GATE Petroleum 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 partial derivative $\frac{\partial f}{\partial x}$ of $f(x,y)$.

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

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

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

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

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

4. State Euler's theorem for a homogeneous function $f(x,y)$ of degree $n$.

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

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

   Let $D=f_{xx}f_{yy}-f_{xy}^{2}$. 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.

6. What is the necessary condition for a critical (stationary) point of $f(x,y)$?

   Both first partial derivatives vanish: $$\frac{\partial f}{\partial x}=0\quad\text{and}\quad\frac{\partial f}{\partial y}=0.$$

7. Briefly describe the method of Lagrange multipliers for constrained optimization.

   To extremize $f$ subject to $g=0$, solve $\nabla f=\lambda\nabla g$ together with $g=0$, where $\lambda$ is the Lagrange multiplier. The solutions give candidate constrained extrema.

8. Define the gradient $\nabla\phi$ 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 is a vector pointing in the direction of greatest increase of $\phi$.

9. Define the directional derivative of $\phi$ in the direction of unit vector $\hat{n}$.

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

10. Define the divergence of a vector field $\vec{F}=F_1\hat{i}+F_2\hat{j}+F_3\hat{k}$.

    $$\nabla\cdot\vec{F}=\frac{\partial F_1}{\partial x}+\frac{\partial F_2}{\partial y}+\frac{\partial F_3}{\partial z}.$$ It is a scalar measuring net outward flux per unit volume.

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

    $$\nabla\times\vec{F}=\begin{vmatrix}\hat{i}&\hat{j}&\hat{k}\\[2pt]\frac{\partial}{\partial x}&\frac{\partial}{\partial y}&\frac{\partial}{\partial z}\\[2pt]F_1&F_2&F_3\end{vmatrix}.$$ It is a vector measuring local rotation.

12. What does it mean for a vector field to be solenoidal and irrotational?

    Solenoidal: $\nabla\cdot\vec{F}=0$ (divergence-free). Irrotational (conservative): $\nabla\times\vec{F}=\vec{0}$, so $\vec{F}=\nabla\phi$ for some scalar potential $\phi$.

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

    $$\nabla\times(\nabla\phi)=\vec{0},\qquad \nabla\cdot(\nabla\times\vec{F})=0.$$ The curl of a gradient and the divergence of a curl always vanish.

14. Express $\nabla\cdot(\nabla\phi)$ and state what operator it is.

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

15. State the identity for $\nabla\cdot(\phi\vec{F})$.

    $$\nabla\cdot(\phi\vec{F})=\phi(\nabla\cdot\vec{F})+\vec{F}\cdot(\nabla\phi).$$

16. State the curl–curl vector identity $\nabla\times(\nabla\times\vec{F})$.

    $$\nabla\times(\nabla\times\vec{F})=\nabla(\nabla\cdot\vec{F})-\nabla^{2}\vec{F}.$$

17. What is a line integral $\int_C \vec{F}\cdot d\vec{r}$, and what does it represent physically?

    $$\int_C \vec{F}\cdot d\vec{r}=\int_C (F_1\,dx+F_2\,dy+F_3\,dz),$$ representing the work done by force field $\vec{F}$ along curve $C$.

18. For a conservative field $\vec{F}=\nabla\phi$, what is $\int_C \vec{F}\cdot d\vec{r}$ from $A$ to $B$?

    It is path-independent: $$\int_C \vec{F}\cdot d\vec{r}=\phi(B)-\phi(A),$$ and equals $0$ around any closed loop.

19. State Green's theorem.

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

20. State the Gauss divergence theorem.

    For a closed surface $S$ enclosing volume $V$: $$\oiint_S \vec{F}\cdot \hat{n}\,dS=\iiint_V (\nabla\cdot\vec{F})\,dV.$$ It converts a surface flux integral into a volume integral.

21. State Stokes' theorem.

    For an open surface $S$ bounded by closed curve $C$: $$\oint_C \vec{F}\cdot d\vec{r}=\iint_S (\nabla\times\vec{F})\cdot \hat{n}\,dS.$$ It relates a line integral to a surface curl integral.

22. How is a first-order ODE $\frac{dy}{dx}=f(x,y)$ solved by separation of variables?

    If it can be written $g(y)\,dy=h(x)\,dx$, integrate both sides: $$\int g(y)\,dy=\int h(x)\,dx + C.$$

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

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

24. What is the condition for the equation $M\,dx+N\,dy=0$ to be exact, and how is it solved?

    Exact if $\frac{\partial M}{\partial y}=\frac{\partial N}{\partial x}$. Then there exists $F$ with $F_x=M$, $F_y=N$, and the solution is $F(x,y)=C$.

What this deck covers

Engineering Mathematics flashcards FAQ