n: 2 sin x − sin 2x + (2/3) sin 3x − …. For f(x) = x², a₀/2 = π²/3 and aₙ = 4(−1)ⁿ/n². For the square wave equal to −1 on (−π, 0) and +1 on (0, π), bₙ = 4/(nπ) for odd n and zero for even n.
At a jump discontinuity the Fourier series converges to the average of the left and right values (the Dirichlet conditions). Near the jump the partial sums overshoot by about 9% of the jump however many terms are taken — the Gibbs phenomenon.
A function given only on (0, L) can be extended as an even function to give a half-range cosine series, or as an odd function to give a half-range sine series, with coefficients (2/L)∫₀ᴸ f cos(nπx/L) dx or (2/L)∫₀ᴸ f sin(nπx/L) dx. Half-range sine series are exactly what the heat and wave equations with fixed ends require (Chapter 10). Fourier analysis is also how engineers decompose vibration signals from engines and the periodic loads on rotating farm machinery.
The del operator is ∇ = i ∂/∂x + j ∂/∂y + k ∂/∂z. Applied to a scalar field φ it gives the gradient ∇φ, a vector pointing in the direction of fastest increase of φ, with magnitude equal to that rate. The gradient is normal to the level surfaces φ = constant, so the unit normal to a surface φ = c is ∇φ/|∇φ|. The directional derivative of φ in the direction of a vector a is ∇φ · a/|a| — the vector must be made a unit vector first. For φ = xyz at (1, 2, 3), ∇φ = (yz, xz, xy) = (6, 3, 2), and the greatest rate of increase is √49 = 7.
The divergence of a vector field F = F₁i + F₂j + F₃k is the scalar ∇ · F = ∂F₁/∂x + ∂F₂/∂y + ∂F₃/∂z. It measures net outflow per unit volume at a point: positive at a source, negative at a sink. A field with zero divergence everywhere is solenoidal; an incompressible fluid velocity field is solenoidal, which is the continuity equation of fluid mechanics.
The curl ∇ × F is the vector given by the determinant with rows (i, j, k), (∂/∂x, ∂/∂y, ∂/∂z) and (F₁, F₂, F₃). It measures local rotation; for a rigid rotation with angular velocity ω the curl is 2ω. A field with zero curl is irrotational. An irrotational field on a simply connected region is conservative: F = ∇φ for a scalar potential φ, found by integrating component by component and matching the leftover functions.
Two identities hold for any smooth fields: curl(grad φ) = 0 and div(curl F) = 0. The divergence of a gradient is the Laplacian, ∇²φ = φₓₓ + φ_yy + φ_zz, which is not zero in general; fields with ∇²φ = 0 are called harmonic. Groundwater flow in a uniform aquifer, steady heat conduction and potential flow all satisfy Laplace's equation.
A line integral ∫_C F · dr adds up the tangential component of F along a curve C. Parametrise the curve, x = x(t), y = y(t), z = z(t), substitute, and integrate in t between the parameter values at the ends. In mechanics it is the work done by a force along a path. If F = ∇φ, the line integral depends only on the end points and equals φ(end) − φ(start); around any closed path it is zero. That is what 'conservative' means.
A double integral ∬f dA over a region is evaluated as an iterated integral; the limits of the inner integral may depend on the outer variable, and reversing the order means redrawing the region and rewriting the limits, not just swapping them. ∫₀¹∫₀ˣ xy dy dx = 1/8. In polar coordinates dA = r dr dθ — the factor r is the Jacobian and is often forgotten. A triple integral ∭f dV computes volumes, masses and moments; in cylindrical coordinates dV = r dr dθ dz and in spherical coordinates dV = ρ² sin φ dρ dφ dθ. The tetrahedron cut from the first octant by x + y + z = 1 has volume 1/6.
A surface integral ∬_S F · n dS is the flux of F through S, where n is the unit normal chosen outward for a closed surface. For flow, it is the volume of fluid crossing the surface per unit time.
The three theorems all say that an integral of a derivative over a region equals an integral of the function over its boundary.
