GATE Textile Engineering ENGINEERING MATHEMATICS Syllabus

State Leibniz's rule for the $n$-th derivative of a product $(uv)$.

$(uv)^{(n)} = \sum_{k=0}^{n} \binom{n}{k} u^{(n-k)} v^{(k)}$.

Give the $n$-th derivative of $y = e^{ax}$.

$\dfrac{d^{n}y}{dx^{n}} = a^{n} e^{ax}$.

Give the $n$-th derivative of $y = \sin(ax)$.

$\dfrac{d^{n}y}{dx^{n}} = a^{n} \sin\!\left(ax + \dfrac{n\pi}{2}\right)$.

For $z = f(x, y)$, what is the total differential $dz$?

$dz = \dfrac{\partial z}{\partial x}\,dx + \dfrac{\partial z}{\partial y}\,dy$.

State the condition under which mixed partial derivatives are equal (Clairaut/Schwarz theorem).

If $f_{xy}$ and $f_{yx}$ are continuous in a region, then $f_{xy} = f_{yx}$.

State Euler's theorem for a homogeneous function $f(x,y)$ of degree $n$.

$x\dfrac{\partial f}{\partial x} + y\dfrac{\partial f}{\partial y} = n f$.

What is the first-order (necessary) condition for a local extremum of $f(x)$ at an interior point?

$f'(x) = 0$ (a stationary/critical point).

State the second-derivative test for a single-variable function at a critical point $x_{0}$.

If $f''(x_{0}) > 0$ it is a local minimum; if $f''(x_{0}) < 0$ a local maximum; if $f''(x_{0}) = 0$ the test is inconclusive.

Give the second-derivative (Hessian) test for $f(x,y)$ using $D = f_{xx}f_{yy} - f_{xy}^{2}$ at a critical point.

If $D>0$ and $f_{xx}>0$: minimum; $D>0$ and $f_{xx}<0$: maximum; $D<0$: saddle point; $D=0$: inconclusive.

In the method of Lagrange multipliers, what equation is solved to extremize $f$ subject to constraint $g=0$?

$\nabla f = \lambda \nabla g$ together with $g = 0$.

For a function $z = f(x,y)$, write the approximate error $\Delta z$ in terms of errors $\Delta x$ and $\Delta y$.

$\Delta z \approx \dfrac{\partial f}{\partial x}\Delta x + \dfrac{\partial f}{\partial y}\Delta y$.

Give the linear (tangent-line) approximation of $f$ near $x = a$.

$f(x) \approx f(a) + f'(a)(x - a)$.

State the Fundamental Theorem of Calculus (evaluation form) for $\int_{a}^{b} f(x)\,dx$.

If $F'(x) = f(x)$, then $\int_{a}^{b} f(x)\,dx = F(b) - F(a)$.

How is an improper integral $\int_{a}^{\infty} f(x)\,dx$ defined and when does it converge?

$\int_{a}^{\infty} f(x)\,dx = \lim_{b \to \infty} \int_{a}^{b} f(x)\,dx$; it converges if this limit exists and is finite.

For which values of $p$ does $\int_{1}^{\infty} \dfrac{dx}{x^{p}}$ converge?

It converges if $p > 1$ and diverges if $p \leq 1$.

Define the Gamma function and state its factorial property.

$\Gamma(n) = \int_{0}^{\infty} x^{n-1} e^{-x}\,dx$, with $\Gamma(n+1) = n\,\Gamma(n)$ and $\Gamma(n+1) = n!$ for non-negative integers.

What does it mean for a sequence $\{a_{n}\}$ to converge to a limit $L$?

For every $\varepsilon > 0$ there is $N$ such that $n > N$ implies $|a_{n} - L| < \varepsilon$; i.e. $\lim_{n \to \infty} a_{n} = L$.

State the $n$-th term test (divergence test) for the series $\sum a_{n}$.

If $\lim_{n \to \infty} a_{n} \neq 0$, the series diverges. (If the limit is $0$, the test is inconclusive.)

