GATE Civil 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. What is a Maclaurin series, and write the expansion of $e^{x}$.

   A Maclaurin series is a Taylor series about $a = 0$. $$e^{x} = \sum_{n=0}^{\infty} \frac{x^{n}}{n!} = 1 + x + \frac{x^{2}}{2!} + \frac{x^{3}}{3!} + \cdots$$

2. Write the Maclaurin series expansions of $\sin x$ and $\cos x$.

   $$\sin x = x - \frac{x^{3}}{3!} + \frac{x^{5}}{5!} - \cdots, \qquad \cos x = 1 - \frac{x^{2}}{2!} + \frac{x^{4}}{4!} - \cdots$$

3. State the Fundamental Theorem of Calculus (evaluation form).

   If $F'(x) = f(x)$, then $$\int_{a}^{b} f(x)\,dx = F(b) - F(a)$$

4. State the integration by parts formula.

   $$\int u\,dv = uv - \int v\,du$$

5. Write the formula for the area between two curves $y = f(x)$ and $y = g(x)$ from $x=a$ to $x=b$ (with $f \geq g$).

   $$A = \int_{a}^{b} \big[f(x) - g(x)\big]\,dx$$

6. State the disk method formula for the volume of a solid of revolution about the $x$-axis.

   $$V = \pi \int_{a}^{b} \big[f(x)\big]^{2}\,dx$$

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

   $$\frac{\partial f}{\partial x} = \lim_{h \to 0} \frac{f(x+h, y) - f(x, y)}{h}$$ the rate of change of $f$ with respect to $x$ holding $y$ constant.

8. State the total derivative (total differential) of $z = f(x, y)$.

   $$dz = \frac{\partial f}{\partial x}\,dx + \frac{\partial f}{\partial y}\,dy$$

9. For $z = f(x,y)$ with $x = x(t)$, $y = y(t)$, write the chain rule for $\frac{dz}{dt}$.

   $$\frac{dz}{dt} = \frac{\partial f}{\partial x}\frac{dx}{dt} + \frac{\partial f}{\partial y}\frac{dy}{dt}$$

10. Define the gradient of a scalar field $\phi(x,y,z)$.

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

11. Define the divergence of a vector field $\vec{F} = F_{1}\hat{i} + F_{2}\hat{j} + F_{3}\hat{k}$.

    $$\nabla \cdot \vec{F} = \frac{\partial F_{1}}{\partial x} + \frac{\partial F_{2}}{\partial y} + \frac{\partial F_{3}}{\partial z}$$ (a scalar).

12. Write the curl of a vector field $\vec{F}$ in determinant form.

    $$\nabla \times \vec{F} = \begin{vmatrix} \hat{i} & \hat{j} & \hat{k} \\ \frac{\partial}{\partial x} & \frac{\partial}{\partial y} & \frac{\partial}{\partial z} \\ F_{1} & F_{2} & F_{3} \end{vmatrix}$$

13. State the two key vector identities: $\nabla \cdot (\nabla \times \vec{F})$ and $\nabla \times (\nabla \phi)$.

    $$\nabla \cdot (\nabla \times \vec{F}) = 0, \qquad \nabla \times (\nabla \phi) = \vec{0}$$ Divergence of a curl is zero; curl of a gradient is the zero vector.

14. Write the formula for the directional derivative of $\phi$ in the direction of unit vector $\hat{a}$.

    $$D_{\hat{a}}\phi = \nabla \phi \cdot \hat{a}$$ It is maximum when $\hat{a}$ is along $\nabla\phi$, equal to $|\nabla\phi|$.

15. State the Green's theorem relating a line integral to a double integral.

    $$\oint_{C} (P\,dx + Q\,dy) = \iint_{R} \left( \frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y} \right) dx\,dy$$

16. State the Divergence (Gauss) theorem.

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

17. 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 $C$ to the surface integral of the curl.

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

    Standard form: $\frac{dy}{dx} + P(x)y = Q(x)$. Integrating factor: $$\mu = e^{\int P(x)\,dx}$$ Solution: $y\,\mu = \int Q\,\mu\,dx + C$.

19. What is the condition for the equation $M\,dx + N\,dy = 0$ to be exact?

    $$\frac{\partial M}{\partial y} = \frac{\partial N}{\partial x}$$

20. Write the standard form of a Bernoulli equation and the substitution used to solve it.

    $\frac{dy}{dx} + P(x)y = Q(x)\,y^{n}$. Substitute $v = y^{1-n}$ to convert it into a linear ODE in $v$.

21. For a higher-order linear ODE with constant coefficients, how is the complementary function found when the auxiliary equation has distinct real roots $m_{1}, m_{2}$?

    $$y_{c} = C_{1}e^{m_{1}x} + C_{2}e^{m_{2}x}$$ For a repeated root $m$: $y_{c} = (C_{1} + C_{2}x)e^{mx}$; for complex roots $\alpha \pm i\beta$: $y_{c} = e^{\alpha x}(C_{1}\cos\beta x + C_{2}\sin\beta x)$.

22. What substitution converts the Euler–Cauchy equation $x^{2}y'' + axy' + by = 0$ into a constant-coefficient ODE?

    Substitute $x = e^{t}$ (i.e. $t = \ln x$), which transforms it into a linear ODE with constant coefficients in $t$. Equivalently, try a solution $y = x^{m}$.

23. Distinguish an initial value problem (IVP) from a boundary value problem (BVP).

    An IVP specifies all conditions at a single value of the independent variable (e.g. $y(0)$ and $y'(0)$). A BVP specifies conditions at two or more different points (e.g. $y(0)$ and $y(L)$).

24. Write the Fourier series of a function $f(x)$ of period $2L$, including the Euler formula for $a_{n}$.

    $$f(x) = \frac{a_{0}}{2} + \sum_{n=1}^{\infty}\left(a_{n}\cos\frac{n\pi x}{L} + b_{n}\sin\frac{n\pi x}{L}\right)$$ with $$a_{n} = \frac{1}{L}\int_{-L}^{L} f(x)\cos\frac{n\pi x}{L}\,dx$$

What this deck covers

Engineering Mathematics flashcards FAQ