Green's theorem in the plane: ∮_C (P dx + Q dy) = ∬_R (∂Q/∂x − ∂P/∂y) dA, with C traversed anticlockwise (region on the left). Taking P = −y/2, Q = x/2 gives area = (1/2)∮(x dy − y dx), the principle behind the planimeter. ∮(x dy − y dx) around a circle of radius 2 is therefore 2 × 4π = 8π.
Stokes' theorem: ∮_C F · dr = ∬_S (∇ × F) · n dS, where S is any surface bounded by C, with the normal oriented by the right-hand rule relative to the direction of C. Green's theorem is Stokes' theorem for a flat surface in the xy-plane.
Gauss's divergence theorem: ∯_S F · n dS = ∭_V ∇ · F dV, where S is the closed surface bounding V and n points outward. For F = xi + yj + zk, ∇ · F = 3, so the outward flux through any closed surface is three times the enclosed volume — 108π for a sphere of radius 3, and 3 for the unit cube. This theorem converts the local continuity equation into a whole-volume mass balance and back, which is why it underlies every conservation law used in hydrology, drying and heat transfer.
In an exam, use a theorem whenever the direct integral is long and the derivative is simple: a constant curl or divergence reduces the work to an area or a volume.
An ordinary differential equation (ODE) relates a function of one variable to its derivatives. Its order is that of the highest derivative; its degree is the power of that highest derivative once the equation is cleared of radicals and fractions in the derivatives. (d²y/dx²)³ + (dy/dx)⁴ + y = 0 has order 2 and degree 3. An ODE is linear if y and its derivatives appear only to the first power and are not multiplied together. The general solution of an nth-order ODE contains n arbitrary constants; an initial or boundary condition fixes them to give a particular solution. Dropping the constant, or adding it only at the end after a division, is the most expensive slip in this chapter.
A separable equation can be written g(y) dy = f(x) dx; integrate both sides and add one constant. The exponential law dy/dt = −ky gives y = y₀e^(−kt); the time to fall to half is ln 2/k (13.9 days for k = 0.05 per day). Newton's law of cooling, dT/dt = −k(T − Tₐ), gives T − Tₐ = (T₀ − Tₐ)e^(−kt): it is the excess over the surrounding temperature that decays exponentially, not the temperature itself.
The thin-layer drying models used for grain follow the same structure. The simplest, often called the Lewis or Newton model, takes dM/dt = −k(M − Mₑ), with Mₑ the equilibrium moisture content, so the moisture ratio MR = (M − Mₑ)/(M₀ − Mₑ) = e^(−kt). The drying constant k depends on the crop, air temperature, humidity and velocity; it is fitted to measured data rather than taken from a table.
The standard form is dy/dx + P(x)y = Q(x). Multiply by the integrating factor IF = e^(∫P dx); the left side becomes d(y·IF)/dx, so y·IF = ∫Q·IF dx + C. The constant is added at the integration, before dividing by IF. For dy/dx + y/x = x², IF = x, so xy = x⁴/4 + C and y = x³/4 + C/x.
Mixing problems are linear first-order equations. A tank of volume V with inflow and outflow rate q, inflow concentration cᵢₙ, and well-mixed contents obeys dm/dt = q cᵢₙ − (q/V)m. With clean inflow the mass decays as m = m₀e^(−qt/V). The ratio V/q is the residence time; the same model describes flushing of a fertigation tank or a reservoir.
A Bernoulli equation dy/dx + P y = Q yⁿ (n ≠ 0, 1) becomes linear under v = y^(1−n). The logistic equation dN/dt = rN(1 − N/K) is both separable and Bernoulli with n = 2; its solution rises in an S-curve to the carrying capacity K.
M(x, y) dx + N(x, y) dy = 0 is exact if ∂M/∂y = ∂N/∂x; then it is the total differential of some F(x, y), and the solution is F = C. Find F by integrating M with respect to x (treating y as constant) and adding the integral, with respect to y, of only those terms of N that do not contain x. Integrating both M and N fully and adding counts the mixed terms twice. If the test fails, an integrating factor depending on x alone exists when (M_y − Nₓ)/N is a function of x only, and one depending on y alone when (Nₓ − M_y)/M is a function of y only.
