GATE Aerospace Engineering Engineering Mathematics Syllabus

What three conditions must hold for $f$ to be continuous at $x=a$?

$f(a)$ is defined, $\lim_{x\to a}f(x)$ exists, and $\lim_{x\to a}f(x)=f(a)$.

State the limit definition of the derivative $f'(x)$.

$f'(x)=\lim_{h\to 0}\dfrac{f(x+h)-f(x)}{h}$, provided the limit exists.

What is the relationship between differentiability and continuity of a function?

Differentiability at a point implies continuity there, but continuity does not imply differentiability (e.g. $f(x)=|x|$ is continuous but not differentiable at $x=0$).

State the standard limits $\lim_{x\to 0}\frac{\sin x}{x}$ and $\lim_{x\to 0}\frac{1-\cos x}{x^{2}}$.

$\lim_{x\to 0}\dfrac{\sin x}{x}=1$ and $\lim_{x\to 0}\dfrac{1-\cos x}{x^{2}}=\dfrac{1}{2}$.

State L'Hôpital's rule for the indeterminate form $\frac{0}{0}$ or $\frac{\infty}{\infty}$.

$\lim_{x\to a}\dfrac{f(x)}{g(x)}=\lim_{x\to a}\dfrac{f'(x)}{g'(x)}$, provided the right-hand limit exists and the form is indeterminate.

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 $f'(c)=\dfrac{f(b)-f(a)}{b-a}$.

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 $u=g(x)$.

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.

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

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

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.

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.

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.

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.

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).

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).

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).

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 $\phi$.

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 $\nabla\phi$.

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

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

State the integration by parts formula.

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

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.

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.

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 $R$.

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 $V$.

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 $S$.

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.

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

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

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 $u(x,y)=C$.

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 $\dfrac{dv}{dx}+(1-n)P(x)v=(1-n)Q(x)$.