GATE Agricultural Engineering Engineering Mathematics Syllabus

Define a homogeneous function of degree $n$.

$f(x,y)$ is homogeneous of degree $n$ if $f(tx,ty)=t^{n} f(x,y)$ for all $t>0$.

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

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

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

$x^{2}f_{xx}+2xy\,f_{xy}+y^{2}f_{yy}=n(n-1)f$.

Write the formula for the total differential of $z=f(x,y)$.

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

State the chain rule for $\dfrac{dz}{dt}$ when $z=f(x,y)$ with $x=x(t)$, $y=y(t)$.

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

What is the necessary condition for a stationary (critical) point of $f(x,y)$?

Both first partials vanish: $f_{x}=0$ and $f_{y}=0$ simultaneously.

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

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

Describe the Lagrange multiplier method for extremizing $f(x,y)$ subject to $g(x,y)=0$.

Form $L=f-\lambda g$ and solve $\nabla f=\lambda\nabla g$ together with $g=0$, i.e. $f_{x}=\lambda g_{x}$, $f_{y}=\lambda g_{y}$, $g=0$.

Define convergence of an infinite series $\sum_{n=1}^{\infty} a_{n}$.

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

State the necessary (n-th term) condition for convergence of $\sum a_{n}$.

If $\sum a_{n}$ converges then $\lim_{n\to\infty} a_{n}=0$. Equivalently, if $\lim_{n\to\infty} a_{n}\neq 0$, the series diverges (not a sufficient test for convergence).

State D'Alembert's ratio test for $\sum a_{n}$ with positive terms.

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

State Cauchy's root 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 $L=1$ the test is inconclusive.

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 $p\leq 1$ (the harmonic series $p=1$ diverges).

Write the Fourier series of a function $f(x)$ with period $2\pi$ on $(-\pi,\pi)$.

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

State the Euler formulas for the Fourier coefficients $a_{0}$, $a_{n}$, $b_{n}$ on $(-\pi,\pi)$.

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

In a Fourier series, which coefficients vanish for an even function and which for an odd function?

For an even function all $b_{n}=0$ (cosine series only). For an odd function $a_{0}=0$ and all $a_{n}=0$ (sine series only).

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

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

What is the Maclaurin series, and how does it relate to the Taylor series?

The Maclaurin series is the Taylor series expanded about $a=0$: $f(x)=\sum_{n=0}^{\infty}\dfrac{f^{(n)}(0)}{n!}x^{n}$.

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 $\cos x$.

$\sin x=\sum_{n=0}^{\infty}\dfrac{(-1)^{n}x^{2n+1}}{(2n+1)!}=x-\dfrac{x^{3}}{3!}+\dfrac{x^{5}}{5!}-\cdots$ and $\cos x=\sum_{n=0}^{\infty}\dfrac{(-1)^{n}x^{2n}}{(2n)!}=1-\dfrac{x^{2}}{2!}+\dfrac{x^{4}}{4!}-\cdots$

Distinguish a scalar point function from a vector point function.

A scalar point function assigns a scalar to each point in space, e.g. temperature $\phi(x,y,z)$. A vector point function assigns a vector to each point, e.g. velocity $\vec{F}(x,y,z)$.

Write the formula for the derivative of a vector function $\vec{r}(t)$ and state its geometric meaning.

$\dfrac{d\vec{r}}{dt}=\lim_{\Delta t\to 0}\dfrac{\vec{r}(t+\Delta t)-\vec{r}(t)}{\Delta t}$; it is tangent to the curve traced by $\vec{r}(t)$.

Define the vector differential operator del ($\nabla$) in Cartesian coordinates.

$\nabla=\vec{i}\dfrac{\partial}{\partial x}+\vec{j}\dfrac{\partial}{\partial y}+\vec{k}\dfrac{\partial}{\partial z}$.

Define the gradient of a scalar field $\phi$ and state what it represents.

$\nabla\phi=\vec{i}\dfrac{\partial \phi}{\partial x}+\vec{j}\dfrac{\partial \phi}{\partial y}+\vec{k}\dfrac{\partial \phi}{\partial z}$. It points in the direction of maximum rate of increase of $\phi$ and is normal to level surfaces $\phi=\text{const}$.

Define the divergence of a vector field $\vec{F}=F_{1}\vec{i}+F_{2}\vec{j}+F_{3}\vec{k}$.

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

Define the curl of a vector field $\vec{F}$ and state its result type.

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

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

Solenoidal: $\nabla\cdot\vec{F}=0$ (divergence-free). Irrotational: $\nabla\times\vec{F}=\vec{0}$ (curl-free).

State the two standard identities: divergence of a curl, and curl of a gradient.

$\nabla\cdot(\nabla\times\vec{F})=0$ and $\nabla\times(\nabla\phi)=\vec{0}$.

Express the Laplacian of a scalar $\phi$ using the del operator.

$\nabla^{2}\phi=\nabla\cdot(\nabla\phi)=\dfrac{\partial^{2}\phi}{\partial x^{2}}+\dfrac{\partial^{2}\phi}{\partial y^{2}}+\dfrac{\partial^{2}\phi}{\partial z^{2}}$.

How is the directional derivative of $\phi$ in the direction of a unit vector $\hat{a}$ computed using the gradient?

$D_{\hat{a}}\phi=\nabla\phi\cdot\hat{a}$, the component of the gradient along $\hat{a}$. It is maximum when $\hat{a}$ is along $\nabla\phi$, with value $|\nabla\phi|$.