An equation dy/dx = f(x, y) in which f is homogeneous of degree zero (it can be written as a function of y/x) is solved by y = vx, so dy/dx = v + x dv/dx, which separates. For dy/dx = (x + y)/x this gives dv/dx = 1/x, v = ln x + C, y = x ln x + Cx.
Not every first-order equation has a closed-form solution. When none of these forms fits, Chapter 11's numerical methods take over.
A linear second-order ODE with constant coefficients, ay'' + by' + cy = f(x), has general solution y = y_c + y_p: the complementary function y_c solves the equation with f = 0, and y_p is any one particular integral. Try y = e^(mx) in the homogeneous equation to get the auxiliary equation am² + bm + c = 0. Three cases: real distinct roots m₁, m₂ give y_c = C₁e^(m₁x) + C₂e^(m₂x); a repeated root m gives (C₁ + C₂x)e^(mx); complex roots α ± iβ give e^(αx)(C₁ cos βx + C₂ sin βx). Two solutions are independent if their Wronskian W = y₁y₂' − y₂y₁' is non-zero.
The spring-mass-damper m x'' + c x' + k x = 0 is the model for suspended seats, sprayer booms and machine mounts. Its undamped natural frequency is ωₙ = √(k/m) rad/s (divide by 2π for hertz). Critical damping occurs at c_c = 2√(km); the damping ratio ζ = c/c_c. ζ > 1 gives overdamped return without oscillation, ζ = 1 the fastest non-oscillating return, and ζ < 1 decaying oscillation.
The Cauchy-Euler equation x²y'' + axy' + by = 0 has variable coefficients but is solved by y = xᵐ, giving m(m − 1) + am + b = 0; equivalently the substitution x = eᶻ turns it into a constant-coefficient equation in z.
The method of undetermined coefficients guesses y_p in the form of the forcing: a polynomial for a polynomial, Ae^(ax) for e^(ax), A cos ωx + B sin ωx for either sine or cosine. The operator shortcut for f = e^(ax) is y_p = e^(ax)/F(a), where F(m) is the auxiliary polynomial, valid when F(a) ≠ 0. When the forcing already solves the homogeneous equation (resonance), multiply the trial by x (by x² if the root is double). For y'' + y = sin x the result is y_p = −(x/2) cos x, whose amplitude grows with time — the mathematical picture of resonance that machine designers avoid by keeping forcing frequencies away from natural frequencies.
Variation of parameters works for any forcing: with y_c = C₁y₁ + C₂y₂, the particular integral is y_p = −y₁∫(y₂f/W) dx + y₂∫(y₁f/W) dx, for the equation in the normalised form y'' + p y' + q y = f.
The constants are fixed only after y_p has been added to y_c. Applying the initial conditions to y_c alone is a frequent error.
The Laplace transform of f(t), t ≥ 0, is F(s) = L{f} = ∫₀^∞ e^(−st) f(t) dt. It turns a linear ODE with constant coefficients into an algebraic equation in s, with the initial conditions built in. Core pairs: L{1} = 1/s; L{tⁿ} = n!/s^(n+1); L{e^(at)} = 1/(s − a); L{sin ωt} = ω/(s² + ω²); L{cos ωt} = s/(s² + ω²); L{sinh at} = a/(s² − a²); L{cosh at} = s/(s² − a²).
Rules: linearity; first shifting, L{e^(at)f(t)} = F(s − a); second shifting, L{f(t − a)u(t − a)} = e^(−as)F(s), so L{u(t − a)} = e^(−as)/s and L{δ(t − a)} = e^(−as); derivatives, L{f'} = sF − f(0) and L{f''} = s²F − s f(0) − f'(0); multiplication by t, L{t f(t)} = −F'(s); integration, L{∫₀ᵗ f dτ} = F(s)/s; and convolution, L{f * g} = F(s)G(s). The initial value theorem gives f(0⁺) = lim(s→∞) sF(s), and the final value theorem gives f(∞) = lim(s→0) sF(s), valid only if the limit of f exists (all poles of sF(s) in the left half-plane).
