GATE Electrical Engineering Engineering Mathematics Syllabus

State the Mean Value Theorem for integrals.

If $f$ is continuous on $[a,b]$, there exists $c\in[a,b]$ such that $$\int_a^b f(x)\,dx = f(c)\,(b-a).$$

State Leibniz's rule for differentiating an integral with variable limits.

$$\frac{d}{dx}\int_{a(x)}^{b(x)} f(x,t)\,dt = f(x,b(x))\,b'(x) - f(x,a(x))\,a'(x) + \int_{a(x)}^{b(x)} \frac{\partial f}{\partial x}\,dt.$$

What is an improper integral, and when does it converge?

An integral with an infinite limit or an unbounded integrand, defined as a limit, e.g. $\int_a^\infty f\,dx = \lim_{b\to\infty}\int_a^b f\,dx$. It converges if the limit exists and is finite.

For what values of $p$ does $\int_1^\infty \frac{dx}{x^{p}}$ converge?

It converges if and only if $p > 1$, with value $\frac{1}{p-1}$; it diverges for $p \leq 1$.

Define the Gamma function and give its key recurrence.

$$\Gamma(n) = \int_0^\infty x^{n-1}e^{-x}\,dx, \quad n>0.$$ It satisfies $\Gamma(n+1) = n\,\Gamma(n)$, and $\Gamma(n+1)=n!$ for non-negative integers; $\Gamma\!\left(\tfrac12\right)=\sqrt{\pi}$.

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

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

State Euler's theorem for a homogeneous function of degree $n$.

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

What is the necessary condition for an interior extremum of $f(x,y)$?

The first-order conditions $\frac{\partial f}{\partial x}=0$ and $\frac{\partial f}{\partial y}=0$ must hold (a critical/stationary point).

State the second-derivative test for $f(x,y)$ using the discriminant.

Let $D = f_{xx}f_{yy} - f_{xy}^{2}$ at a critical point. 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.

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 is a double integral over region $R$ used to find area, and how do limits work?

The area is $\iint_R dA$. For a region the inner integral is over one variable with the other held to describe the boundary; e.g. $\int_{x_1}^{x_2}\int_{y_1(x)}^{y_2(x)} f\,dy\,dx$.

State the Jacobian formula for changing variables in a double integral.

$$\iint_R f(x,y)\,dx\,dy = \iint_{R'} f\left|\frac{\partial(x,y)}{\partial(u,v)}\right| du\,dv,$$ where $\frac{\partial(x,y)}{\partial(u,v)}$ is the Jacobian determinant.

Write the area element $dA$ in polar coordinates for a double integral.

$$dA = r\,dr\,d\theta.$$ So $\iint f(x,y)\,dA = \iint f(r\cos\theta, r\sin\theta)\,r\,dr\,d\theta.$

State the Euler formulas for the Fourier coefficients of $f(x)$ on $[-\pi,\pi]$.

$$a_0 = \frac{1}{\pi}\int_{-\pi}^{\pi} f(x)\,dx,\quad a_n = \frac{1}{\pi}\int_{-\pi}^{\pi} f(x)\cos nx\,dx,\quad b_n = \frac{1}{\pi}\int_{-\pi}^{\pi} f(x)\sin nx\,dx.$$

Write the general Fourier series representation of a periodic function $f(x)$.

$$f(x) = \frac{a_0}{2} + \sum_{n=1}^{\infty}\left(a_n\cos nx + b_n\sin nx\right).$$

What does the Fourier series of an even function contain, and of an odd function?

An even function has only cosine terms (all $b_n=0$); an odd function has only sine terms (all $a_n=0$ including $a_0$).

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

$f$ is periodic, single-valued, has a finite number of maxima/minima and discontinuities in one period, and is absolutely integrable over a period. At a jump, the series converges to the average $\frac{f(x^+)+f(x^-)}{2}$.

Define the gradient $\nabla \phi$ of a scalar field $\phi$.

$$\nabla \phi = \frac{\partial \phi}{\partial x}\hat{i} + \frac{\partial \phi}{\partial y}\hat{j} + \frac{\partial \phi}{\partial z}\hat{k}.$$ It points in the direction of steepest increase of $\phi$.

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

$$\nabla\cdot\vec{F} = \frac{\partial F_x}{\partial x}+\frac{\partial F_y}{\partial y}+\frac{\partial F_z}{\partial z}, \qquad \nabla\times\vec{F} = \begin{vmatrix}\hat{i}&\hat{j}&\hat{k}\\ \partial_x&\partial_y&\partial_z\\ F_x&F_y&F_z\end{vmatrix}.$$

State two key vector identities involving curl of a gradient and divergence of a curl.

$$\nabla\times(\nabla\phi) = \vec{0}, \qquad \nabla\cdot(\nabla\times\vec{F}) = 0.$$

State the vector identity for the divergence of a gradient (Laplacian).

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

How is the directional derivative of $\phi$ in the direction of unit vector $\hat{a}$ computed?

$$D_{\hat{a}}\phi = \nabla\phi \cdot \hat{a}.$$ It is maximum, equal to $|\nabla\phi|$, in the direction of $\nabla\phi$.

What is the maximum value of the directional derivative of $\phi$ at a point, and its direction?

The maximum value is $|\nabla\phi|$, attained in the direction of the gradient $\nabla\phi$ itself.

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

$$\int_C \vec{F}\cdot d\vec{r} = \int_C (F_x\,dx + F_y\,dy + F_z\,dz),$$ representing work done by $\vec{F}$ along $C$.

When is a vector field $\vec{F}$ conservative, and what does it imply for line integrals?

$\vec{F}$ is conservative if $\nabla\times\vec{F}=\vec{0}$ (and the domain is simply connected); then $\vec{F}=\nabla\phi$, the line integral is path-independent, and $\oint_C \vec{F}\cdot d\vec{r}=0$.

State Green's theorem in the plane.

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

State Stokes's 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 any 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 divergence integrated over the enclosed volume $V$.

Contrast what Stokes's theorem and Gauss's divergence theorem each convert/relate.

Stokes's theorem relates a line integral (closed curve) to a surface integral of the curl. Gauss's divergence theorem relates a closed surface integral (flux) to a volume integral of the divergence.