GATE Marine Engineering Engineering Mathematics Syllabus

State the Cayley-Hamilton theorem.

Every square matrix satisfies its own characteristic equation. If the characteristic polynomial is $p(\lambda) = 0$, then $p(A) = 0$ (the zero matrix).

If $\lambda$ is an eigenvalue of $A$, what are the eigenvalues of $A^{-1}$, $A^{k}$, and $A^{T}$?

$A^{-1}$ has eigenvalue $\frac{1}{\lambda}$; $A^{k}$ has eigenvalue $\lambda^{k}$; $A^{T}$ has the same eigenvalue $\lambda$ as $A$.

What can be said about the eigenvalues of a symmetric (real) matrix and a skew-symmetric matrix?

Real symmetric matrices have all real eigenvalues. Skew-symmetric matrices have eigenvalues that are zero or purely imaginary.

For the gradient of a scalar field $\phi$, write the operator $\nabla\phi$ in Cartesian coordinates.

$$\nabla\phi = \frac{\partial \phi}{\partial x}\hat{i} + \frac{\partial \phi}{\partial y}\hat{j} + \frac{\partial \phi}{\partial z}\hat{k}$$

What is the physical/geometric significance of the gradient $\nabla\phi$?

$\nabla\phi$ points in the direction of maximum rate of increase of $\phi$, its magnitude is that maximum rate, and it is normal to the level surface $\phi = \text{constant}$.

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 the outflow/source density.

Define the curl of a vector field $\vec{F}$ and state its physical meaning.

$$\nabla \times \vec{F} = \begin{vmatrix} \hat{i} & \hat{j} & \hat{k} \\ \frac{\partial}{\partial x} & \frac{\partial}{\partial y} & \frac{\partial}{\partial z} \\ F_{1} & F_{2} & F_{3} \end{vmatrix}$$ It measures the rotation/circulation of the field.

When is a vector field $\vec{F}$ called solenoidal, and when is it irrotational?

Solenoidal if $\nabla \cdot \vec{F} = 0$ (divergence-free). Irrotational (conservative) if $\nabla \times \vec{F} = 0$.

What are the values of $\nabla \times (\nabla\phi)$ and $\nabla \cdot (\nabla \times \vec{F})$?

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

Write the Laplacian of a scalar field $\phi$ in terms of the del operator and in Cartesian form.

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

State the chain rule for $z = f(x,y)$ where $x = x(t)$ and $y = y(t)$.

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

State the chain rule for $z = f(u,v)$ where $u = u(x,y)$ and $v = v(x,y)$, giving $\frac{\partial z}{\partial x}$.

$$\frac{\partial z}{\partial x} = \frac{\partial z}{\partial u}\frac{\partial u}{\partial x} + \frac{\partial z}{\partial v}\frac{\partial v}{\partial x}$$

What is the directional derivative of a scalar field $\phi$ in the direction of unit vector $\hat{a}$?

$$D_{\hat{a}}\phi = \nabla\phi \cdot \hat{a}$$ It gives the rate of change of $\phi$ along $\hat{a}$.

In what direction is the directional derivative of $\phi$ maximum, and what is its maximum value?

It is maximum in the direction of $\nabla\phi$, and the maximum value equals $|\nabla\phi|$.

Define a line integral of a vector field $\vec{F}$ along a curve $C$.

$$\int_{C} \vec{F} \cdot d\vec{r} = \int_{C} (F_{1}\,dx + F_{2}\,dy + F_{3}\,dz)$$ It represents work done by $\vec{F}$ along $C$.

When is a line integral $\int_{C}\vec{F}\cdot d\vec{r}$ independent of the path?

When $\vec{F}$ is conservative, i.e. $\nabla \times \vec{F} = \vec{0}$ and $\vec{F} = \nabla\phi$. Then $\int_{C}\vec{F}\cdot d\vec{r} = \phi(B) - \phi(A)$, depending only on endpoints.