Worked example: y'' + 3y' + 2y = 4 with y(0) = y'(0) = 0. Transforming, (s² + 3s + 2)Y = 4/s, so Y = 4/[s(s + 1)(s + 2)] = 2/s − 4/(s + 1) + 2/(s + 2), and y = 2 − 4e^(−t) + 2e^(−2t). Check: y(0) = 0, y'(0) = 4 − 4 = 0, and y → 2 as t → ∞, which the final value theorem confirms: sY → 4/2 = 2.
Inverse transforms rely on partial fractions and completing the square. The step and impulse functions make the Laplace method the natural tool for switched inputs — a pump that starts at t = 2 s, a sudden load on a hitch — and it is the foundation of transfer functions in control engineering.
A partial differential equation (PDE) involves partial derivatives of a function of two or more variables. The second-order linear equation A u_xx + B u_xy + C u_yy + (lower-order terms) = 0 is classified by the discriminant B² − 4AC: negative is elliptic, zero is parabolic, positive is hyperbolic. The three model equations are the Laplace equation u_xx + u_yy = 0 (elliptic: steady states, such as steady seepage under a structure or steady temperature in a plate), the heat or diffusion equation u_t = α u_xx (parabolic: transient heat conduction, moisture diffusion in a grain kernel, solute spreading), and the wave equation u_tt = c² u_xx (hyperbolic: vibrating strings and shafts, pressure surges in pipes).
The type decides what data make a problem well posed: elliptic equations need conditions on the whole boundary; parabolic equations need an initial condition and boundary conditions; hyperbolic equations need an initial displacement, an initial velocity and boundary conditions.
For the heat equation on a rod 0 < x < L with both ends at zero temperature, try u = X(x)T(t). Substituting gives X''/X = T'/(αT) = −λ, a constant, because a function of x can equal a function of t only if both are constant. The boundary conditions X(0) = X(L) = 0 allow only λ = (nπ/L)², with X = sin(nπx/L), and then T = e^(−α(nπ/L)²t). The solution is u = Σbₙ sin(nπx/L) e^(−α(nπ/L)²t), where the bₙ are the half-range sine coefficients of the initial temperature. Higher modes die out faster, by the factor n², so after a short time the profile is essentially the first sine mode decaying with time constant L²/(απ²). For a 1 m rod with α = 1 × 10⁻⁵ m²/s that is about 10,100 s, close to three hours.
For the wave equation on a string fixed at both ends, the same method gives modes sin(nπx/L) oscillating at angular frequency nπc/L, that is at frequencies f = nc/(2L). The wave speed is c = √(T/ρ) for a string of tension T and mass per unit length ρ. On an infinite string d'Alembert's solution is u = [f(x + ct) + f(x − ct)]/2 + (1/2c)∫g(s) ds over (x − ct, x + ct), where f is the initial displacement and g the initial velocity.
At steady state the heat equation reduces to Laplace's equation. In one dimension u'' = 0, so the temperature varies linearly between the end values. In two dimensions, separation of variables on a rectangle gives products of sines and hyperbolic sines.
When the geometry or boundary conditions are irregular, PDEs are solved numerically. Replacing second derivatives by central differences on a square grid of spacing h turns Laplace's equation into the five-point rule: each interior node value is the average of its four neighbours. Solving the resulting linear system — directly, or by repeated averaging (Gauss-Seidel iteration) — gives the field. The same idea underlies flow nets for seepage and many groundwater models.
For the heat equation, the explicit scheme u(i, n+1) = u(i, n) + r[u(i+1, n) − 2u(i, n) + u(i−1, n)], with r = αΔt/Δx², is stable only when r ≤ 1/2. Choosing the time step too large for the grid makes the computed temperatures oscillate and blow up, a limit of the same kind as the Euler stability limit in Chapter 11. Implicit schemes such as Crank-Nicolson remove the limit at the cost of solving a linear system each step.
