GATE Aerospace 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 chain rule for $y=f(g(x))$.

   $\dfrac{dy}{dx}=f'(g(x))\cdot g'(x)$, i.e. $\dfrac{dy}{dx}=\dfrac{dy}{du}\cdot\dfrac{du}{dx}$ with $u=g(x)$.

2. Define the 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.

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

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

4. What is the necessary (first-order) condition for a local extremum of $f(x,y)$?

   Both first partial derivatives vanish: $\dfrac{\partial f}{\partial x}=0$ and $\dfrac{\partial f}{\partial y}=0$, giving a stationary (critical) point.

5. State the second-derivative (Hessian) test for $f(x,y)$ at a critical point 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.

6. For a single-variable function, give the second-derivative test for maxima and minima.

   At a critical point $x_{0}$ where $f'(x_{0})=0$: if $f''(x_{0})>0$ it is a local minimum; if $f''(x_{0})<0$ it is a local maximum; if $f''(x_{0})=0$ the test fails.

7. Define the gradient of a scalar field $\phi(x,y,z)$ and state its physical meaning.

   $\nabla\phi=\dfrac{\partial\phi}{\partial x}\hat{i}+\dfrac{\partial\phi}{\partial y}\hat{j}+\dfrac{\partial\phi}{\partial z}\hat{k}$. It points in the direction of maximum rate of increase of $\phi$ and is normal to level surfaces $\phi=$ const.

8. Define the divergence of a vector field $\vec{F}=(F_1,F_2,F_3)$ and its physical meaning.

   $\nabla\cdot\vec{F}=\dfrac{\partial F_1}{\partial x}+\dfrac{\partial F_2}{\partial y}+\dfrac{\partial F_3}{\partial z}$ (a scalar). It measures the net outward flux per unit volume (source/sink strength).

9. Define the curl of a vector field $\vec{F}$.

   $\nabla\times\vec{F}=\begin{vmatrix}\hat{i}&\hat{j}&\hat{k}\\ \partial_x&\partial_y&\partial_z\\ F_1&F_2&F_3\end{vmatrix}$ (a vector measuring local rotation/circulation of the field).

10. State the standard identities $\nabla\times(\nabla\phi)$ and $\nabla\cdot(\nabla\times\vec{F})$.

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

11. What is a solenoidal field and what is an irrotational (conservative) field?

    Solenoidal: $\nabla\cdot\vec{F}=0$ (no sources/sinks). Irrotational: $\nabla\times\vec{F}=\vec{0}$, meaning $\vec{F}=\nabla\phi$ for some scalar potential $\phi$.

12. State the formula for the directional derivative of $\phi$ in the direction of unit vector $\hat{u}$.

    $D_{\hat{u}}\phi=\nabla\phi\cdot\hat{u}$. It is maximum (equal to $|\nabla\phi|$) when $\hat{u}$ is along $\nabla\phi$.

13. State the fundamental theorem of calculus relating the definite integral to an antiderivative.

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

14. State the integration by parts formula.

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

15. Define a line integral of a vector field $\vec{F}$ along a curve $C$ and what it represents.

    $\displaystyle\int_{C}\vec{F}\cdot d\vec{r}$, representing the work done by $\vec{F}$ along $C$. For a conservative field $\vec{F}=\nabla\phi$ it equals $\phi(\text{end})-\phi(\text{start})$ and is path-independent.

16. Write the surface integral expressing the flux of $\vec{F}$ through a surface $S$.

    $\displaystyle\iint_{S}\vec{F}\cdot\hat{n}\,dS=\iint_{S}\vec{F}\cdot d\vec{S}$, where $\hat{n}$ is the outward unit normal.

17. State Green's theorem in the plane.

    $\displaystyle\oint_{C}(P\,dx+Q\,dy)=\iint_{R}\left(\dfrac{\partial Q}{\partial x}-\dfrac{\partial P}{\partial y}\right)dx\,dy$, where $C$ is the positively oriented boundary of region $R$.

18. State the Gauss (divergence) theorem.

    $\displaystyle\iint_{S}\vec{F}\cdot\hat{n}\,dS=\iiint_{V}(\nabla\cdot\vec{F})\,dV$, converting outward flux through closed surface $S$ into a volume integral over $V$.

19. State Stokes' theorem.

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

20. Compare Green's, Stokes', and Gauss' theorems by what they relate.

    Green: line integral around a plane curve to a double (area) integral. Stokes: line integral around a space curve to the surface integral of the curl (3-D generalization of Green's). Gauss: closed surface flux integral to a volume integral of the divergence.

21. What makes a first-order ODE $\frac{dy}{dx}=f(x,y)$ separable, and how is it solved?

    It is separable if it can be written $g(y)\,dy=h(x)\,dx$. Solve by integrating both sides: $\int g(y)\,dy=\int h(x)\,dx + C$.

22. Give the standard form of a first-order linear ODE and its integrating factor.

    $\dfrac{dy}{dx}+P(x)\,y=Q(x)$ with integrating factor $\mu=e^{\int P\,dx}$. Solution: $y\,\mu=\displaystyle\int Q\,\mu\,dx + C$.

23. What is the condition for the first-order ODE $M\,dx+N\,dy=0$ to be exact, and what is its solution form?

    Exact iff $\dfrac{\partial M}{\partial y}=\dfrac{\partial N}{\partial x}$. Then there exists $u(x,y)$ with $\dfrac{\partial u}{\partial x}=M,\ \dfrac{\partial u}{\partial y}=N$, and the solution is $u(x,y)=C$.

24. What is a Bernoulli equation (a first-order nonlinear ODE) and how is it reduced to linear form?

    $\dfrac{dy}{dx}+P(x)y=Q(x)y^{n}$, $n\neq 0,1$. Substitute $v=y^{1-n}$, which transforms it into the linear ODE $\dfrac{dv}{dx}+(1-n)P(x)v=(1-n)Q(x)$.

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Engineering Mathematics flashcards FAQ