GATE Electrical 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 condition for equality of mixed partial derivatives (Clairaut/Schwarz theorem).

   If the mixed second partials are continuous, then $$\frac{\partial^2 f}{\partial x\,\partial y} = \frac{\partial^2 f}{\partial y\,\partial x}.$$

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

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

3. What is the necessary condition for an interior extremum of $f(x,y)$?

   The first-order conditions $\frac{\partial f}{\partial x}=0$ and $\frac{\partial f}{\partial y}=0$ must hold (a critical/stationary point).

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

   Let $D = f_{xx}f_{yy} - f_{xy}^{2}$ at a critical point. If $D>0$ and $f_{xx}>0$: local minimum; $D>0$ and $f_{xx}<0$: local maximum; $D<0$: saddle point; $D=0$: test inconclusive.

5. Describe 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. $f_x = \lambda g_x$, $f_y = \lambda g_y$.

6. How is a double integral over region $R$ used to find area, and how do limits work?

   The area is $\iint_R dA$. For a region the inner integral is over one variable with the other held to describe the boundary; e.g. $\int_{x_1}^{x_2}\int_{y_1(x)}^{y_2(x)} f\,dy\,dx$.

7. State the Jacobian formula for changing variables in a double integral.

   $$\iint_R f(x,y)\,dx\,dy = \iint_{R'} f\left|\frac{\partial(x,y)}{\partial(u,v)}\right| du\,dv,$$ where $\frac{\partial(x,y)}{\partial(u,v)}$ is the Jacobian determinant.

8. Write the area element $dA$ in polar coordinates for a double integral.

   $$dA = r\,dr\,d\theta.$$ So $\iint f(x,y)\,dA = \iint f(r\cos\theta, r\sin\theta)\,r\,dr\,d\theta.$

9. State the Euler formulas for the Fourier coefficients of $f(x)$ on $[-\pi,\pi]$.

   $$a_0 = \frac{1}{\pi}\int_{-\pi}^{\pi} f(x)\,dx,\quad a_n = \frac{1}{\pi}\int_{-\pi}^{\pi} f(x)\cos nx\,dx,\quad b_n = \frac{1}{\pi}\int_{-\pi}^{\pi} f(x)\sin nx\,dx.$$

10. Write the general Fourier series representation of a periodic function $f(x)$.

    $$f(x) = \frac{a_0}{2} + \sum_{n=1}^{\infty}\left(a_n\cos nx + b_n\sin nx\right).$$

11. What does the Fourier series of an even function contain, and of an odd function?

    An even function has only cosine terms (all $b_n=0$); an odd function has only sine terms (all $a_n=0$ including $a_0$).

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

    $f$ is periodic, single-valued, has a finite number of maxima/minima and discontinuities in one period, and is absolutely integrable over a period. At a jump, the series converges to the average $\frac{f(x^+)+f(x^-)}{2}$.

13. Define the gradient $\nabla \phi$ of a scalar field $\phi$.

    $$\nabla \phi = \frac{\partial \phi}{\partial x}\hat{i} + \frac{\partial \phi}{\partial y}\hat{j} + \frac{\partial \phi}{\partial z}\hat{k}.$$ It points in the direction of steepest increase of $\phi$.

14. Define the divergence and curl of a vector field $\vec{F}$.

    $$\nabla\cdot\vec{F} = \frac{\partial F_x}{\partial x}+\frac{\partial F_y}{\partial y}+\frac{\partial F_z}{\partial z}, \qquad \nabla\times\vec{F} = \begin{vmatrix}\hat{i}&\hat{j}&\hat{k}\\ \partial_x&\partial_y&\partial_z\\ F_x&F_y&F_z\end{vmatrix}.$$

15. State two key vector identities involving curl of a gradient and divergence of a curl.

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

16. State the vector identity for the divergence of a gradient (Laplacian).

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

17. How is the directional derivative of $\phi$ in the direction of unit vector $\hat{a}$ computed?

    $$D_{\hat{a}}\phi = \nabla\phi \cdot \hat{a}.$$ It is maximum, equal to $|\nabla\phi|$, in the direction of $\nabla\phi$.

18. What is the maximum value of the directional derivative of $\phi$ at a point, and its direction?

    The maximum value is $|\nabla\phi|$, attained in the direction of the gradient $\nabla\phi$ itself.

19. Define a line integral of a vector field $\vec{F}$ along a curve $C$.

    $$\int_C \vec{F}\cdot d\vec{r} = \int_C (F_x\,dx + F_y\,dy + F_z\,dz),$$ representing work done by $\vec{F}$ along $C$.

20. When is a vector field $\vec{F}$ conservative, and what does it imply for line integrals?

    $\vec{F}$ is conservative if $\nabla\times\vec{F}=\vec{0}$ (and the domain is simply connected); then $\vec{F}=\nabla\phi$, the line integral is path-independent, and $\oint_C \vec{F}\cdot d\vec{r}=0$.

21. State Green's theorem in the plane.

    $$\oint_C (P\,dx + Q\,dy) = \iint_R \left(\frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y}\right)dx\,dy,$$ for a positively oriented simple closed curve $C$ bounding region $R$.

22. State Stokes's 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 the surface integral of the curl over any surface $S$ bounded by $C$.

23. State the Gauss divergence theorem.

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

24. Contrast what Stokes's theorem and Gauss's divergence theorem each convert/relate.

    Stokes's theorem relates a line integral (closed curve) to a surface integral of the curl. Gauss's divergence theorem relates a closed surface integral (flux) to a volume integral of the divergence.

What this deck covers

Engineering Mathematics flashcards FAQ