Many engineering equations cannot be solved by algebra: the normal depth in Manning's equation, the friction factor in the Colebrook equation, the well drawdown in a Theis-type relation. Numerical root finding solves f(x) = 0 by iteration.
The bisection method starts from an interval [a, b] with f(a) and f(b) of opposite sign, so a root lies between them if f is continuous. Take the midpoint m; keep whichever half still shows a sign change; repeat. After n steps the bracket has length (b − a)/2ⁿ, so reaching a tolerance ε needs n ≥ log₂[(b − a)/ε]. Bisection always converges, but only linearly: each step gains one binary digit. For x³ − x − 2 on [1, 2], the midpoints are 1.5, 1.75, 1.625, … approaching the root 1.5214.
Newton-Raphson uses the tangent line: xₙ₊₁ = xₙ − f(xₙ)/f'(xₙ). Near a simple root it converges quadratically — the number of correct digits roughly doubles each step. For x³ − 2x − 5 = 0 from x₀ = 2: x₁ = 2.1, x₂ = 2.094568, x₃ = 2.094551, already correct to six places. It can fail when f'(xₙ) is zero or small (the tangent shoots far away), when the starting value is poor, or when it cycles; at a repeated root it slows to linear. Applied to x² − N it gives the square-root iteration xₙ₊₁ = (xₙ + N/xₙ)/2. The secant method replaces f' by a difference quotient and converges with order about 1.62 without needing derivatives; the regula falsi (false position) method is a bracketing variant.
Numerical integration is how field data become totals: runoff volume from a hydrograph, cross-section area from depth soundings, energy from a power trace, area of an irregular plot from offsets.
With n equal intervals of width h and ordinates y₀, …, yₙ, the composite trapezoidal rule is T = (h/2)[y₀ + yₙ + 2(y₁ + … + yₙ₋₁)]. Its error is proportional to h² times the second derivative, so halving h cuts the error by about four, and it over-estimates the integral of a convex function. Simpson's 1/3 rule fits parabolas through pairs of intervals and needs n even: S = (h/3)[y₀ + yₙ + 4(odd-numbered ordinates) + 2(even-numbered interior ordinates)]. Its error is proportional to h⁴ times the fourth derivative, so it is exact for cubics and halving h cuts the error by about sixteen. Simpson's 3/8 rule, (3h/8)[y₀ + 3y₁ + 3y₂ + y₃] per group of three intervals, needs n a multiple of three and is useful to finish off an odd number of intervals.
Example: ∫₀¹ e^(−x²) dx with h = 0.25 (n = 4). Trapezoidal gives 0.742984; Simpson gives 0.746855; the exact value is 0.746824. The commonest exam errors are using h instead of h/2 (or h/3), swapping the 4 and 2 weights, and applying Simpson's 1/3 rule to an odd number of intervals. Check units too: a hydrograph read in m³/s at hourly intervals needs h = 3,600 s. Richardson extrapolation combines results for h and h/2 to cancel the leading error term; applied repeatedly to the trapezoidal rule it becomes Romberg integration.
For y' = f(x, y) with y(x₀) = y₀, Euler's method steps forward along the tangent: yₙ₊₁ = yₙ + h f(xₙ, yₙ). Its local error per step is O(h²) and its global error O(h), so halving h roughly halves the error. For y' = −y, y(0) = 1, h = 0.1, Euler gives 0.9 against the exact 0.904837. Euler's method is also conditionally stable: for y' = −λy it requires h < 2/λ, which becomes severe for stiff problems with large λ.
Heun's method (modified Euler, a second-order Runge-Kutta method) predicts with Euler and corrects with the average slope: y* = yₙ + h f(xₙ, yₙ), then yₙ₊₁ = yₙ + (h/2)[f(xₙ, yₙ) + f(xₙ₊₁, y*)]. Its global error is O(h²).
