GATE Mining Engineering Engineering Mathematics Syllabus

State the total differential of $z=f(x,y)$.

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

State Rolle's theorem.

If $f$ is continuous on $[a,b]$, differentiable on $(a,b)$, and $f(a)=f(b)$, then there exists $c\in(a,b)$ with $f'(c)=0$.

State Lagrange's Mean Value Theorem.

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

List the common indeterminate forms encountered in limits.

$\dfrac{0}{0}$, $\dfrac{\infty}{\infty}$, $0\cdot\infty$, $\infty-\infty$, $0^{0}$, $1^{\infty}$, and $\infty^{0}$.

State L'Hospital's rule for the form $\dfrac{0}{0}$ or $\dfrac{\infty}{\infty}$.

If $\lim \dfrac{f(x)}{g(x)}$ is of form $\dfrac{0}{0}$ or $\dfrac{\infty}{\infty}$, then $\lim \dfrac{f(x)}{g(x)} = \lim \dfrac{f'(x)}{g'(x)}$, provided the latter limit exists.

Evaluate $\lim_{x\to 0} \dfrac{\sin x}{x}$ using L'Hospital's rule.

It is of form $\dfrac{0}{0}$; differentiating gives $\lim_{x\to 0} \dfrac{\cos x}{1} = 1$.

State the first-derivative and second-derivative tests for a local extremum of $f(x)$.

A critical point satisfies $f'(x)=0$. By the second-derivative test: if $f''(x)>0$ it is a local minimum, if $f''(x)<0$ a local maximum; if $f''(x)=0$ the test is inconclusive.

State the second-derivative test for extrema of $f(x,y)$ using the Hessian discriminant $D$.

With $D = f_{xx}f_{yy} - (f_{xy})^{2}$ at a critical point: if $D>0$ and $f_{xx}>0$ it is a local minimum; if $D>0$ and $f_{xx}<0$ a local maximum; if $D<0$ a saddle point; if $D=0$ inconclusive.

What is the method of Lagrange multipliers for constrained optimization?

To extremize $f(x,y)$ subject to $g(x,y)=0$, solve $\nabla f = \lambda \nabla g$ together with $g=0$, i.e. $f_{x}=\lambda g_{x}$, $f_{y}=\lambda g_{y}$, $g=0$.

State Taylor's theorem (series expansion) of $f(x)$ about $x=a$.

$f(x) = f(a) + f'(a)(x-a) + \dfrac{f''(a)}{2!}(x-a)^{2} + \cdots + \dfrac{f^{(n)}(a)}{n!}(x-a)^{n} + R_{n}$.

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

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

Write the Maclaurin series for $\sin x$ and $\cos x$.

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

Define convergence of an infinite series $\sum a_{n}$.

The series converges if the sequence of partial sums $S_{N} = \sum_{n=1}^{N} a_{n}$ approaches a finite limit as $N\to\infty$.

State the necessary (but not sufficient) condition for convergence of $\sum a_{n}$.

If $\sum a_{n}$ converges then $\lim_{n\to\infty} a_{n} = 0$. The converse fails (e.g. the harmonic series $\sum \frac{1}{n}$ diverges although $a_n\to 0$).

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

Let $L = \lim_{n\to\infty} \left| \dfrac{a_{n+1}}{a_{n}} \right|$. 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.

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

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

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

State the root test (Cauchy's test) for $\sum a_{n}$.

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

State Leibniz's test for an alternating series $\sum (-1)^{n} b_{n}$.

If $b_{n}>0$, $b_{n}$ is monotonically decreasing, and $\lim_{n\to\infty} b_{n}=0$, then the alternating series converges.

Write the Fourier series of a periodic function $f(x)$ with period $2L$.

$f(x) = \dfrac{a_{0}}{2} + \sum_{n=1}^{\infty}\left[ a_{n}\cos\dfrac{n\pi x}{L} + b_{n}\sin\dfrac{n\pi x}{L} \right]$.

Give the Euler formulas for the Fourier coefficients $a_{0}$, $a_{n}$, $b_{n}$ over $(-L,L)$.

$a_{0}=\dfrac{1}{L}\int_{-L}^{L} f(x)\,dx$, $a_{n}=\dfrac{1}{L}\int_{-L}^{L} f(x)\cos\dfrac{n\pi x}{L}\,dx$, $b_{n}=\dfrac{1}{L}\int_{-L}^{L} f(x)\sin\dfrac{n\pi x}{L}\,dx$.

What does the Fourier series contain for an even function and for an odd function?

An even function has only cosine terms (all $b_{n}=0$); an odd function has only sine terms (all $a_{n}=0$ and $a_{0}=0$).

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

$\nabla \phi = \dfrac{\partial \phi}{\partial x}\hat{i} + \dfrac{\partial \phi}{\partial y}\hat{j} + \dfrac{\partial \phi}{\partial z}\hat{k}$; it points in the direction of maximum increase of $\phi$.

Define the divergence of a vector field $\vec{F}=(F_{1},F_{2},F_{3})$.

$\nabla\cdot\vec{F} = \dfrac{\partial F_{1}}{\partial x} + \dfrac{\partial F_{2}}{\partial y} + \dfrac{\partial F_{3}}{\partial z}$ (a scalar).

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

$\nabla\times\vec{F} = \begin{vmatrix} \hat{i} & \hat{j} & \hat{k} \\ \dfrac{\partial}{\partial x} & \dfrac{\partial}{\partial y} & \dfrac{\partial}{\partial z} \\ F_{1} & F_{2} & F_{3} \end{vmatrix}$ (a vector).

What characterizes a solenoidal field and an irrotational (conservative) field?

A field is solenoidal if $\nabla\cdot\vec{F}=0$ (divergence-free); it is irrotational/conservative if $\nabla\times\vec{F}=\vec{0}$, in which case $\vec{F}=\nabla\phi$ for some scalar potential $\phi$.

What is the value 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 sufficiently smooth fields).

Distinguish line, surface and volume integrals of vector/scalar fields.

A line integral $\int_{C}\vec{F}\cdot d\vec{r}$ integrates along a curve (e.g. work done); a surface integral $\iint_{S}\vec{F}\cdot \hat{n}\,dS$ integrates flux through a surface; a volume integral $\iiint_{V} f\,dV$ integrates over a 3D region.

State Green's theorem in the plane.

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

State Stokes' theorem and the Gauss (divergence) theorem.

Stokes: $\oint_{C}\vec{F}\cdot d\vec{r} = \iint_{S} (\nabla\times\vec{F})\cdot \hat{n}\,dS$ relates a line integral to a surface curl integral. Gauss: $\iint_{S}\vec{F}\cdot \hat{n}\,dS = \iiint_{V} (\nabla\cdot\vec{F})\,dV$ relates outward flux to the volume integral of divergence.