GATE Mechanical Engineering Engineering Mathematics Syllabus

List the standard indeterminate forms.

The seven indeterminate forms are $\frac{0}{0}$, $\frac{\infty}{\infty}$, $0 \cdot \infty$, $\infty - \infty$, $0^{0}$, $1^{\infty}$, and $\infty^{0}$.

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

If $\lim \frac{f(x)}{g(x)}$ gives $\frac{0}{0}$ or $\frac{\infty}{\infty}$, then $$\lim \frac{f(x)}{g(x)} = \lim \frac{f'(x)}{g'(x)}$$ provided the right-hand limit exists.

How do you evaluate a limit of the form $1^{\infty}$, e.g. $\lim f(x)^{g(x)}$?

Take the logarithm: let $L = \lim f^{g}$, then $\ln L = \lim g(x)\ln f(x)$, which is a $0 \cdot \infty$ form. A useful shortcut: if $f \to 1$, $\lim f^{g} = e^{\lim g(f-1)}$.

State the Fundamental Theorem of Calculus (evaluation part).

If $F$ is an antiderivative of continuous $f$ on $[a,b]$, then $$\int_{a}^{b} f(x)\,dx = F(b) - F(a).$$ Also $\frac{d}{dx}\int_{a}^{x} f(t)\,dt = f(x)$.

Define an improper integral and what it means for it to converge.

An improper integral has an infinite limit or an unbounded integrand, evaluated as a limit, e.g. $\int_{a}^{\infty} f\,dx = \lim_{t \to \infty}\int_{a}^{t} f\,dx$. It converges if this limit is finite, otherwise it diverges.

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

It converges if $p > 1$ (value $\frac{1}{p-1}$) and diverges if $p \leq 1$. Note the opposite holds for $\int_{0}^{1}\frac{dx}{x^{p}}$, which converges for $p < 1$.

What property must a definite integral satisfy under $\int_{a}^{b} f(x)\,dx = \int_{a}^{b} f(a+b-x)\,dx$?

This is the king property of definite integrals: replacing $x$ by $a+b-x$ leaves the value unchanged. It is widely used to simplify integrals with symmetric limits.

Define the double integral $\iint_{R} f(x,y)\,dA$ geometrically.

It represents the volume under the surface $z = f(x,y)$ over the region $R$ in the $xy$-plane. It is evaluated as an iterated integral $\int\!\int f(x,y)\,dy\,dx$ over the region's bounds.

How does the area element transform when changing a double integral to polar coordinates?

With $x = r\cos\theta$, $y = r\sin\theta$, the area element becomes $dA = r\,dr\,d\theta$. The factor $r$ is the Jacobian $\left|\frac{\partial(x,y)}{\partial(r,\theta)}\right|$.

What is the Jacobian for a change of variables in a double integral?

For $x = x(u,v)$, $y = y(u,v)$, $$J = \frac{\partial(x,y)}{\partial(u,v)} = \begin{vmatrix} x_{u} & x_{v} \\ y_{u} & y_{v} \end{vmatrix},$$ and $dx\,dy = |J|\,du\,dv$.

How is a triple integral used to compute the volume of a solid region $V$?

The volume is $$\text{Vol} = \iiint_{V} dV = \iiint_{V} dx\,dy\,dz,$$ taking $f = 1$. The mass is obtained by integrating density $\rho(x,y,z)$ over $V$.

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

It is the derivative of $f$ with respect to $x$ treating $y$ as constant: $$\frac{\partial f}{\partial x} = \lim_{h \to 0} \frac{f(x+h, y) - f(x,y)}{h}.$$

State Clairaut's (Schwarz's) theorem on mixed partial derivatives.

If the mixed second partials are continuous, the order of differentiation does not matter: $$\frac{\partial^{2} f}{\partial x \partial y} = \frac{\partial^{2} f}{\partial y \partial x}.$$

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

$$dz = \frac{\partial f}{\partial x}\,dx + \frac{\partial f}{\partial y}\,dy.$$ If $x$ and $y$ depend on $t$, then $\frac{dz}{dt} = \frac{\partial f}{\partial x}\frac{dx}{dt} + \frac{\partial f}{\partial y}\frac{dy}{dt}$ (chain rule).