The classical fourth-order Runge-Kutta method (RK4) evaluates four slopes per step: k₁ = h f(xₙ, yₙ); k₂ = h f(xₙ + h/2, yₙ + k₁/2); k₃ = h f(xₙ + h/2, yₙ + k₂/2); k₄ = h f(xₙ + h, yₙ + k₃); then yₙ₊₁ = yₙ + (k₁ + 2k₂ + 2k₃ + k₄)/6. Its global error is O(h⁴). For y' = −y with h = 0.1 it gives 0.9048375, correct to seven decimal places. Mistakes to avoid: evaluating k₂ at the full step, dividing by 4 instead of 6, and mixing the convention k = h·f with k = f (then the update uses h/6).
Higher-order equations are handled by rewriting them as first-order systems: y'' = g(x, y, y') becomes y' = v, v' = g(x, y, v), and each variable is stepped together. Practical codes adapt the step size to keep an error estimate within a tolerance; the Runge-Kutta-Fehlberg pair is the classic example. Drying curves, reservoir routing, crop-growth and soil-water balance models are all integrated this way.
You will use every chapter here in later courses, often without being told. Hydrology uses series and integration to turn rainfall and runoff records into volumes; soil and water engineering uses the Laplace equation for seepage and drain spacing; farm power uses second-order ODEs for vibration and eigenvalues for natural frequencies; process engineering uses first-order ODEs for drying and cooling and the heat equation for thermal processing. Learn each method with a checking habit: the trace-and-determinant check for eigenvalues, substituting a solution back into its ODE, the final value theorem on a Laplace answer, and an order-of-magnitude estimate before trusting a numerical result.
After five years the mathematics shows up in spreadsheets and design calculations rather than proofs. You fit a drying constant to a dryer trial by linearising ln(MR) against time; you integrate a hydrograph or a canal cross-section from field readings with the trapezoidal or Simpson's rule; you solve Manning's equation for normal depth by Newton-Raphson; you check a seat or boom mount for resonance by comparing the forcing frequency with √(k/m). The practical skill is recognising which model a problem reduces to, knowing the error of the method, and choosing a step size or interval that is fine enough without wasting effort.
A senior engineer reviews other people's numbers. The mathematics becomes judgement: is the model's time step inside the stability limit; does the simulated drying curve approach the right equilibrium; is a numerical groundwater model mass-conserving by the divergence theorem; does the optimisation of a fleet or an irrigation schedule respect every constraint, and what do the Lagrange multipliers say about which constraint is costing money. You decide when a closed-form solution is good enough for a decision and when a full numerical model is justified, and you insist on sensitivity checks using differentials or repeated runs before a design goes out.
At research level these tools become the building blocks of new models. Coupled heat and moisture transfer in grain beds, soil-water movement under the Richards equation, soil-tool interaction and machine dynamics are systems of non-linear PDEs and ODEs solved by finite-difference, finite-element or finite-volume methods whose accuracy and stability rest on the analysis in Chapters 10 and 11. Eigenvalue analysis underlies modal testing of machine structures and stability of control systems; Fourier analysis underlies vibration and signal processing in precision agriculture. The research task is to choose the model, prove or test its well-posedness and convergence, quantify uncertainty, and validate it against field data.
This manual covers the sixteen units of the degree course only. The GATE AG Section 1 syllabus also lists complex variables (analytic functions, Cauchy's theorems, residues) and probability and statistics (distributions, mean, variance, correlation and regression); neither is covered here, and ICAR AIEEA-PG Unit I and NCEES FE also examine statistics. Proofs are sketched, not given in full. Matrix computation for large systems (LU and iterative solvers), multistep ODE methods (Adams, Milne), finite-element methods and stiff-equation solvers are named but not developed. Worked examples are kept to exam scale; field-scale applications appear only as illustrations.
Original practice questions written to the syllabus and pattern of the exams named in each question's examTags. They are not past papers and do not reproduce any board's questions. Every numerical answer and every numerical distractor was computed with a script (Python with SymPy/NumPy) before it was written.
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