GATE Agricultural 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 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.

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

3. 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.

4. 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).

5. 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.

6. 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.

7. 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).

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

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

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

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

10. 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).

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

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

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

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

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

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

14. 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 $\cos x=\sum_{n=0}^{\infty}\dfrac{(-1)^{n}x^{2n}}{(2n)!}=1-\dfrac{x^{2}}{2!}+\dfrac{x^{4}}{4!}-\cdots$

15. 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 $\vec{F}(x,y,z)$.

16. 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 $\vec{r}(t)$.

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

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

18. 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 $\phi=\text{const}$.

19. 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).

20. 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.

21. 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).

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

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

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

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

24. 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 $|\nabla\phi|$.

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Engineering Mathematics flashcards FAQ