GATE E&C Engineering Engineering Mathematics Syllabus

State Lagrange's mean value theorem.

If $f$ is continuous on $[a,b]$ and differentiable on $(a,b)$, then there exists $c\in(a,b)$ such that $f'(c)=\dfrac{f(b)-f(a)}{b-a}$.

State Cauchy's mean value theorem.

If $f,g$ are continuous on $[a,b]$, differentiable on $(a,b)$, and $g'(x)\neq 0$, then there exists $c\in(a,b)$ with $\dfrac{f'(c)}{g'(c)}=\dfrac{f(b)-f(a)}{g(b)-g(a)}$.

State the fundamental theorem of calculus (both parts).

Part 1: if $F(x)=\int_a^x f(t)\,dt$ then $F'(x)=f(x)$. Part 2: if $F'=f$ then $\int_a^b f(x)\,dx=F(b)-F(a)$.

State Leibniz's rule for differentiating an integral with variable limits.

$\dfrac{d}{dx}\int_{a(x)}^{b(x)} f(x,t)\,dt = f(x,b(x))\,b'(x) - f(x,a(x))\,a'(x) + \int_{a(x)}^{b(x)} \frac{\partial f}{\partial x}\,dt$.

State the mean value theorem for integrals.

If $f$ is continuous on $[a,b]$, there exists $c\in[a,b]$ such that $\int_a^b f(x)\,dx = f(c)\,(b-a)$; the average value of $f$ is $\frac{1}{b-a}\int_a^b f(x)\,dx$.

Define an improper integral of the first kind and its convergence.

An integral over an infinite interval, e.g. $\int_a^{\infty} f(x)\,dx=\lim_{t\to\infty}\int_a^{t} f(x)\,dx$. It converges if this limit exists and is finite; otherwise it diverges.

For what values of $p$ does $\int_1^{\infty}\frac{dx}{x^{p}}$ converge?

It converges iff $p>1$ (value $\frac{1}{p-1}$) and diverges for $p\leq 1$. Conversely $\int_0^{1}\frac{dx}{x^{p}}$ converges iff $p<1$.

Define the Gamma function and give $\Gamma(n)$ for positive integers.

$\Gamma(n)=\int_0^{\infty} x^{n-1}e^{-x}\,dx$ for $n>0$. For positive integers $\Gamma(n)=(n-1)!$, and $\Gamma\!\left(\tfrac{1}{2}\right)=\sqrt{\pi}$.

Define the Beta function and its relation to the Gamma function.

$B(m,n)=\int_0^{1} x^{m-1}(1-x)^{n-1}\,dx=\dfrac{\Gamma(m)\Gamma(n)}{\Gamma(m+n)}$ for $m,n>0$.

Define a partial derivative $\frac{\partial f}{\partial x}$ of $f(x,y)$.

$\dfrac{\partial f}{\partial x}=\lim_{h\to 0}\dfrac{f(x+h,y)-f(x,y)}{h}$ — differentiate with respect to $x$ holding $y$ constant.

State the condition for equality of mixed partial derivatives (Clairaut/Schwarz theorem).

If $f_{xy}$ and $f_{yx}$ are continuous in a neighborhood of a point, then $\dfrac{\partial^2 f}{\partial x\,\partial y}=\dfrac{\partial^2 f}{\partial y\,\partial x}$.

State Euler's theorem for homogeneous functions.

If $f(x,y)$ is homogeneous of degree $n$, then $x\dfrac{\partial f}{\partial x}+y\dfrac{\partial f}{\partial y}=n f$.

Give the total differential of $z=f(x,y)$ and the chain rule for $\frac{dz}{dt}$.

$dz=\dfrac{\partial f}{\partial x}dx+\dfrac{\partial f}{\partial y}dy$; if $x=x(t),y=y(t)$ then $\dfrac{dz}{dt}=\dfrac{\partial f}{\partial x}\dfrac{dx}{dt}+\dfrac{\partial f}{\partial y}\dfrac{dy}{dt}$.

State the necessary condition for a local extremum of $f(x,y)$.

At an interior extremum (stationary point) both first partials vanish: $f_x=0$ and $f_y=0$ simultaneously.

State the second-derivative (Hessian) test for $f(x,y)$.

Let $D=f_{xx}f_{yy}-(f_{xy})^2$ at a stationary point. If $D>0$ and $f_{xx}>0$: local minimum; $D>0$ and $f_{xx}<0$: local maximum; $D<0$: saddle point; $D=0$: test inconclusive.

