GATE Metallurgical Engineering Engineering Mathematics Flashcards

24 sample cards from the Engineering Mathematics deck

Sampled from the end of the deck, so these are different cards from the ones shown on the syllabus page.

1. Give the Euler formulas for the Fourier coefficients $a_0$, $a_n$, $b_n$ of a $2\pi$-periodic function.

   $$a_0 = \frac{1}{\pi}\int_{-\pi}^{\pi} f(x)\,dx,\quad a_n = \frac{1}{\pi}\int_{-\pi}^{\pi} f(x)\cos nx\,dx,\quad b_n = \frac{1}{\pi}\int_{-\pi}^{\pi} f(x)\sin nx\,dx.$$

2. What form does a Fourier series take for an even function and for an odd function?

   Even function: only cosine terms (a Fourier cosine series, all $b_n=0$). Odd function: only sine terms (a Fourier sine series, all $a_n=0$ and $a_0=0$).

3. State Dirichlet's conditions for the convergence of a Fourier series.

   $f(x)$ must be periodic, single-valued, with a finite number of maxima/minima and a finite number of finite discontinuities in one period, and be absolutely integrable over a period. At a jump, the series converges to the average of the left and right limits.

4. Define the gradient of a scalar field $\phi(x,y,z)$.

   $$\nabla\phi = \frac{\partial\phi}{\partial x}\hat{i} + \frac{\partial\phi}{\partial y}\hat{j} + \frac{\partial\phi}{\partial z}\hat{k}.$$ It points in the direction of greatest increase of $\phi$ and is normal to level surfaces.

5. Define the divergence of a vector field $\vec{F} = F_1\hat{i} + F_2\hat{j} + F_3\hat{k}$.

   $$\nabla\cdot\vec{F} = \frac{\partial F_1}{\partial x} + \frac{\partial F_2}{\partial y} + \frac{\partial F_3}{\partial z}.$$ It is a scalar measuring the net outflow (source/sink strength) per unit volume.

6. Define the curl of a vector field $\vec{F}$.

   $$\nabla\times\vec{F} = \begin{vmatrix} \hat{i} & \hat{j} & \hat{k} \\ \frac{\partial}{\partial x} & \frac{\partial}{\partial y} & \frac{\partial}{\partial z} \\ F_1 & F_2 & F_3 \end{vmatrix}.$$ It is a vector measuring the local rotation of the field.

7. What does it mean for a vector field $\vec{F}$ to be solenoidal, and what for it to be irrotational?

   Solenoidal: $\nabla\cdot\vec{F} = 0$ (divergence-free, no sources/sinks). Irrotational (conservative): $\nabla\times\vec{F} = \vec{0}$, so $\vec{F} = \nabla\phi$ for some scalar potential $\phi$.

8. What are the values of $\nabla\cdot(\nabla\times\vec{F})$ and $\nabla\times(\nabla\phi)$?

   Both are identically zero: $\nabla\cdot(\nabla\times\vec{F}) = 0$ and $\nabla\times(\nabla\phi) = \vec{0}$, for any sufficiently smooth $\vec{F}$ and $\phi$.

9. State the line integral that gives the work done by a force field $\vec{F}$ along a curve $C$, and when it is path-independent.

   Work $= \int_C \vec{F}\cdot d\vec{r}$. It is path-independent if and only if $\vec{F}$ is conservative ($\vec{F}=\nabla\phi$), in which case $\int_C \vec{F}\cdot d\vec{r} = \phi(B) - \phi(A)$.

10. State Green's theorem in the plane.

    $$\oint_C (P\,dx + Q\,dy) = \iint_R \left(\frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y}\right) dx\,dy,$$ where $C$ is the positively oriented boundary of region $R$.

11. State the Gauss divergence theorem.

    $$\iint_S \vec{F}\cdot \hat{n}\,dS = \iiint_V (\nabla\cdot\vec{F})\,dV.$$ The flux of $\vec{F}$ out of a closed surface $S$ equals the volume integral of its divergence over the enclosed region $V$.

12. State Stokes' theorem.

    $$\oint_C \vec{F}\cdot d\vec{r} = \iint_S (\nabla\times\vec{F})\cdot \hat{n}\,dS.$$ The circulation around closed curve $C$ equals the flux of the curl through any surface $S$ bounded by $C$.

