GATE E&C 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. Define the Gamma function and give $\Gamma(n)$ for positive integers.

   $\Gamma(n)=\int_0^{\infty} x^{n-1}e^{-x}\,dx$ for $n>0$. For positive integers $\Gamma(n)=(n-1)!$, and $\Gamma\!\left(\tfrac{1}{2}\right)=\sqrt{\pi}$.

2. Define the Beta function and its relation to the Gamma function.

   $B(m,n)=\int_0^{1} x^{m-1}(1-x)^{n-1}\,dx=\dfrac{\Gamma(m)\Gamma(n)}{\Gamma(m+n)}$ for $m,n>0$.

3. Define a partial derivative $\frac{\partial f}{\partial x}$ of $f(x,y)$.

   $\dfrac{\partial f}{\partial x}=\lim_{h\to 0}\dfrac{f(x+h,y)-f(x,y)}{h}$ — differentiate with respect to $x$ holding $y$ constant.

4. State the condition for equality of mixed partial derivatives (Clairaut/Schwarz theorem).

   If $f_{xy}$ and $f_{yx}$ are continuous in a neighborhood of a point, then $\dfrac{\partial^2 f}{\partial x\,\partial y}=\dfrac{\partial^2 f}{\partial y\,\partial x}$.

5. State Euler's theorem for homogeneous functions.

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

6. Give the total differential of $z=f(x,y)$ and the chain rule for $\frac{dz}{dt}$.

   $dz=\dfrac{\partial f}{\partial x}dx+\dfrac{\partial f}{\partial y}dy$; if $x=x(t),y=y(t)$ then $\dfrac{dz}{dt}=\dfrac{\partial f}{\partial x}\dfrac{dx}{dt}+\dfrac{\partial f}{\partial y}\dfrac{dy}{dt}$.

7. State the necessary condition for a local extremum of $f(x,y)$.

   At an interior extremum (stationary point) both first partials vanish: $f_x=0$ and $f_y=0$ simultaneously.

8. State the second-derivative (Hessian) test for $f(x,y)$.

   Let $D=f_{xx}f_{yy}-(f_{xy})^2$ at a stationary 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.

9. Describe the method of Lagrange multipliers for constrained extrema.

   To extremize $f(x,y)$ subject to $g(x,y)=0$, solve $\nabla f=\lambda\nabla g$ together with $g=0$, i.e. $f_x=\lambda g_x$, $f_y=\lambda g_y$, $g=0$, for the stationary points.

10. How do you reverse the order of integration in a double integral, and why?

    Sketch the region, then re-express the limits with the other variable outer. The value is unchanged (Fubini's theorem) when $f$ is continuous on the region; reversing often simplifies an otherwise hard integral.

11. State the Jacobian for changing variables in a double integral.

    $\iint_R f\,dx\,dy=\iint_{R'} f\,|J|\,du\,dv$ where $J=\dfrac{\partial(x,y)}{\partial(u,v)}=\begin{vmatrix} x_u & x_v \\ y_u & y_v \end{vmatrix}$.

12. Give the area element in polar coordinates for a double integral.

    $dx\,dy=r\,dr\,d\theta$, so $\iint f(x,y)\,dx\,dy=\iint f(r\cos\theta,r\sin\theta)\,r\,dr\,d\theta$.

13. Give the volume elements for cylindrical and spherical coordinates.

    Cylindrical: $dV=r\,dr\,d\theta\,dz$. Spherical: $dV=\rho^{2}\sin\phi\,d\rho\,d\phi\,d\theta$ (with $\phi$ the polar angle).

14. State Green's theorem in the plane.

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

15. State the Gauss divergence theorem.

    $\iiint_V (\nabla\cdot\vec{F})\,dV=\oiint_S \vec{F}\cdot\hat{n}\,dS$, relating the volume integral of the divergence to the outward flux through the closed surface $S$.

16. State Stokes' theorem.

    $\oint_C \vec{F}\cdot d\vec{r}=\iint_S (\nabla\times\vec{F})\cdot\hat{n}\,dS$, relating the line integral around boundary curve $C$ to the surface integral of the curl over $S$.

17. When is a line integral $\int_C \vec{F}\cdot d\vec{r}$ independent of path?

    When $\vec{F}$ is conservative, i.e. $\vec{F}=\nabla\phi$ for some scalar potential $\phi$ (equivalently $\nabla\times\vec{F}=\vec{0}$ in a simply connected region). Then $\int_C\vec{F}\cdot d\vec{r}=\phi(B)-\phi(A)$.

18. State the Taylor series of $f(x)$ about $x=a$.

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

19. Give the Maclaurin series for $e^{x}$, $\sin x$ and $\cos x$.

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

20. How do you solve a first-order linear ODE $\frac{dy}{dx}+P(x)y=Q(x)$?

    Use the integrating factor $\mu=e^{\int P\,dx}$. Then $(\mu y)'=\mu Q$, giving $y=\dfrac{1}{\mu}\left(\int \mu Q\,dx + C\right)$.

21. What is an exact first-order ODE and its condition?

    $M\,dx+N\,dy=0$ is exact if $\dfrac{\partial M}{\partial y}=\dfrac{\partial N}{\partial x}$. Then there is $F$ with $F_x=M$, $F_y=N$, and the solution is $F(x,y)=C$.

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

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

23. For a higher-order linear ODE with constant coefficients, how is the homogeneous solution found from the auxiliary equation?

    From the auxiliary (characteristic) equation: distinct real roots $m$ give $e^{mx}$; a repeated root $m$ (multiplicity $k$) gives $(c_1+c_2 x+\cdots+c_k x^{k-1})e^{mx}$; complex roots $\alpha\pm i\beta$ give $e^{\alpha x}(c_1\cos\beta x+c_2\sin\beta x)$.

24. How do you solve a Cauchy–Euler equation $x^{2}y''+axy'+by=0$?

    Assume $y=x^{m}$, giving the indicial equation $m(m-1)+am+b=0$. Alternatively substitute $x=e^{t}$ (so $t=\ln x$) to convert it into a constant-coefficient linear ODE in $t$.

What this deck covers

Engineering Mathematics flashcards FAQ