GATE Biomedical Engineering Engineering Mathematics Syllabus

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

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

State Euler's theorem for a homogeneous function $f(x,y)$ of degree $n$.

If $f$ is homogeneous of degree $n$, then $$x\frac{\partial f}{\partial x} + y\frac{\partial f}{\partial y} = n f.$$

State the second-derivative test for maxima/minima of $f(x,y)$.

At a stationary 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 fails.

What is the necessary condition for a stationary (critical) point of $f(x,y)$?

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

Describe the method of Lagrange multipliers for constrained optimization.

To extremize $f(x,y)$ subject to $g(x,y) = 0$, solve $\nabla f = \lambda \nabla g$ together with $g = 0$; i.e. $f_{x} = \lambda g_{x}$, $f_{y} = \lambda g_{y}$.

How does changing the order of integration help in a double integral, and what stays invariant?

Swapping the order can simplify evaluation, but the region of integration and the value of the integral remain unchanged; the new limits must describe the same region.

State the Jacobian used to change variables from Cartesian to polar coordinates in a double integral.

With $x = r\cos\theta$, $y = r\sin\theta$, the Jacobian is $J = r$, so $dx\,dy = r\,dr\,d\theta$.

How is the area of a region $R$ expressed as a double integral?

$$\text{Area} = \iint_{R} dx\,dy.$$

How is the volume under a surface $z = f(x,y)$ over region $R$ expressed?

$$V = \iint_{R} f(x,y)\,dx\,dy.$$

Write the Fourier series of a function $f(x)$ with period $2\pi$.

$$f(x) = \frac{a_{0}}{2} + \sum_{n=1}^{\infty}\left(a_{n}\cos nx + b_{n}\sin nx\right).$$

Give the Euler formulas for the Fourier coefficients on $(-\pi, \pi)$.

$$a_{n} = \frac{1}{\pi}\int_{-\pi}^{\pi} f(x)\cos nx\,dx, \quad b_{n} = \frac{1}{\pi}\int_{-\pi}^{\pi} f(x)\sin nx\,dx.$$

For Fourier series, what terms survive if $f(x)$ is even versus odd?

If $f$ is even: only cosine terms (all $b_{n} = 0$). If $f$ is odd: only sine terms (all $a_{n} = 0$ and $a_{0} = 0$).

State the Dirichlet conditions for the convergence of a Fourier series.

$f$ must be periodic, single-valued, piecewise continuous (finite discontinuities), and have a finite number of maxima and minima in one period. At a jump it converges to the average of left and right limits.

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.

What is the Laplacian expressed via gradient and divergence?

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

Define a solenoidal and an irrotational vector field.

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

What does it mean for a line integral $\int_{C} \vec{F}\cdot d\vec{r}$ to be path-independent?

It is path-independent if $\vec{F}$ is conservative, i.e. $\vec{F} = \nabla \phi$ for some scalar $\phi$ (equivalently $\nabla \times \vec{F} = \vec{0}$). Then $\int_{C}\vec{F}\cdot d\vec{r} = \phi(B) - \phi(A)$.

Write the general form of a line integral of a vector field along curve $C$.

$$\int_{C} \vec{F}\cdot d\vec{r} = \int_{C}\left(F_{1}\,dx + F_{2}\,dy + F_{3}\,dz\right).$$

State Green's theorem in the plane.

$$\oint_{C}\left(P\,dx + Q\,dy\right) = \iint_{R}\left(\frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y}\right)dx\,dy,$$ where $C$ is the positively oriented boundary of region $R$.

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 a surface integral over $S$ bounded by $C$.

State the Gauss divergence theorem.

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

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

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

What condition makes $M\,dx + N\,dy = 0$ an exact differential equation?

It is exact if $\dfrac{\partial M}{\partial y} = \dfrac{\partial N}{\partial x}$. Then the solution is $\int M\,dx + \int(\text{terms of } N \text{ free of } x)\,dy = C$.

How is a Bernoulli equation $\frac{dy}{dx} + P y = Q y^{n}$ solved?

Divide by $y^{n}$ and substitute $v = y^{1-n}$, which converts it into the linear equation $\dfrac{dv}{dx} + (1-n)P v = (1-n)Q$.

For a homogeneous linear ODE with constant coefficients, what solution arises from real distinct roots $m_{1}, m_{2}$ of the auxiliary equation?

$$y = C_{1}e^{m_{1}x} + C_{2}e^{m_{2}x}.$$

For a constant-coefficient linear ODE, what is the solution for a repeated root $m$ (multiplicity 2) and for complex roots $\alpha \pm i\beta$?

Repeated root: $y = (C_{1} + C_{2}x)e^{mx}$. Complex roots: $y = e^{\alpha x}(C_{1}\cos\beta x + C_{2}\sin\beta x)$.

What is the general solution structure of a non-homogeneous linear ODE?

$y = y_{c} + y_{p}$, where $y_{c}$ is the complementary function (solution of the homogeneous equation) and $y_{p}$ is a particular integral satisfying the full equation.

How is Cauchy-Euler equation $x^{2}\frac{d^{2}y}{dx^{2}} + a x\frac{dy}{dx} + b y = 0$ solved?

Substitute $x = e^{t}$ (so $t = \ln x$), converting it to a constant-coefficient ODE in $t$; equivalently try $y = x^{m}$ to get an auxiliary equation in $m$.

In the method of separation of variables for a PDE, what is the basic assumption?

The solution is assumed to be a product of single-variable functions, e.g. $u(x,t) = X(x)\,T(t)$, which separates the PDE into ODEs each equal to a separation constant.

How are second-order linear PDEs $A u_{xx} + B u_{xy} + C u_{yy} + \ldots = 0$ classified, and give one physical example of each.

By the discriminant $B^{2} - 4AC$: $<0$ elliptic (e.g. Laplace equation), $=0$ parabolic (e.g. heat/diffusion equation), $>0$ hyperbolic (e.g. wave equation).