13. How do you classify a first-order ODE as linear, and give its standard form?

    A first-order ODE is linear if it is degree-one in $y$ and $y'$: standard form $\frac{dy}{dx} + P(x)\,y = Q(x)$. Non-linear ODEs contain higher powers or products of $y$ and its derivatives.

14. Give the integrating factor and general solution of the linear ODE $\frac{dy}{dx} + P(x)y = Q(x)$.

    Integrating factor $\mu = e^{\int P\,dx}$. Solution: $$y\,\mu = \int Q\,\mu\,dx + C,\quad\text{i.e.}\quad y = \frac{1}{\mu}\left(\int Q\mu\,dx + C\right).$$

15. What is the condition for the first-order ODE $M(x,y)\,dx + N(x,y)\,dy = 0$ to be exact?

    It is exact if $\dfrac{\partial M}{\partial y} = \dfrac{\partial N}{\partial x}$. Then there exists $F$ with $F_x = M$, $F_y = N$, and the solution is $F(x,y) = C$.

16. Write the form of a Bernoulli equation and the substitution used to linearize it.

    Bernoulli: $\frac{dy}{dx} + P(x)y = Q(x)y^{n}$, $n\neq 0,1$. Substitute $v = y^{1-n}$, which reduces it to a linear first-order ODE in $v$.

17. For a higher-order linear homogeneous ODE with constant coefficients, how do you form the complementary function from the roots of the auxiliary equation?

    Distinct real roots $m_1,m_2$: $C_1e^{m_1 x}+C_2e^{m_2 x}$. Repeated root $m$ (twice): $(C_1+C_2 x)e^{mx}$. Complex roots $\alpha\pm i\beta$: $e^{\alpha x}(C_1\cos\beta x + C_2\sin\beta x)$.

18. What is the general solution structure of a non-homogeneous linear ODE?

    $y = y_c + y_p$, the sum of the complementary function $y_c$ (general solution of the homogeneous equation) and a particular integral $y_p$ (any solution of the full non-homogeneous equation).

19. How is a Cauchy–Euler equation $x^{2}y'' + a x y' + b y = 0$ solved?

    Assume $y = x^{m}$, giving an auxiliary (indicial) equation $m(m-1) + am + b = 0$. Alternatively substitute $x = e^{t}$ to convert it into a constant-coefficient linear ODE in $t$.

20. Define the Laplace transform of a function $f(t)$.

    $$\mathcal{L}\{f(t)\} = F(s) = \int_{0}^{\infty} e^{-st}f(t)\,dt,$$ defined for values of $s$ where the integral converges.

21. Give the Laplace transforms of $1$, $t^{n}$, $e^{at}$, $\sin at$, and $\cos at$.

    $\mathcal{L}\{1\}=\frac{1}{s}$; $\mathcal{L}\{t^{n}\}=\frac{n!}{s^{n+1}}$; $\mathcal{L}\{e^{at}\}=\frac{1}{s-a}$; $\mathcal{L}\{\sin at\}=\frac{a}{s^{2}+a^{2}}$; $\mathcal{L}\{\cos at\}=\frac{s}{s^{2}+a^{2}}$.

22. State the Laplace transform of a derivative $f'(t)$ and the first shifting theorem.

    $\mathcal{L}\{f'(t)\} = sF(s) - f(0)$. First shifting theorem: $\mathcal{L}\{e^{at}f(t)\} = F(s-a)$.

23. Classify the three standard second-order PDEs (Laplace, heat, wave) by name and order.

    All are linear second-order PDEs. Laplace: elliptic, steady-state. Heat (diffusion): parabolic, time-dependent. Wave: hyperbolic, time-dependent. Classification of $Au_{xx}+Bu_{xy}+Cu_{yy}+\dots$ uses the discriminant $B^{2}-4AC$ ($<0$ elliptic, $=0$ parabolic, $>0$ hyperbolic).

24. Write the one-dimensional heat equation and the one-dimensional wave equation.

    Heat equation: $$\frac{\partial u}{\partial t} = c^{2}\frac{\partial^{2} u}{\partial x^{2}}.$$ Wave equation: $$\frac{\partial^{2} u}{\partial t^{2}} = c^{2}\frac{\partial^{2} u}{\partial x^{2}}.$$ Laplace's equation (2D): $\frac{\partial^{2}u}{\partial x^{2}} + \frac{\partial^{2}u}{\partial y^{2}} = 0$.

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