GATE Mining 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. 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.

2. 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.

3. 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. $f_{x}=\lambda g_{x}$, $f_{y}=\lambda g_{y}$, $g=0$.

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

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

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

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

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

7. 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 $N\to\infty$.

8. 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 $a_n\to 0$).

9. 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.

10. 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.

11. 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 $p\leq 1$.

12. 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.

13. 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.

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

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

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

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

16. 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 $a_{0}=0$).

17. 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 $\phi$.

18. 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).

19. 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).

20. 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 $\phi$.

21. 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).

22. 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.

23. 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 $R$.

24. 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.

What this deck covers

Engineering Mathematics flashcards FAQ