GATE Aerospace 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 Aerospace Engineering, not a summary of it.

1. 1

   Linear Algebra

   5 topics

   • Vector algebra
   • Matrix algebra
   • Systems of linear equations
   • Rank of a matrix
   • Eigen values and eigen vectors
2. 2

   Calculus

   12 topics

   • Functions of single variable
   • Limits
   • Continuity and differentiability
   • Mean value theorem
   • Chain rule
   • Partial derivatives
   • Maxima and minima
   • Gradient, divergence and curl
   • Directional derivatives
   • Integration
   • Line, surface and volume integrals
   • Theorems of Stokes, Gauss and Green
3. 3

   Differential Equations

   3 topics

   • First order linear and nonlinear differential equations
   • Higher order linear ODEs with constant coefficients
   • Partial differential equations and separation of variables methods
4. 4

   Special Topics

   6 topics

   • Fourier Series
   • Laplace Transforms
   • Numerical methods for linear and nonlinear algebraic equations
   • Numerical integration and differentiation
   • Complex analysis
   • Probability and statistics

Engineering Mathematics flashcards for GATE Aerospace Engineering

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

1. How is the scalar (dot) product of two vectors $\vec{a}$ and $\vec{b}$ defined geometrically, and what does it yield?

   $\vec{a}\cdot\vec{b}=|\vec{a}||\vec{b}|\cos\theta$, where $\theta$ is the angle between them. It yields a scalar and is zero when the vectors are perpendicular.

2. How is the vector (cross) product $\vec{a}\times\vec{b}$ defined, and what is its magnitude?

   $\vec{a}\times\vec{b}=|\vec{a}||\vec{b}|\sin\theta\,\hat{n}$, a vector perpendicular to both. Its magnitude $|\vec{a}||\vec{b}|\sin\theta$ equals the area of the parallelogram formed by $\vec{a}$ and $\vec{b}$.

3. What does the scalar triple product $\vec{a}\cdot(\vec{b}\times\vec{c})$ represent geometrically?

   It equals the (signed) volume of the parallelepiped with edges $\vec{a},\vec{b},\vec{c}$. If it is zero, the three vectors are coplanar.

4. State the vector triple product (BAC-CAB) identity.

   $\vec{a}\times(\vec{b}\times\vec{c})=\vec{b}(\vec{a}\cdot\vec{c})-\vec{c}(\vec{a}\cdot\vec{b})$

5. What condition on two nonzero vectors makes them parallel versus perpendicular?

   Parallel: $\vec{a}\times\vec{b}=\vec{0}$ (equivalently $\vec{a}=\lambda\vec{b}$). Perpendicular: $\vec{a}\cdot\vec{b}=0$.

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

   Symmetric: $A^{T}=A$. Skew-symmetric: $A^{T}=-A$ (its diagonal entries are all zero).

7. What is an orthogonal matrix, and what is its key property regarding its inverse?

   A square matrix $Q$ with $Q^{T}Q=QQ^{T}=I$, so $Q^{-1}=Q^{T}$. Its determinant is $\pm 1$ and it preserves lengths.

8. State the formula for the inverse of a nonsingular matrix $A$ using its adjugate.

   $A^{-1}=\dfrac{1}{\det(A)}\,\operatorname{adj}(A)$, valid when $\det(A)\neq 0$.

9. How do determinants and inverses behave under matrix products: $\det(AB)$ and $(AB)^{-1}$?

   $\det(AB)=\det(A)\det(B)$ and $(AB)^{-1}=B^{-1}A^{-1}$.

10. Define the rank of a matrix.

    The rank is the maximum number of linearly independent rows (equivalently columns), equal to the order of the largest nonzero minor, equal to the number of nonzero rows in its row-echelon form.

11. For an $n\times n$ matrix $A$, relate full rank to invertibility and the determinant.

    $A$ is invertible (nonsingular) iff $\operatorname{rank}(A)=n$ iff $\det(A)\neq 0$. If $\operatorname{rank}(A)<n$, $A$ is singular.

12. State the consistency conditions for the linear system $A\vec{x}=\vec{b}$ in terms of ranks.

    Consistent iff $\operatorname{rank}(A)=\operatorname{rank}([A|\vec{b}])$. If this common rank $r$ equals the number of unknowns $n$, the solution is unique; if $r<n$, there are infinitely many solutions. If $\operatorname{rank}(A)\neq\operatorname{rank}([A|\vec{b}])$, the system is inconsistent (no solution).

13. For a homogeneous system $A\vec{x}=\vec{0}$ with $n$ unknowns, when does a nontrivial solution exist?

    It always has the trivial solution. A nontrivial solution exists iff $\operatorname{rank}(A)<n$; for a square $A$ this means $\det(A)=0$. The number of independent solutions is $n-\operatorname{rank}(A)$.

14. State the rank-nullity theorem for an $m\times n$ matrix $A$.

    $\operatorname{rank}(A)+\operatorname{nullity}(A)=n$, where nullity is the dimension of the null space (solution space of $A\vec{x}=\vec{0}$).

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

    A scalar $\lambda$ and nonzero vector $\vec{x}$ satisfying $A\vec{x}=\lambda\vec{x}$. Eigenvalues are the roots of the characteristic equation $\det(A-\lambda I)=0$.

16. How do the sum and product of all eigenvalues of $A$ relate to the matrix?

    Sum of eigenvalues $=\operatorname{trace}(A)$ (sum of diagonal entries); product of eigenvalues $=\det(A)$.

17. State the Cayley-Hamilton theorem.

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

18. How do the eigenvalues of $A^{-1}$, $A^{k}$, and $A^{T}$ relate to the eigenvalues $\lambda$ of $A$?

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

19. What can be said about the eigenvalues of a real symmetric matrix and a triangular matrix?

    A real symmetric matrix has all real eigenvalues and orthogonal eigenvectors. A triangular (or diagonal) matrix has eigenvalues equal to its diagonal entries.

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

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

21. What three conditions must hold for $f$ to be continuous at $x=a$?

    $f(a)$ is defined, $\lim_{x\to a}f(x)$ exists, and $\lim_{x\to a}f(x)=f(a)$.

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