GATE Civil Engineering Engineering Mathematics Syllabus

Engineering Mathematics syllabus — full chapter and topic list

Expand any chapter to see its topics and sub-topics. This is the whole examinable outline for Engineering Mathematics in GATE Civil Engineering, not a summary of it.

1. 1

   Linear Algebra

   3 topics

   • Matrix algebra
   • Systems of linear equations
   • Eigen values and Eigen vectors
2. 2

   Calculus

   10 topics

   • Functions of single variable
   • Limit, continuity and differentiability
   • Mean value theorems, local maxima and minima
   • Taylor series
   • Evaluation of definite and indefinite integrals, application of definite integral to obtain area and volume
   • Partial derivatives
   • Total derivative
   • Gradient, Divergence and Curl, Vector identities
   • Directional derivatives
   • Line, Surface and Volume integrals
3. 3

   Ordinary Differential Equation (ODE)

   4 topics

   • First order (linear and non-linear) equations
   • Higher order linear equations with constant coefficients
   • Euler-Cauchy equations
   • Initial and boundary value problems
4. 4

   Partial Differential Equation (PDE)

   5 topics

   • Fourier series
   • Separation of variables
   • Solutions of one-dimensional diffusion equation
   • First and second order one-dimensional wave equation
   • Two-dimensional Laplace equation
5. 5

   Probability and Statistics

   5 topics

   • Sampling theorems
   • Conditional probability
   • Descriptive statistics – Mean, median, mode and standard deviation
   • Random Variables – Discrete and Continuous, Poisson and Normal Distribution
   • Linear regression
6. 6

   Numerical Methods

   6 topics

   • Error analysis
   • Numerical solutions of linear and non-linear algebraic equations
   • Newton’s and Lagrange polynomials
   • Numerical differentiation
   • Integration by trapezoidal and Simpson’s rule
   • Single and multi-step methods for first order differential equations

Engineering Mathematics flashcards for GATE Civil Engineering

21 of 51 cards from the Engineering Mathematics deck — real questions with worked answers.

1. What is a symmetric matrix, and what condition defines it?

   A square matrix $A$ is symmetric if it equals its own transpose: $A = A^{T}$, i.e. $a_{ij} = a_{ji}$ for all $i,j$.

2. Define a skew-symmetric matrix and state a key property of its diagonal elements.

   A square matrix is skew-symmetric if $A^{T} = -A$, i.e. $a_{ij} = -a_{ji}$. All diagonal elements are zero ($a_{ii} = 0$).

3. What is an orthogonal matrix and what is the value of its determinant?

   A square matrix $A$ is orthogonal if $A^{T}A = AA^{T} = I$, so $A^{-1} = A^{T}$. Its determinant is $\pm 1$.

4. State the formula for the inverse of a non-singular matrix $A$ in terms of its adjoint.

   $$A^{-1} = \frac{1}{\det(A)}\,\operatorname{adj}(A), \quad \det(A) \neq 0$$

5. What does the rank of a matrix represent?

   The rank is the maximum number of linearly independent rows (or columns), equal to the order of the largest non-zero minor / number of non-zero rows in row echelon form.

6. State the property relating determinants of a product: $\det(AB) = ?$

   $$\det(AB) = \det(A)\,\det(B)$$

7. For a matrix product, what is $(AB)^{T}$ and $(AB)^{-1}$?

   $(AB)^{T} = B^{T}A^{T}$ and $(AB)^{-1} = B^{-1}A^{-1}$ (reversal of order).

8. Using Cramer's rule, how is $x_{i}$ found for the system $Ax = b$?

   $$x_{i} = \frac{\det(A_{i})}{\det(A)}$$ where $A_{i}$ is $A$ with its $i$-th column replaced by $b$ (requires $\det(A) \neq 0$).

9. State the Rouché–Capelli consistency condition for a linear system $Ax = b$.

   The system is consistent iff $\operatorname{rank}(A) = \operatorname{rank}([A\,|\,b])$. If this common rank equals the number of unknowns $n$, the solution is unique; if less than $n$, infinitely many solutions exist.

10. For a homogeneous system $Ax = 0$, when does a non-trivial solution exist?

    A non-trivial solution exists iff $\det(A) = 0$ (i.e. $\operatorname{rank}(A) < n$, the number of unknowns).

11. Define an eigenvalue and eigenvector of a square matrix $A$.

    A scalar $\lambda$ and non-zero vector $x$ satisfying $Ax = \lambda x$. $\lambda$ is the eigenvalue and $x$ the corresponding eigenvector.

12. What is the characteristic equation used to find eigenvalues?

    $$\det(A - \lambda I) = 0$$

13. State the relationship between the trace of a matrix and its eigenvalues.

    The sum of the eigenvalues equals the trace: $\sum \lambda_{i} = \operatorname{trace}(A) = \sum a_{ii}$.

14. How does the product of all eigenvalues relate to the determinant?

    $$\prod_{i} \lambda_{i} = \det(A)$$

15. State the Cayley–Hamilton theorem.

    Every square matrix satisfies its own characteristic equation: if $p(\lambda) = \det(A - \lambda I)$, then $p(A) = 0$.

16. What are the eigenvalues of a triangular (or diagonal) matrix?

    They are exactly the diagonal entries of the matrix.

17. What can you say about the eigenvalues of a real symmetric matrix?

    They are always real, and eigenvectors corresponding to distinct eigenvalues are orthogonal.

18. If $\lambda$ is an eigenvalue of $A$, what are the eigenvalues of $A^{-1}$ and $A^{k}$?

    $A^{-1}$ has eigenvalue $\frac{1}{\lambda}$; $A^{k}$ has eigenvalue $\lambda^{k}$ (with the same eigenvector).

19. State the formal $\varepsilon$–$\delta$ definition of $\lim_{x \to a} f(x) = L$.

    For every $\varepsilon > 0$ there exists $\delta > 0$ such that $0 < |x - a| < \delta \implies |f(x) - L| < \varepsilon$.

20. State the three conditions for a function $f$ to be continuous at $x = a$.

    (1) $f(a)$ is defined; (2) $\lim_{x \to a} f(x)$ exists; (3) $\lim_{x \to a} f(x) = f(a)$.

21. What is the relationship between differentiability and continuity at a point?

    Differentiability implies continuity, but continuity does not imply differentiability (e.g. $f(x)=|x|$ at $x=0$).

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