State Euler's theorem for a homogeneous function of degree $n$.

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

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

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

What is the Maclaurin series, and give the expansion of $e^{x}$?

A Maclaurin series is a Taylor series about $a = 0$. For example $$e^{x} = \sum_{n=0}^{\infty} \frac{x^{n}}{n!} = 1 + x + \frac{x^{2}}{2!} + \frac{x^{3}}{3!} + \cdots$$

Write the Taylor expansion of $f(x,y)$ about $(a,b)$ up to first order.

$$f(x,y) \approx f(a,b) + (x-a)f_{x}(a,b) + (y-b)f_{y}(a,b) + \frac{1}{2!}\left[(x-a)^{2}f_{xx} + 2(x-a)(y-b)f_{xy} + (y-b)^{2}f_{yy}\right] + \cdots$$

State the second-derivative test for maxima/minima of a single-variable function.

At a critical point where $f'(c) = 0$: if $f''(c) > 0$ it is a local minimum; if $f''(c) < 0$ it is a local maximum; if $f''(c) = 0$ the test is inconclusive.

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

At a critical point ($f_{x} = f_{y} = 0$), let $D = f_{xx}f_{yy} - (f_{xy})^{2}$. If $D > 0$ and $f_{xx} > 0$: minimum; $D > 0$ and $f_{xx} < 0$: maximum; $D < 0$: saddle point; $D = 0$: inconclusive.

What is the method of Lagrange multipliers used for?

To find extrema of $f(x,y)$ subject to a constraint $g(x,y) = 0$. Solve $\nabla f = \lambda \nabla g$ together with the constraint, i.e. $f_{x} = \lambda g_{x}$, $f_{y} = \lambda g_{y}$, $g = 0$.

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

$$f(x) = \frac{a_{0}}{2} + \sum_{n=1}^{\infty}\left(a_{n}\cos nx + b_{n}\sin nx\right),$$ where $a_{n} = \frac{1}{\pi}\int_{-\pi}^{\pi} f(x)\cos nx\,dx$ and $b_{n} = \frac{1}{\pi}\int_{-\pi}^{\pi} f(x)\sin nx\,dx$.

What does the Fourier series reduce to for even and for odd functions?

For an even function only cosine terms survive ($b_{n} = 0$), giving a Fourier cosine series. For an odd function only sine terms survive ($a_{0} = a_{n} = 0$), giving a Fourier sine series.

State Dirichlet's conditions for the existence of a Fourier series.

$f(x)$ must be periodic, single-valued, have a finite number of maxima/minima and a finite number of finite discontinuities in one period, and be absolutely integrable over a period.

To what value does a Fourier series converge at a point of discontinuity $x_{0}$?

It converges to the average of the left- and right-hand limits: $$\frac{1}{2}\left[f(x_{0}^{-}) + f(x_{0}^{+})\right].$$

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 is a vector pointing in the direction of greatest rate of increase of $\phi$, 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 outward flux per unit volume (source/sink strength).

Define the curl of a vector field $\vec{F}$ and what it represents.

$$\nabla \times \vec{F} = \begin{vmatrix} \hat{i} & \hat{j} & \hat{k} \\ \frac{\partial}{\partial x} & \frac{\partial}{\partial y} & \frac{\partial}{\partial z} \\ F_{1} & F_{2} & F_{3} \end{vmatrix}.$$ It is a vector measuring the rotation (circulation density) of the field.

State two important vector identities involving curl and divergence that always equal zero.

The curl of a gradient is zero: $\nabla \times (\nabla \phi) = \vec{0}$. The divergence of a curl is zero: $\nabla \cdot (\nabla \times \vec{F}) = 0$.

What are solenoidal and irrotational (conservative) vector fields?

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

Give the formula for the directional derivative of $\phi$ in the direction of unit vector $\hat{a}$.

$$D_{\hat{a}}\phi = \nabla \phi \cdot \hat{a}.$$ It is maximum in the direction of $\nabla \phi$ with value $|\nabla \phi|$, and zero along directions tangent to a level surface.