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GATE Mechanical Engineering Engineering Mathematics Flashcards
51 question-and-answer cards covering Engineering Mathematics as it is examined in GATE Mechanical Engineering. 24 of them are printed below, taken from across the deck — no signup, no paywall on the preview.
24 sample cards from the Engineering Mathematics deck
Sampled from the end of the deck, so these are different cards from the ones shown on the syllabus page.
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.
What this deck covers
The Engineering Mathematics deck follows the GATE Mechanical Engineering Engineering Mathematics syllabus — 6 chapters and 27 topics — so questions land on material that is genuinely examinable rather than trivia around it. That works out to roughly 8.5 cards per chapter.
Answers are written to be recallable, not just readable — averaging about 182 characters, which is long enough to carry the reasoning and short enough to say out loud.
A deck like this earns its keep on the second and third pass. Read the syllabus first so you know the shape of the subject, then use the cards to find the specific facts that have not stuck.
Engineering Mathematics flashcards FAQ
How many Engineering Mathematics flashcards are in this GATE Mechanical Engineering deck?
51 cards. This page previews 24 of them, sampled evenly across the deck so you can judge the difficulty before installing anything.
Are these GATE Mechanical Engineering flashcards free?
Yes. The preview here is free to read with no signup, and the full 51-card deck is free inside the Examius app.
What do the Engineering Mathematics cards cover?
They follow the GATE Mechanical Engineering Engineering Mathematics syllabus — 6 chapters and 27 topics — so the questions track what is actually examinable.
How should I use these flashcards?
Read the syllabus first so you know the shape of the subject, then drill the deck. Examius schedules each card with spaced repetition, so cards you keep missing come back sooner and ones you know drift further apart.