GATE Mechanical 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 Mechanical 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

   5 topics

   • Functions of single variable
     • Limit
     • Continuity and differentiability
     • Mean value theorems
     • Indeterminate forms
   • Definite and improper integrals
     • Double and triple integrals
   • Partial derivatives
     • Total derivative
     • Taylor series (in one and two variables)
     • Maxima and minima
   • Fourier series
   • Gradient, divergence and curl
     • Vector identities
     • Directional derivatives
     • Line, surface and volume integrals
     • Applications of Gauss, Stokes and Green’s theorems
3. 3

   Differential equations

   6 topics

   • First order equations (linear and nonlinear)
   • Higher order linear differential equations with constant coefficients
   • Euler-Cauchy equation
   • Initial and boundary value problems
   • Laplace transforms
   • Solutions of heat, wave and Laplace's equations
4. 4

   Complex variables

   4 topics

   • Analytic functions
   • Cauchy-Riemann equations
   • Cauchy’s integral theorem and integral formula
   • Taylor and Laurent series
5. 5

   Probability and Statistics

   6 topics

   • Definitions of probability
   • Sampling theorems
   • Conditional probability
   • Mean, median, mode and standard deviation
   • Random variables
   • Binomial, Poisson and normal distributions
6. 6

   Numerical Methods

   3 topics

   • Numerical solutions of linear and non-linear algebraic equations
   • Integration by trapezoidal and Simpson’s rules
   • Single and multi-step methods for differential equations

Engineering Mathematics flashcards for GATE Mechanical Engineering

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

1. What is the rank of a matrix?

   The rank is the number of linearly independent rows (or columns), equivalently the order of the largest non-zero minor. For an $m \times n$ matrix, $\text{rank} \leq \min(m,n)$.

2. Define a symmetric matrix and a skew-symmetric matrix.

   A matrix $A$ is symmetric if $A^{T} = A$ (so $a_{ij} = a_{ji}$), and skew-symmetric if $A^{T} = -A$ (so $a_{ij} = -a_{ji}$, forcing diagonal entries to be $0$).

3. When is a square matrix $A$ invertible (non-singular)?

   $A$ is invertible iff $\det(A) \neq 0$, equivalently it has full rank, and its inverse is $A^{-1} = \frac{1}{\det(A)}\,\text{adj}(A)$.

4. State the key properties of determinants for $\det(AB)$ and $\det(A^{T})$.

   $\det(AB) = \det(A)\det(B)$, $\det(A^{T}) = \det(A)$, $\det(A^{-1}) = \frac{1}{\det(A)}$, and for an $n \times n$ matrix $\det(kA) = k^{n}\det(A)$.

5. What is an orthogonal matrix and what is its key determinant property?

   A matrix $Q$ is orthogonal if $Q^{T}Q = QQ^{T} = I$, so $Q^{-1} = Q^{T}$. Its determinant satisfies $\det(Q) = \pm 1$.

6. Define eigenvalues and eigenvectors of a square matrix $A$.

   A scalar $\lambda$ and non-zero vector $\vec{x}$ satisfying $A\vec{x} = \lambda \vec{x}$ are an eigenvalue and corresponding eigenvector. They are found from the characteristic equation $\det(A - \lambda I) = 0$.

7. How are the trace and determinant of $A$ related to its eigenvalues $\lambda_{1}, \dots, \lambda_{n}$?

   The trace equals the sum of eigenvalues: $\text{tr}(A) = \sum_{i} \lambda_{i}$, and the determinant equals their product: $\det(A) = \prod_{i} \lambda_{i}$.

8. State the Cayley–Hamilton theorem.

   Every square matrix satisfies its own characteristic equation. If $p(\lambda) = \det(A - \lambda I)$, then $p(A) = 0$. This can be used to compute $A^{-1}$ and powers of $A$.

9. What are the eigenvalues of $A^{-1}$, $A^{T}$, and $A^{k}$ in terms of eigenvalues of $A$?

   If $\lambda$ is an eigenvalue of $A$: $A^{-1}$ has eigenvalue $\frac{1}{\lambda}$, $A^{k}$ has eigenvalue $\lambda^{k}$, and $A^{T}$ has the same eigenvalue $\lambda$ (eigenvectors may differ).

10. What can be said about the eigenvalues of a real symmetric matrix?

    All eigenvalues of a real symmetric matrix are real, and eigenvectors corresponding to distinct eigenvalues are orthogonal. Such a matrix is always diagonalizable by an orthogonal matrix.

11. State the Rank–Nullity (consistency) condition for the system $A\vec{x} = \vec{b}$.

    The system is consistent iff $\text{rank}(A) = \text{rank}([A|\vec{b}])$. If this common rank equals $n$ (number of unknowns) there is a unique solution; if it is less than $n$ there are infinitely many solutions.

12. When does a homogeneous system $A\vec{x} = \vec{0}$ have non-trivial solutions?

    It always has the trivial solution $\vec{x} = \vec{0}$. Non-trivial solutions exist iff $\text{rank}(A) < n$ (number of unknowns), equivalently $\det(A) = 0$ for a square $A$.

13. What is the number of independent solutions (dimension of the null space) of $A\vec{x} = \vec{0}$?

    By the rank–nullity theorem, the number of linearly independent solutions is $n - \text{rank}(A)$, where $n$ is the number of unknowns.

14. Define the limit of a function: $\lim_{x \to a} f(x) = L$.

    The limit is $L$ if for every $\varepsilon > 0$ there exists $\delta > 0$ such that $0 < |x - a| < \delta \implies |f(x) - L| < \varepsilon$. The limit exists iff the left- and right-hand limits are equal.

15. What is the condition for a function $f$ to be continuous at a point $x = a$?

    $f$ is continuous at $a$ if (1) $f(a)$ is defined, (2) $\lim_{x \to a} f(x)$ exists, and (3) $\lim_{x \to a} f(x) = f(a)$.

16. State the relationship between differentiability and continuity.

    If $f$ is differentiable at a point, then it is continuous there. The converse is false: continuity does not imply differentiability (e.g. $f(x) = |x|$ at $x = 0$).

17. State the definition of the derivative of $f$ at $x$ as a limit.

    $$f'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h}$$ provided the limit exists.

18. State Rolle's theorem.

    If $f$ is continuous on $[a,b]$, differentiable on $(a,b)$, and $f(a) = f(b)$, then there exists at least one $c \in (a,b)$ with $f'(c) = 0$.

19. State Lagrange's Mean Value Theorem.

    If $f$ is continuous on $[a,b]$ and differentiable on $(a,b)$, then there exists $c \in (a,b)$ such that $$f'(c) = \frac{f(b) - f(a)}{b - a}.$$

20. State Cauchy's Mean Value Theorem.

    If $f$ and $g$ are continuous on $[a,b]$, differentiable on $(a,b)$, and $g'(x) \neq 0$, then there exists $c \in (a,b)$ with $$\frac{f'(c)}{g'(c)} = \frac{f(b) - f(a)}{g(b) - g(a)}.$$

21. List the standard indeterminate forms.

    The seven indeterminate forms are $\frac{0}{0}$, $\frac{\infty}{\infty}$, $0 \cdot \infty$, $\infty - \infty$, $0^{0}$, $1^{\infty}$, and $\infty^{0}$.

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