GATE Marine 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. When is a vector field $\vec{F}$ called solenoidal, and when is it irrotational?

   Solenoidal if $\nabla \cdot \vec{F} = 0$ (divergence-free). Irrotational (conservative) if $\nabla \times \vec{F} = 0$.

2. What are the values of $\nabla \times (\nabla\phi)$ and $\nabla \cdot (\nabla \times \vec{F})$?

   Both are zero: $\nabla \times (\nabla\phi) = \vec{0}$ (curl of a gradient) and $\nabla \cdot (\nabla \times \vec{F}) = 0$ (divergence of a curl).

3. Write the Laplacian of a scalar field $\phi$ in terms of the del operator and in Cartesian form.

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

4. State the chain rule for $z = f(x,y)$ where $x = x(t)$ and $y = y(t)$.

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

5. State the chain rule for $z = f(u,v)$ where $u = u(x,y)$ and $v = v(x,y)$, giving $\frac{\partial z}{\partial x}$.

   $$\frac{\partial z}{\partial x} = \frac{\partial z}{\partial u}\frac{\partial u}{\partial x} + \frac{\partial z}{\partial v}\frac{\partial v}{\partial x}$$

6. What is the directional derivative of a scalar field $\phi$ in the direction of unit vector $\hat{a}$?

   $$D_{\hat{a}}\phi = \nabla\phi \cdot \hat{a}$$ It gives the rate of change of $\phi$ along $\hat{a}$.

7. In what direction is the directional derivative of $\phi$ maximum, and what is its maximum value?

   It is maximum in the direction of $\nabla\phi$, and the maximum value equals $|\nabla\phi|$.

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

   $$\int_{C} \vec{F} \cdot d\vec{r} = \int_{C} (F_{1}\,dx + F_{2}\,dy + F_{3}\,dz)$$ It represents work done by $\vec{F}$ along $C$.

9. When is a line integral $\int_{C}\vec{F}\cdot d\vec{r}$ independent of the path?

   When $\vec{F}$ is conservative, i.e. $\nabla \times \vec{F} = \vec{0}$ and $\vec{F} = \nabla\phi$. Then $\int_{C}\vec{F}\cdot d\vec{r} = \phi(B) - \phi(A)$, depending only on endpoints.

10. State Green's theorem relating a line integral to a double (surface) integral in the plane.

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

11. State Stokes' theorem relating a line integral to a surface integral.

    $$\oint_{C}\vec{F}\cdot d\vec{r} = \iint_{S}(\nabla \times \vec{F})\cdot \hat{n}\,dS$$ where $C$ is the boundary of surface $S$.

12. State the Gauss divergence theorem relating a surface integral to a volume integral.

    $$\iint_{S}\vec{F}\cdot \hat{n}\,dS = \iiint_{V}(\nabla \cdot \vec{F})\,dV$$ where $S$ is the closed surface bounding volume $V$.

13. How is the volume of a region $V$ expressed as a triple (volume) integral?

    $$\text{Volume} = \iiint_{V} dV = \iiint_{V} dx\,dy\,dz$$

14. Distinguish between linear and non-linear equations.

    A linear equation has unknowns appearing only to the first power with no products of unknowns (e.g. $ax + by = c$). A non-linear equation contains powers $\geq 2$, products, or transcendental functions of the unknowns (e.g. $x^{2} + xy = 1$).

15. Write the Newton-Raphson iteration formula for solving a non-linear equation $f(x) = 0$.

    $$x_{n+1} = x_{n} - \frac{f(x_{n})}{f'(x_{n})}$$ It has quadratic convergence near a simple root.

16. Write the iteration formula of the bisection (or secant) method; give the secant method formula for $f(x)=0$.

    Secant method: $$x_{n+1} = x_{n} - f(x_{n})\frac{x_{n} - x_{n-1}}{f(x_{n}) - f(x_{n-1})}$$ It needs two starting points and no derivative.

17. Define the order and degree of a differential equation.

    Order = order of the highest derivative present. Degree = power of the highest-order derivative after the equation is made free of radicals and fractions in derivatives.

18. Write the standard form of a first-order linear differential equation and its integrating factor.

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

19. What is the condition for a first-order equation $M\,dx + N\,dy = 0$ to be exact, and its solution method?

    Exact if $\dfrac{\partial M}{\partial y} = \dfrac{\partial N}{\partial x}$. Solution: find $\phi$ with $\frac{\partial \phi}{\partial x}=M$, $\frac{\partial \phi}{\partial y}=N$, giving $\phi(x,y)=C$.

20. How is a variable-separable first-order ODE solved?

    Rewrite as $f(y)\,dy = g(x)\,dx$, then integrate both sides: $$\int f(y)\,dy = \int g(x)\,dx + C$$

21. For a higher-order linear ODE with constant coefficients, what is the auxiliary (characteristic) equation and the complementary function for distinct real roots $m_1, m_2$?

    Auxiliary equation: replace $D^{k}$ by $m^{k}$ and set $= 0$. For distinct real roots: $$y_{c} = C_{1}e^{m_{1}x} + C_{2}e^{m_{2}x}$$

22. For a constant-coefficient linear ODE, what is the form of the complementary function for repeated real roots $m, m$ and for complex roots $\alpha \pm i\beta$?

    Repeated roots $m$: $y_{c} = (C_{1} + C_{2}x)e^{mx}$. Complex roots $\alpha \pm i\beta$: $y_{c} = e^{\alpha x}(C_{1}\cos\beta x + C_{2}\sin\beta x)$.

23. How are second-order linear PDEs $A\phi_{xx} + B\phi_{xy} + C\phi_{yy} + \ldots = 0$ classified using the discriminant?

    Using $B^{2} - 4AC$: if $> 0$ hyperbolic (e.g. wave equation); if $= 0$ parabolic (e.g. heat equation); if $< 0$ elliptic (e.g. Laplace equation).

24. Write the one-dimensional heat (diffusion) equation and the Laplace equation in two dimensions.

    Heat equation: $$\frac{\partial u}{\partial t} = c^{2}\frac{\partial^{2} u}{\partial x^{2}}$$ Laplace equation: $$\frac{\partial^{2} u}{\partial x^{2}} + \frac{\partial^{2} u}{\partial y^{2}} = 0$$

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Engineering Mathematics flashcards FAQ