State the ratio test for convergence of $\sum a_{n}$.

Let $L = \lim_{n \to \infty} \left|\dfrac{a_{n+1}}{a_{n}}\right|$. Converges if $L < 1$, diverges if $L > 1$, inconclusive if $L = 1$.

State the root test for convergence of $\sum a_{n}$.

Let $L = \lim_{n \to \infty} \sqrt[n]{|a_{n}|}$. Converges if $L < 1$, diverges if $L > 1$, inconclusive if $L = 1$.

State the $p$-series test for $\sum_{n=1}^{\infty} \dfrac{1}{n^{p}}$.

Converges if $p > 1$, diverges if $p \leq 1$.

State the comparison test for positive-term series.

If $0 \le a_{n} \le b_{n}$: convergence of $\sum b_{n}$ implies convergence of $\sum a_{n}$; divergence of $\sum a_{n}$ implies divergence of $\sum b_{n}$.

How is the radius of convergence $R$ of a power series $\sum a_{n} x^{n}$ found via the ratio test?

$R = \lim_{n \to \infty} \left|\dfrac{a_{n}}{a_{n+1}}\right|$; the series converges for $|x| < R$.

Write the Taylor series expansion of $f(x)$ about $x = a$.

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

State the Maclaurin series for $e^{x}$.

$e^{x} = \sum_{n=0}^{\infty} \dfrac{x^{n}}{n!} = 1 + x + \dfrac{x^{2}}{2!} + \dfrac{x^{3}}{3!} + \cdots$

State the Maclaurin series for $\sin x$.

$\sin x = \sum_{n=0}^{\infty} \dfrac{(-1)^{n} x^{2n+1}}{(2n+1)!} = x - \dfrac{x^{3}}{3!} + \dfrac{x^{5}}{5!} - \cdots$

Write the general first-order linear ODE and its integrating factor.

$\dfrac{dy}{dx} + P(x)y = Q(x)$; integrating factor $\mu = e^{\int P(x)\,dx}$, giving $y\,\mu = \int Q\,\mu\,dx + C$.

What is the condition for the first-order ODE $M\,dx + N\,dy = 0$ to be exact?

$\dfrac{\partial M}{\partial y} = \dfrac{\partial N}{\partial x}$.

What substitution reduces a Bernoulli equation $\dfrac{dy}{dx} + P y = Q y^{n}$ to linear form?

Substitute $v = y^{1-n}$, which converts it into a first-order linear ODE in $v$.

For the higher-order linear ODE with constant coefficients, how are the complementary roots used when the auxiliary equation has real distinct roots $m_{1}, m_{2}$?

The complementary function is $y_{c} = C_{1}e^{m_{1}x} + C_{2}e^{m_{2}x}$.

For a constant-coefficient linear ODE, give the complementary function when the auxiliary equation has complex roots $m = \alpha \pm i\beta$.

$y_{c} = e^{\alpha x}\left(C_{1}\cos\beta x + C_{2}\sin\beta x\right)$.

State the standard form of the Euler-Cauchy (equidimensional) equation and the substitution that solves it.

$x^{2}\dfrac{d^{2}y}{dx^{2}} + a x \dfrac{dy}{dx} + b y = 0$; substitute $x = e^{t}$ (or try $y = x^{m}$) to obtain a constant-coefficient ODE.

How is a second-order linear PDE $A u_{xx} + B u_{xy} + C u_{yy} + \cdots = 0$ classified?

By the discriminant $B^{2} - 4AC$: hyperbolic if $> 0$, parabolic if $= 0$, elliptic if $< 0$.

Write the one-dimensional wave equation and identify its type.

$\dfrac{\partial^{2} u}{\partial t^{2}} = c^{2}\dfrac{\partial^{2} u}{\partial x^{2}}$; it is a hyperbolic PDE.

Write the one-dimensional heat (diffusion) equation and identify its type.

$\dfrac{\partial u}{\partial t} = \alpha \dfrac{\partial^{2} u}{\partial x^{2}}$; it is a parabolic PDE.