GATE Petroleum 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 Petroleum Engineering, not a summary of it.

1. 1

   Linear Algebra

   3 topics

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

   Calculus

   14 topics

   • Functions of single variable
   • Limit, continuity and differentiability
   • Taylor series
   • Mean value theorems
   • Evaluation of definite and improper integrals
   • Partial derivatives
   • Total derivative
   • Maxima and minima
   • Gradient
   • Divergence and Curl
   • Vector identities
   • Directional derivatives
   • Line, Surface and Volume integrals
   • Stokes, Gauss and Green’s theorems
3. 3

   Differential Equations

   6 topics

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

   Complex Variables

   3 topics

   • Complex number
   • Polar form of complex number
   • Triangle inequality
5. 5

   Probability and Statistics

   6 topics

   • Definitions of probability and sampling theorems
   • Conditional probability
   • Mean, median, mode and standard deviation
   • Random variables
   • Poisson, Normal and Binomial distributions
   • Linear regression analysis
6. 6

   Numerical Methods

   3 topics

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

Engineering Mathematics flashcards for GATE Petroleum Engineering

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

1. What is the necessary and sufficient condition for a square matrix $A$ to be invertible (non-singular)?

   Its determinant must be non-zero, $\det(A) \neq 0$. Equivalently, $A$ has full rank and its columns (rows) are linearly independent.

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

   $$A^{-1} = \frac{1}{\det(A)}\,\operatorname{adj}(A)$$ where $\operatorname{adj}(A)$ is the transpose of the cofactor matrix.

3. For a $2\times 2$ matrix $A=\begin{pmatrix} a & b \\ c & d \end{pmatrix}$, give $\det(A)$ and $A^{-1}$.

   $\det(A)=ad-bc$, and $$A^{-1}=\frac{1}{ad-bc}\begin{pmatrix} d & -b \\ -c & a \end{pmatrix}.$$

4. What is the rank of a matrix, and how is it found?

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

5. State the properties of determinants: effect of (a) swapping two rows, (b) multiplying a row by scalar $k$, (c) a repeated row.

   (a) Swapping two rows multiplies the determinant by $-1$. (b) Multiplying one row by $k$ multiplies the determinant by $k$. (c) Two identical rows give $\det = 0$.

6. State the Cayley–Hamilton theorem.

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

7. How many solutions does a system $A\vec{x}=\vec{b}$ have based on rank? State the consistency conditions.

   With $n$ unknowns: if $\operatorname{rank}(A)=\operatorname{rank}([A|b])=n$, unique solution; if both ranks equal but $<n$, infinitely many; if $\operatorname{rank}(A)\neq\operatorname{rank}([A|b])$, no solution (inconsistent).

8. For a homogeneous system $A\vec{x}=\vec{0}$, when does a non-trivial solution exist?

   A non-trivial solution exists if and only if $\det(A)=0$, i.e. $\operatorname{rank}(A)<n$ (number of unknowns). Otherwise only $\vec{x}=\vec{0}$.

9. State Cramer's rule for solving $A\vec{x}=\vec{b}$.

   For a non-singular $A$, $$x_i=\frac{\det(A_i)}{\det(A)}$$ where $A_i$ is $A$ with its $i$-th column replaced by $\vec{b}$.

10. Define the eigenvalues and eigenvectors of a square matrix $A$.

    A scalar $\lambda$ and non-zero vector $\vec{x}$ satisfying $A\vec{x}=\lambda\vec{x}$. The eigenvalues solve the characteristic equation $\det(A-\lambda I)=0$; each $\vec{x}$ is the corresponding eigenvector.

11. For an $n\times n$ matrix, relate the sum and product of eigenvalues to the matrix.

    Sum of eigenvalues $=\operatorname{trace}(A)=\sum a_{ii}$. Product of eigenvalues $=\det(A)$.

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

    $A^{-1}$ has eigenvalues $\frac{1}{\lambda}$; $A^{k}$ has eigenvalues $\lambda^{k}$; $A^{T}$ has the same eigenvalues $\lambda$ as $A$.

13. What are the eigenvalues of (a) a triangular matrix and (b) a symmetric real matrix?

    (a) The diagonal entries themselves. (b) All real, and eigenvectors for distinct eigenvalues are orthogonal.

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

    For every $\varepsilon>0$ there exists $\delta>0$ such that $0<|x-a|<\delta \implies |f(x)-L|<\varepsilon$. The left- and right-hand limits must both equal $L$.

15. State the condition for a function $f$ to be continuous at $x=a$.

    $f$ is continuous at $a$ if $\lim_{x\to a} f(x)=f(a)$; i.e. $f(a)$ exists, the limit exists, and they are equal.

16. What is the relationship between differentiability and continuity?

    Differentiability implies continuity, but continuity does not imply differentiability (e.g. $f(x)=|x|$ is continuous yet not differentiable 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. Write the Taylor series expansion of $f(x)$ about $x=a$.

    $$f(x)=\sum_{n=0}^{\infty}\frac{f^{(n)}(a)}{n!}(x-a)^{n}=f(a)+f'(a)(x-a)+\frac{f''(a)}{2!}(x-a)^{2}+\cdots$$

19. What is a Maclaurin series, and give the expansion of $e^{x}$.

    A Taylor series about $a=0$. $$e^{x}=\sum_{n=0}^{\infty}\frac{x^{n}}{n!}=1+x+\frac{x^{2}}{2!}+\frac{x^{3}}{3!}+\cdots$$

20. Give the Maclaurin series for $\sin x$ and $\cos x$.

    $$\sin x=x-\frac{x^{3}}{3!}+\frac{x^{5}}{5!}-\cdots,\qquad \cos x=1-\frac{x^{2}}{2!}+\frac{x^{4}}{4!}-\cdots$$

21. State Rolle's theorem.

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

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