GATE Civil Engineering Engineering Mathematics Syllabus

What is the relationship between differentiability and continuity at a point?

Differentiability implies continuity, but continuity does not imply differentiability (e.g. $f(x)=|x|$ at $x=0$).

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

$$f'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h}$$

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

If $\lim \frac{f(x)}{g(x)}$ is of form $\frac{0}{0}$ or $\frac{\infty}{\infty}$, then $$\lim \frac{f(x)}{g(x)} = \lim \frac{f'(x)}{g'(x)}$$ provided the latter exists.

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 the Lagrange Mean Value Theorem.

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

State the second derivative test for local extrema of $f(x)$ at a critical point $c$ where $f'(c)=0$.

If $f''(c) > 0$, $c$ is a local minimum; if $f''(c) < 0$, a local maximum; if $f''(c) = 0$, the test is inconclusive.

Write the Taylor series expansion of $f(x)$ about the point $x = a$.

$$f(x) = \sum_{n=0}^{\infty} \frac{f^{(n)}(a)}{n!}(x-a)^{n} = f(a) + f'(a)(x-a) + \frac{f''(a)}{2!}(x-a)^{2} + \cdots$$

What is a Maclaurin series, and write the expansion of $e^{x}$.

A Maclaurin series is a Taylor series about $a = 0$. $$e^{x} = \sum_{n=0}^{\infty} \frac{x^{n}}{n!} = 1 + x + \frac{x^{2}}{2!} + \frac{x^{3}}{3!} + \cdots$$

Write the Maclaurin series expansions of $\sin x$ and $\cos x$.

$$\sin x = x - \frac{x^{3}}{3!} + \frac{x^{5}}{5!} - \cdots, \qquad \cos x = 1 - \frac{x^{2}}{2!} + \frac{x^{4}}{4!} - \cdots$$

State the Fundamental Theorem of Calculus (evaluation form).

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

State the integration by parts formula.

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

Write the formula for the area between two curves $y = f(x)$ and $y = g(x)$ from $x=a$ to $x=b$ (with $f \geq g$).

$$A = \int_{a}^{b} \big[f(x) - g(x)\big]\,dx$$

State the disk method formula for the volume of a solid of revolution about the $x$-axis.

$$V = \pi \int_{a}^{b} \big[f(x)\big]^{2}\,dx$$

Define the partial derivative $\frac{\partial f}{\partial x}$ of $f(x,y)$.

$$\frac{\partial f}{\partial x} = \lim_{h \to 0} \frac{f(x+h, y) - f(x, y)}{h}$$ the rate of change of $f$ with respect to $x$ holding $y$ constant.

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

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

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

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

Define the gradient 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 points in the direction of maximum increase of $\phi$.

Define the divergence of a vector field $\vec{F} = F_{1}\hat{i} + F_{2}\hat{j} + F_{3}\hat{k}$.

$$\nabla \cdot \vec{F} = \frac{\partial F_{1}}{\partial x} + \frac{\partial F_{2}}{\partial y} + \frac{\partial F_{3}}{\partial z}$$ (a scalar).

Write the curl of a vector field $\vec{F}$ in determinant form.

$$\nabla \times \vec{F} = \begin{vmatrix} \hat{i} & \hat{j} & \hat{k} \\ \frac{\partial}{\partial x} & \frac{\partial}{\partial y} & \frac{\partial}{\partial z} \\ F_{1} & F_{2} & F_{3} \end{vmatrix}$$

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

$$\nabla \cdot (\nabla \times \vec{F}) = 0, \qquad \nabla \times (\nabla \phi) = \vec{0}$$ Divergence of a curl is zero; curl of a gradient is the zero vector.

Write the formula for the directional derivative of $\phi$ in the direction of unit vector $\hat{a}$.

$$D_{\hat{a}}\phi = \nabla \phi \cdot \hat{a}$$ It is maximum when $\hat{a}$ is along $\nabla\phi$, equal to $|\nabla\phi|$.

State the Green's theorem relating a line integral to a double integral.

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

State the Divergence (Gauss) theorem.

$$\iint_{S} \vec{F} \cdot \hat{n}\,dS = \iiint_{V} (\nabla \cdot \vec{F})\,dV$$ relating the surface flux to the volume integral of divergence.

State Stokes' theorem.

$$\oint_{C} \vec{F} \cdot d\vec{r} = \iint_{S} (\nabla \times \vec{F}) \cdot \hat{n}\,dS$$ relating a line integral around $C$ to the surface integral of the curl.

What is the standard form and integrating factor of a first-order linear ODE?

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

What is the condition for the equation $M\,dx + N\,dy = 0$ to be exact?

$$\frac{\partial M}{\partial y} = \frac{\partial N}{\partial x}$$

Write the standard form of a Bernoulli equation and the substitution used to solve it.

$\frac{dy}{dx} + P(x)y = Q(x)\,y^{n}$. Substitute $v = y^{1-n}$ to convert it into a linear ODE in $v$.

For a higher-order linear ODE with constant coefficients, how is the complementary function found when the auxiliary equation has distinct real roots $m_{1}, m_{2}$?

$$y_{c} = C_{1}e^{m_{1}x} + C_{2}e^{m_{2}x}$$ For a repeated root $m$: $y_{c} = (C_{1} + C_{2}x)e^{mx}$; for complex roots $\alpha \pm i\beta$: $y_{c} = e^{\alpha x}(C_{1}\cos\beta x + C_{2}\sin\beta x)$.

What substitution converts the Euler–Cauchy equation $x^{2}y'' + axy' + by = 0$ into a constant-coefficient ODE?

Substitute $x = e^{t}$ (i.e. $t = \ln x$), which transforms it into a linear ODE with constant coefficients in $t$. Equivalently, try a solution $y = x^{m}$.

Distinguish an initial value problem (IVP) from a boundary value problem (BVP).

An IVP specifies all conditions at a single value of the independent variable (e.g. $y(0)$ and $y'(0)$). A BVP specifies conditions at two or more different points (e.g. $y(0)$ and $y(L)$).

Write the Fourier series of a function $f(x)$ of period $2L$, including the Euler formula for $a_{n}$.

$$f(x) = \frac{a_{0}}{2} + \sum_{n=1}^{\infty}\left(a_{n}\cos\frac{n\pi x}{L} + b_{n}\sin\frac{n\pi x}{L}\right)$$ with $$a_{n} = \frac{1}{L}\int_{-L}^{L} f(x)\cos\frac{n\pi x}{L}\,dx$$