GATE Petroleum Engineering Engineering Mathematics Syllabus

State Rolle's theorem.

If $f$ is continuous on $[a,b]$, differentiable on $(a,b)$, and $f(a)=f(b)$, then there exists $c\in(a,b)$ with $f'(c)=0$.

State Lagrange's mean value theorem.

If $f$ is continuous on $[a,b]$ and differentiable on $(a,b)$, then there exists $c\in(a,b)$ such that $$f'(c)=\frac{f(b)-f(a)}{b-a}.$$

State Cauchy's mean value theorem.

If $f,g$ are continuous on $[a,b]$, differentiable on $(a,b)$, and $g'\neq0$, then there is $c\in(a,b)$ with $$\frac{f'(c)}{g'(c)}=\frac{f(b)-f(a)}{g(b)-g(a)}.$$

State the fundamental theorem of calculus (part relating definite integral to antiderivative).

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

What is an improper integral, and what does it mean for it to converge?

An integral with an infinite limit or an unbounded integrand, e.g. $\int_{a}^{\infty} f\,dx=\lim_{t\to\infty}\int_{a}^{t} f\,dx$. It converges if this limit exists and is finite; otherwise it diverges.

For what values of $p$ does $\int_{1}^{\infty}\frac{dx}{x^{p}}$ converge?

It converges if $p>1$ (to $\frac{1}{p-1}$) and diverges if $p\leq 1$.

Define the Gamma function and give $\Gamma(n)$ for positive integer $n$.

$$\Gamma(n)=\int_{0}^{\infty} x^{n-1}e^{-x}\,dx,\qquad \Gamma(n)=(n-1)!\ \text{for integer }n,\quad \Gamma\!\left(\tfrac{1}{2}\right)=\sqrt{\pi}.$$

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.

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

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

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

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

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

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

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.

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

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

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.

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 $\phi$.

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 $\nabla\phi$.

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.

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.

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 $\phi$.

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.

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 $\phi$.

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

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

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

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

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 $C$.

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.

State Green's theorem.

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

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.

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.

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: $$\int g(y)\,dy=\int h(x)\,dx + C.$$

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 $$y\,\mu=\int \mu\,Q\,dx + C.$$

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 $F(x,y)=C$.