State Green's theorem relating a line integral to a double (surface) integral 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 Stokes' theorem relating a line integral to a surface integral.

$$\oint_{C}\vec{F}\cdot d\vec{r} = \iint_{S}(\nabla \times \vec{F})\cdot \hat{n}\,dS$$ where $C$ is the boundary of surface $S$.

State the Gauss divergence theorem relating a surface integral to a volume integral.

$$\iint_{S}\vec{F}\cdot \hat{n}\,dS = \iiint_{V}(\nabla \cdot \vec{F})\,dV$$ where $S$ is the closed surface bounding volume $V$.

How is the volume of a region $V$ expressed as a triple (volume) integral?

$$\text{Volume} = \iiint_{V} dV = \iiint_{V} dx\,dy\,dz$$

Distinguish between linear and non-linear equations.

A linear equation has unknowns appearing only to the first power with no products of unknowns (e.g. $ax + by = c$). A non-linear equation contains powers $\geq 2$, products, or transcendental functions of the unknowns (e.g. $x^{2} + xy = 1$).

Write the Newton-Raphson iteration formula for solving a non-linear equation $f(x) = 0$.

$$x_{n+1} = x_{n} - \frac{f(x_{n})}{f'(x_{n})}$$ It has quadratic convergence near a simple root.

Write the iteration formula of the bisection (or secant) method; give the secant method formula for $f(x)=0$.

Secant method: $$x_{n+1} = x_{n} - f(x_{n})\frac{x_{n} - x_{n-1}}{f(x_{n}) - f(x_{n-1})}$$ It needs two starting points and no derivative.

Define the order and degree of a differential equation.

Order = order of the highest derivative present. Degree = power of the highest-order derivative after the equation is made free of radicals and fractions in derivatives.

Write the standard form of a first-order linear differential equation and its integrating factor.

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

What is the condition for a first-order equation $M\,dx + N\,dy = 0$ to be exact, and its solution method?

Exact if $\dfrac{\partial M}{\partial y} = \dfrac{\partial N}{\partial x}$. Solution: find $\phi$ with $\frac{\partial \phi}{\partial x}=M$, $\frac{\partial \phi}{\partial y}=N$, giving $\phi(x,y)=C$.

How is a variable-separable first-order ODE solved?

Rewrite as $f(y)\,dy = g(x)\,dx$, then integrate both sides: $$\int f(y)\,dy = \int g(x)\,dx + C$$

For a higher-order linear ODE with constant coefficients, what is the auxiliary (characteristic) equation and the complementary function for distinct real roots $m_1, m_2$?

Auxiliary equation: replace $D^{k}$ by $m^{k}$ and set $= 0$. For distinct real roots: $$y_{c} = C_{1}e^{m_{1}x} + C_{2}e^{m_{2}x}$$

For a constant-coefficient linear ODE, what is the form of the complementary function for repeated real roots $m, m$ and for complex roots $\alpha \pm i\beta$?

Repeated roots $m$: $y_{c} = (C_{1} + C_{2}x)e^{mx}$. Complex roots $\alpha \pm i\beta$: $y_{c} = e^{\alpha x}(C_{1}\cos\beta x + C_{2}\sin\beta x)$.

How are second-order linear PDEs $A\phi_{xx} + B\phi_{xy} + C\phi_{yy} + \ldots = 0$ classified using the discriminant?

Using $B^{2} - 4AC$: if $> 0$ hyperbolic (e.g. wave equation); if $= 0$ parabolic (e.g. heat equation); if $< 0$ elliptic (e.g. Laplace equation).

Write the one-dimensional heat (diffusion) equation and the Laplace equation in two dimensions.

Heat equation: $$\frac{\partial u}{\partial t} = c^{2}\frac{\partial^{2} u}{\partial x^{2}}$$ Laplace equation: $$\frac{\partial^{2} u}{\partial x^{2}} + \frac{\partial^{2} u}{\partial y^{2}} = 0$$