GATE Metallurgical Engineering Engineering Mathematics Syllabus

State the $n$-th term test (divergence test) for an infinite series $\sum a_n$.

If $\lim_{n\to\infty} a_n \neq 0$ (or does not exist), the series $\sum a_n$ diverges. If $\lim a_n = 0$, the test is inconclusive.

State the ratio test for convergence of $\sum a_n$.

Let $L = \lim_{n\to\infty}\left|\frac{a_{n+1}}{a_n}\right|$. If $L<1$ the series converges absolutely; if $L>1$ it diverges; if $L=1$ the test is inconclusive.

State the integral test for convergence of a series $\sum_{n=1}^{\infty} a_n$ with $a_n = f(n)$.

If $f(x)$ is positive, continuous and decreasing for $x\geq 1$, then $\sum a_n$ and $\int_{1}^{\infty} f(x)\,dx$ either both converge or both diverge.

For what values of $p$ does the $p$-series $\sum_{n=1}^{\infty}\frac{1}{n^{p}}$ converge?

It converges if $p > 1$ and diverges if $p \leq 1$. (The harmonic series, $p=1$, diverges.)

State the root test for convergence of $\sum a_n$.

Let $L = \lim_{n\to\infty}\sqrt[n]{|a_n|}$. If $L<1$ the series converges absolutely; if $L>1$ it diverges; if $L=1$ the test is inconclusive.

What distinguishes absolute convergence from conditional convergence of a series?

$\sum a_n$ converges absolutely if $\sum |a_n|$ converges. It converges conditionally if $\sum a_n$ converges but $\sum |a_n|$ diverges (e.g. the alternating harmonic series).

Write the Fourier series representation of a function $f(x)$ of period $2\pi$.

$$f(x) = \frac{a_0}{2} + \sum_{n=1}^{\infty}\left(a_n\cos nx + b_n\sin nx\right).$$

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

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

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 $a_0=0$).

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.

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.

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.

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.

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 $\phi$.

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 $\phi$.

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 $\int_C \vec{F}\cdot d\vec{r} = \phi(B) - \phi(A)$.

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 $R$.

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 $V$.

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 $C$.

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.

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

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 $F(x,y) = C$.

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 $v$.

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 $\alpha\pm i\beta$: $e^{\alpha x}(C_1\cos\beta x + C_2\sin\beta x)$.

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).

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 $t$.

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.

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

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

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: $\mathcal{L}\{e^{at}f(t)\} = F(s-a)$.

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).

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): $\frac{\partial^{2}u}{\partial x^{2}} + \frac{\partial^{2}u}{\partial y^{2}} = 0$.