GATE Chemical Engineering Engineering Mathematics Syllabus

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

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

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

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

Give the Maclaurin series for $\sin x$ 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 Rolle's theorem.

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

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 $g'(x) \neq 0$, then there exists $c \in (a,b)$ with $$\frac{f'(c)}{g'(c)} = \frac{f(b) - f(a)}{g(b) - g(a)}.$$

State the fundamental theorem of calculus (evaluation part).

If $F$ is an antiderivative of $f$ (i.e. $F' = f$) and $f$ is continuous on $[a,b]$, then $$\int_{a}^{b} f(x)\,dx = F(b) - F(a).$$

Define an improper integral of the first kind and its convergence.

An integral over an infinite interval, e.g. $\int_{a}^{\infty} f(x)\,dx = \lim_{t \to \infty} \int_{a}^{t} f(x)\,dx$. It converges if the 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 $p \leq 1$.

State the property of a definite integral over symmetric limits for even and odd functions.

$$\int_{-a}^{a} f(x)\,dx = \begin{cases} 2\int_{0}^{a} f(x)\,dx & f \text{ even} \\ 0 & f \text{ odd} \end{cases}$$

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

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

What is a partial derivative of $f(x,y)$ with respect to $x$?

It is the derivative treating $y$ as constant: $$\frac{\partial f}{\partial x} = \lim_{h \to 0} \frac{f(x+h, y) - f(x,y)}{h}.$$

State the condition (Clairaut/Schwarz theorem) for equality of mixed partial derivatives.

If the second partial derivatives are continuous, then the mixed partials are equal: $$\frac{\partial^{2} f}{\partial x \partial y} = \frac{\partial^{2} f}{\partial y \partial x}.$$

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

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

State the chain rule for the total derivative of $z = f(x,y)$ where $x$ and $y$ depend on $t$.

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

State the first-order necessary condition for a local maximum or minimum of $f(x,y)$.

At an interior extremum the first partial derivatives vanish: $$\frac{\partial f}{\partial x} = 0 \quad \text{and} \quad \frac{\partial f}{\partial y} = 0.$$ Such points are called critical (stationary) points.

State the second-derivative (Hessian) test for $f(x,y)$ 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 method of Lagrange multipliers used for?

To find extrema of $f(x,y)$ subject to a constraint $g(x,y) = 0$ by solving $\nabla f = \lambda \nabla g$ together with $g = 0$, where $\lambda$ is the Lagrange multiplier.

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 greatest rate of increase of $\phi$.

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

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

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

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

State the vector identity for the divergence of a curl.

$$\nabla \cdot (\nabla \times \vec{F}) = 0.$$ The divergence of any curl is identically zero.

State the vector identity for the curl of a gradient.

$$\nabla \times (\nabla \phi) = \vec{0}.$$ The curl of any gradient is the zero vector (gradient fields are irrotational).

What is a solenoidal vector field and an irrotational vector field?

A field is solenoidal if $\nabla \cdot \vec{F} = 0$ (divergence-free), and irrotational if $\nabla \times \vec{F} = \vec{0}$ (curl-free, i.e. conservative).

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

$$D_{\hat{a}}\phi = \nabla \phi \cdot \hat{a}.$$ Its maximum value is $|\nabla \phi|$, attained along the direction of $\nabla \phi$.

What is the Laplacian of a scalar field $\phi$, and how is it related to gradient and divergence?

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

State Green's theorem in the plane.

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

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 closed curve $C$ to the surface integral of the curl over a surface $S$ bounded by $C$.

State the Gauss divergence theorem.

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

When is a first-order ODE $M(x,y)\,dx + N(x,y)\,dy = 0$ exact, and how is it solved?

It is 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 $F(x,y) = c$.

Give the integrating factor and general solution form for the linear first-order ODE $\frac{dy}{dx} + P(x)y = Q(x)$.

Integrating factor $\mu = e^{\int P\,dx}$. Solution: $$y \cdot e^{\int P\,dx} = \int Q\,e^{\int P\,dx}\,dx + c.$$