GATE Engineering Mathematics Syllabus

If $A$ is diagonalizable, write the diagonalization formula and what $P$ and $D$ contain.

$A=PDP^{-1}$, where the columns of $P$ are the eigenvectors and $D$ is the diagonal matrix of corresponding eigenvalues.

Using diagonalization $A=PDP^{-1}$, give a formula for $A^{k}$.

$A^{k}=PD^{k}P^{-1}$, where $D^{k}$ is obtained by raising each diagonal eigenvalue to the power $k$.

What can be said about the eigenvalues and diagonalizability of a real symmetric matrix?

A real symmetric matrix has all real eigenvalues and is always orthogonally diagonalizable, with eigenvectors that can be chosen mutually orthonormal.

Define a vector space (the key axioms).

A set $V$ over a field with addition and scalar multiplication that is closed under both operations and satisfies: associativity/commutativity of addition, existence of a zero vector and additive inverses, and distributive/associative/identity laws for scalar multiplication.

Define a subspace of a vector space $V$.

A non-empty subset $W\subseteq V$ that is itself a vector space: it contains $\vec{0}$ and is closed under vector addition and scalar multiplication.

Define linear independence of vectors $\vec{v}_{1},\dots,\vec{v}_{k}$.

They are linearly independent if $c_{1}\vec{v}_{1}+\cdots+c_{k}\vec{v}_{k}=\vec{0}$ implies $c_{1}=\cdots=c_{k}=0$ (only the trivial combination gives zero).

Define a basis of a vector space.

A set of vectors that is both linearly independent and spans the entire space; its number of vectors equals the dimension of the space.

State the Rank–Nullity theorem for a linear map / matrix $A$ with $n$ columns.

$\operatorname{rank}(A)+\operatorname{nullity}(A)=n$, where nullity is the dimension of the null space (solution space of $A\vec{x}=\vec{0}$).

Define the nullity of a matrix $A$.

The nullity is the dimension of the null space (kernel) of $A$, i.e. the number of linearly independent solutions of $A\vec{x}=\vec{0}$, equal to the number of free variables.

State the (precise) definition of $\lim_{x\to a} f(x)=L$.

For every $\varepsilon>0$ there exists $\delta>0$ such that $0<|x-a|<\delta$ implies $|f(x)-L|<\varepsilon$.

Give the three conditions for $f$ to be continuous at $x=a$.

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

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

If $f$ is differentiable at $a$ then it is continuous at $a$; the converse is false (e.g. $f(x)=|x|$ at $x=0$ is continuous but not differentiable).

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

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

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 Cauchy's Mean Value Theorem.

If $f,g$ are continuous on $[a,b]$, differentiable on $(a,b)$, and $g'\neq 0$ on $(a,b)$, then there exists $c\in(a,b)$ with $\dfrac{f'(c)}{g'(c)}=\dfrac{f(b)-f(a)}{g(b)-g(a)}$.

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

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

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

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

State the first-derivative condition for a critical point and the second-derivative test for a local extremum of $f(x)$.

Critical points satisfy $f'(x)=0$. At such a point: $f''(x)>0\Rightarrow$ local minimum, $f''(x)<0\Rightarrow$ local maximum, $f''(x)=0\Rightarrow$ test inconclusive.

For a function $f(x,y)$, state the necessary condition for a local extremum (stationary point).

Both first partial derivatives vanish: $f_{x}=\dfrac{\partial f}{\partial x}=0$ and $f_{y}=\dfrac{\partial f}{\partial y}=0$.

State the second-derivative (Hessian) test for $f(x,y)$ using $D=f_{xx}f_{yy}-f_{xy}^{2}$.

At a stationary point: if $D>0$ and $f_{xx}>0$ it is a local minimum; if $D>0$ and $f_{xx}<0$ a local maximum; if $D<0$ a saddle point; if $D=0$ the test is inconclusive.

Define the partial derivative $\dfrac{\partial f}{\partial x}$ 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 $x$ treating $y$ as constant.

Define the gradient $\nabla f$ of a scalar field $f(x,y,z)$ and state its geometric meaning.

$\nabla f=\left(\dfrac{\partial f}{\partial x},\dfrac{\partial f}{\partial y},\dfrac{\partial f}{\partial z}\right)$. It points in the direction of greatest increase of $f$ and is normal to level surfaces.

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

$D_{\hat{u}}f=\nabla f\cdot\hat{u}$, where $\hat{u}$ is a unit vector. Its maximum value is $|\nabla f|$ (along $\nabla f$).

Define the divergence of a vector field $\vec{F}=(F_{1},F_{2},F_{3})$.

$\operatorname{div}\vec{F}=\nabla\cdot\vec{F}=\dfrac{\partial F_{1}}{\partial x}+\dfrac{\partial F_{2}}{\partial y}+\dfrac{\partial F_{3}}{\partial z}$ — a scalar measuring net outflow per unit volume.

Define the curl of a vector field $\vec{F}$ and what it measures.

$\operatorname{curl}\vec{F}=\nabla\times\vec{F}=\begin{vmatrix}\vec{i}&\vec{j}&\vec{k}\\ \partial_x&\partial_y&\partial_z\\ F_1&F_2&F_3\end{vmatrix}$; it measures the local rotation (circulation density) of the field.

What are the identities $\operatorname{curl}(\nabla f)$ and $\operatorname{div}(\operatorname{curl}\vec{F})$ equal to?

$\nabla\times(\nabla f)=\vec{0}$ and $\nabla\cdot(\nabla\times\vec{F})=0$ (curl of a gradient and divergence of a curl are always zero).

State the Fundamental Theorem of Calculus (evaluation form).

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

Write the integration by parts formula.

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

What is an improper integral, and what does it mean for it to converge? Illustrate with $\int_{1}^{\infty}\frac{1}{x^{p}}\,dx$.

An improper integral has an infinite limit or unbounded integrand; it converges if its limit exists (is finite). $\displaystyle\int_{1}^{\infty}\frac{dx}{x^{p}}$ converges if $p>1$ and diverges if $p\leq 1$.