GATE Biomedical Engineering Engineering Mathematics Flashcards

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.

1. How is the area of a region $R$ expressed as a double integral?

   $$\text{Area} = \iint_{R} dx\,dy.$$

2. How is the volume under a surface $z = f(x,y)$ over region $R$ expressed?

   $$V = \iint_{R} f(x,y)\,dx\,dy.$$

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

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

4. Give the Euler formulas for the Fourier coefficients on $(-\pi, \pi)$.

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

5. For Fourier series, what terms survive if $f(x)$ is even versus odd?

   If $f$ is even: only cosine terms (all $b_{n} = 0$). If $f$ is odd: only sine terms (all $a_{n} = 0$ and $a_{0} = 0$).

6. State the Dirichlet conditions for the convergence of a Fourier series.

   $f$ must be periodic, single-valued, piecewise continuous (finite discontinuities), and have a finite number of maxima and minima in one period. At a jump it converges to the average of left and right limits.

7. State the vector identity for the divergence of a curl.

   $$\nabla \cdot (\nabla \times \vec{F}) = 0.$$ The divergence of any curl is identically zero.

8. State the vector identity for the curl of a gradient.

   $$\nabla \times (\nabla \phi) = \vec{0}.$$ The curl of any gradient is the zero vector.

9. What is the Laplacian expressed via gradient and divergence?

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

10. Define a solenoidal and an irrotational vector field.

    A field $\vec{F}$ is solenoidal if $\nabla \cdot \vec{F} = 0$ (divergence-free), and irrotational if $\nabla \times \vec{F} = \vec{0}$ (curl-free).

11. What does it mean for a line integral $\int_{C} \vec{F}\cdot d\vec{r}$ to be path-independent?

    It is path-independent if $\vec{F}$ is conservative, i.e. $\vec{F} = \nabla \phi$ for some scalar $\phi$ (equivalently $\nabla \times \vec{F} = \vec{0}$). Then $\int_{C}\vec{F}\cdot d\vec{r} = \phi(B) - \phi(A)$.

12. Write the general form of a line integral of a vector field along curve $C$.

    $$\int_{C} \vec{F}\cdot d\vec{r} = \int_{C}\left(F_{1}\,dx + F_{2}\,dy + F_{3}\,dz\right).$$

13. State Green's theorem in the plane.

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

14. State Stokes' theorem.

    $$\oint_{C}\vec{F}\cdot d\vec{r} = \iint_{S}(\nabla \times \vec{F})\cdot \hat{n}\,dS,$$ relating a line integral around closed curve $C$ to a surface integral over $S$ bounded by $C$.

15. State the Gauss divergence theorem.

    $$\iint_{S}\vec{F}\cdot \hat{n}\,dS = \iiint_{V}(\nabla \cdot \vec{F})\,dV,$$ relating flux through closed surface $S$ to the divergence integrated over the enclosed volume $V$.

16. What is the standard form and integrating factor of a first-order linear ODE?

    Standard form: $\dfrac{dy}{dx} + P(x)y = Q(x)$. Integrating factor: $\mathrm{I.F.} = e^{\int P\,dx}$, giving solution $y\cdot\mathrm{I.F.} = \int Q\cdot\mathrm{I.F.}\,dx + C$.

17. What condition makes $M\,dx + N\,dy = 0$ an exact differential equation?

    It is exact if $\dfrac{\partial M}{\partial y} = \dfrac{\partial N}{\partial x}$. Then the solution is $\int M\,dx + \int(\text{terms of } N \text{ free of } x)\,dy = C$.

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

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

19. For a homogeneous linear ODE with constant coefficients, what solution arises from real distinct roots $m_{1}, m_{2}$ of the auxiliary equation?

    $$y = C_{1}e^{m_{1}x} + C_{2}e^{m_{2}x}.$$

20. For a constant-coefficient linear ODE, what is the solution for a repeated root $m$ (multiplicity 2) and for complex roots $\alpha \pm i\beta$?

    Repeated root: $y = (C_{1} + C_{2}x)e^{mx}$. Complex roots: $y = e^{\alpha x}(C_{1}\cos\beta x + C_{2}\sin\beta x)$.

21. What is the general solution structure of a non-homogeneous linear ODE?

    $y = y_{c} + y_{p}$, where $y_{c}$ is the complementary function (solution of the homogeneous equation) and $y_{p}$ is a particular integral satisfying the full equation.

22. How is Cauchy-Euler equation $x^{2}\frac{d^{2}y}{dx^{2}} + a x\frac{dy}{dx} + b y = 0$ solved?

    Substitute $x = e^{t}$ (so $t = \ln x$), converting it to a constant-coefficient ODE in $t$; equivalently try $y = x^{m}$ to get an auxiliary equation in $m$.

23. In the method of separation of variables for a PDE, what is the basic assumption?

    The solution is assumed to be a product of single-variable functions, e.g. $u(x,t) = X(x)\,T(t)$, which separates the PDE into ODEs each equal to a separation constant.

24. How are second-order linear PDEs $A u_{xx} + B u_{xy} + C u_{yy} + \ldots = 0$ classified, and give one physical example of each.

    By the discriminant $B^{2} - 4AC$: $<0$ elliptic (e.g. Laplace equation), $=0$ parabolic (e.g. heat/diffusion equation), $>0$ hyperbolic (e.g. wave equation).

What this deck covers

Engineering Mathematics flashcards FAQ