Describe the method of Lagrange multipliers for constrained extrema.

To extremize $f(x,y)$ subject to $g(x,y)=0$, solve $\nabla f=\lambda\nabla g$ together with $g=0$, i.e. $f_x=\lambda g_x$, $f_y=\lambda g_y$, $g=0$, for the stationary points.

How do you reverse the order of integration in a double integral, and why?

Sketch the region, then re-express the limits with the other variable outer. The value is unchanged (Fubini's theorem) when $f$ is continuous on the region; reversing often simplifies an otherwise hard integral.

State the Jacobian for changing variables in a double integral.

$\iint_R f\,dx\,dy=\iint_{R'} f\,|J|\,du\,dv$ where $J=\dfrac{\partial(x,y)}{\partial(u,v)}=\begin{vmatrix} x_u & x_v \\ y_u & y_v \end{vmatrix}$.

Give the area element in polar coordinates for a double integral.

$dx\,dy=r\,dr\,d\theta$, so $\iint f(x,y)\,dx\,dy=\iint f(r\cos\theta,r\sin\theta)\,r\,dr\,d\theta$.

Give the volume elements for cylindrical and spherical coordinates.

Cylindrical: $dV=r\,dr\,d\theta\,dz$. Spherical: $dV=\rho^{2}\sin\phi\,d\rho\,d\phi\,d\theta$ (with $\phi$ the polar angle).

State Green's theorem in the plane.

$\oint_C (P\,dx+Q\,dy)=\iint_R\left(\dfrac{\partial Q}{\partial x}-\dfrac{\partial P}{\partial y}\right)dx\,dy$, for a positively oriented simple closed curve $C$ bounding region $R$.

State the Gauss divergence theorem.

$\iiint_V (\nabla\cdot\vec{F})\,dV=\oiint_S \vec{F}\cdot\hat{n}\,dS$, relating the volume integral of the divergence to the outward flux through the closed surface $S$.

State Stokes' theorem.

$\oint_C \vec{F}\cdot d\vec{r}=\iint_S (\nabla\times\vec{F})\cdot\hat{n}\,dS$, relating the line integral around boundary curve $C$ to the surface integral of the curl over $S$.

When is a line integral $\int_C \vec{F}\cdot d\vec{r}$ independent of path?

When $\vec{F}$ is conservative, i.e. $\vec{F}=\nabla\phi$ for some scalar potential $\phi$ (equivalently $\nabla\times\vec{F}=\vec{0}$ in a simply connected region). Then $\int_C\vec{F}\cdot d\vec{r}=\phi(B)-\phi(A)$.

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

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

Give the Maclaurin series for $e^{x}$, $\sin x$ and $\cos x$.

$e^{x}=\sum_{n=0}^{\infty}\dfrac{x^{n}}{n!}$, $\sin x=\sum_{n=0}^{\infty}\dfrac{(-1)^n x^{2n+1}}{(2n+1)!}$, $\cos x=\sum_{n=0}^{\infty}\dfrac{(-1)^n x^{2n}}{(2n)!}$.

How do you solve a first-order linear ODE $\frac{dy}{dx}+P(x)y=Q(x)$?

Use the integrating factor $\mu=e^{\int P\,dx}$. Then $(\mu y)'=\mu Q$, giving $y=\dfrac{1}{\mu}\left(\int \mu Q\,dx + C\right)$.

What is an exact first-order ODE and its condition?

$M\,dx+N\,dy=0$ is exact if $\dfrac{\partial M}{\partial y}=\dfrac{\partial N}{\partial x}$. Then there is $F$ with $F_x=M$, $F_y=N$, and the solution is $F(x,y)=C$.

How is a Bernoulli equation $\frac{dy}{dx}+P(x)y=Q(x)y^{n}$ solved?

Divide by $y^{n}$ and substitute $v=y^{1-n}$, which converts it into the linear equation $\dfrac{dv}{dx}+(1-n)P(x)v=(1-n)Q(x)$, solved by an integrating factor.

For a higher-order linear ODE with constant coefficients, how is the homogeneous solution found from the auxiliary equation?

From the auxiliary (characteristic) equation: distinct real roots $m$ give $e^{mx}$; a repeated root $m$ (multiplicity $k$) gives $(c_1+c_2 x+\cdots+c_k x^{k-1})e^{mx}$; complex roots $\alpha\pm i\beta$ give $e^{\alpha x}(c_1\cos\beta x+c_2\sin\beta x)$.

How do you solve a Cauchy–Euler equation $x^{2}y''+axy'+by=0$?

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