# Coordinate Systems and Vector Calculus

 Question 1
The vector function expressed by

$F = a_x(5y - k_1z) + a_y(3z + k_2x) + a_z(k_3y - 4x)$

represents a conservative field, where $a_x, a_y, a_z$ are unit vectors along x, y and z directions, respectively. The values of constants $k_1, k_2, k_3$ are given by:
 A $k_1=3, k_2=3, k_3=7$ B $k_1=3, k_2=8, k_3=5$ C $k_1=4, k_2=5, k_3=3$ D $k_1=0, k_2=0, k_3=0$
GATE EE 2020   Electromagnetic Theory
Question 1 Explanation:
$\bar{F}=(5y-k_{1}Z)\hat{i}+(3z+k_{2}x)\hat{j}+(k_{3}y-4x)\hat{k}$ is conservative field
$\bar{F}$ is irrotational,
\begin{aligned} \triangledown \times \bar{F}&=0\\ \begin{vmatrix} \hat{i} &\hat{j} &\hat{k} \\ \frac{\partial }{\partial x}&\frac{\partial }{\partial y} &\frac{\partial }{\partial z} \\ 5y-k_{1}z &3z-k_{2}x &k_{3}y -4x \end{vmatrix}&=0\\ \hat{i}(k_{3}-3)-\hat{j}(-4+k_{1})+\hat{k}(k_{2}-5)&=0\\ k_{3}-3&=0\\ 4-k_{1}&=0\\ k_{2}-5&=0\\ k_{1}&=4\\ k_{2}&=5\\ k_{3}&=3 \end{aligned}
 Question 2
The figures show diagrammatic representations of vector fields $\vec{X},\vec{Y}$ and $\vec{Z}$ respectively. Which one of the following choices is true?
 A $\bigtriangledown \cdot \vec{X}=0$, $\bigtriangledown \times \vec{Y}\neq 0,\bigtriangledown \times \vec{Z}=0$ B $\bigtriangledown \cdot \vec{X} \neq 0$, $\bigtriangledown \times \vec{Y}=0$, $\bigtriangledown \times \vec{Z} \neq 0$ C $\bigtriangledown \cdot \vec{X}\neq 0$, $\bigtriangledown \times \vec{Y}\neq 0$, $\bigtriangledown \times \vec{Z}\neq 0$ D $\bigtriangledown \cdot \vec{X}=0$, $\bigtriangledown \times \vec{Y}= 0$, $\bigtriangledown \times \vec{Z}=0$
GATE EE 2017-SET-2   Electromagnetic Theory
Question 2 Explanation:
$\vec{X}$ is going away so $\vec{\triangledown } \cdot \vec{X}\neq 0$
$\vec{Y}$ is moving circulator direction so $\vec{\triangledown } \cdot \vec{Y}\neq 0$
$\vec{Z}$ has circular rotation so $\vec{\triangledown } \cdot \vec{Z}\neq 0$
 Question 3
The line integral of the vector field $F = 5xz \hat{i}+ (3x^{2} + 2y) \hat{j} + x^{2}z\hat{k}$ along a path from (0,0,0) to (1,1,1) parametrized by (t, $t^{2}$, t) is _____.
 A 4.41 B 2.26 C 6.56 D 8.34
GATE EE 2016-SET-2   Electromagnetic Theory
Question 3 Explanation:
\begin{aligned} E &=5xZ\bar{i}+(3x^2 +2y)\bar{j}+x^2z\bar{k} \\ &=\int _C \bar{F}\bar{d}r \\ &= \int _C 5xzdx +(3x^2+2y)dy +x^2zdz\\ x&=t, \; y=t^2,\; z=t,\; t=0 \; to \; 1 \\ dx&=dt \\ dy &=2tdt,\; dz =dt\\ &=\int_{0}^{1} 5t^2dt+(3t^2+2t^2)2tdt+t^3dt\\ &= \int_{0}^{1} (5t^2+11t^3)dt\\ &=\left [ \frac{5t^3}{3}+\frac{11t^4}{4} \right ]_0^1=\frac{5}{3}+\frac{11}{4}=4.41 \end{aligned}
 Question 4
In cylindrical coordinate system, the potential produced by a uniform ring charge is given by $\psi = f(r, z)$ , where f is a continuous function of r and z. Let $\vec{E}$ be the resulting electric field. Then the magnitude of $\bigtriangledown \times \vec{E}$
 A increases with r. B is 0. C is 3. D decreases with z.
GATE EE 2016-SET-1   Electromagnetic Theory
Question 4 Explanation:
V is given as static field in time invariant.
Hence, $\triangledown \times E=0$
 Question 5
Match the following.
 A P-2 Q-1 R-4 S-3 B P-4 Q-1 R-3 S-2 C P-4 Q-3 R-1 S-2 D P-3 Q-4 R-2 S-1
GATE EE 2015-SET-2   Electromagnetic Theory
Question 5 Explanation:
Stokes theorem $\oint \vec{A}\cdot dl=\int \int (\triangledown \times A)\cdot \hat{n}ds$
Gauss theorem $\int \int D\cdot ds=Q$
Divergence theorem $\oint \oint A\cdot \hat{n}ds=\int \int \int \triangledown \cdot \bar{A}dV$
Cauchy integral theorem $\oint _cf(z)dz=0$
 Question 6
Consider a function $\vec{f}=\frac{1}{r^{2}}\hat{r}$, where r is the distance from the origin and $\hat{r}$ is the unit vector in the radial direction. The divergence of this function over a sphere of radius R, which includes the origin, is
 A 0 B 2$\pi$ C 4$\pi$ D R$\pi$
GATE EE 2015-SET-1   Electromagnetic Theory
Question 6 Explanation:
$\bar{f}=\frac{1}{r^2}\hat{r}$
From divergence theorem as we know,
\begin{aligned} \int _{vol.} (\triangledown \cdot \bar{f})dV &=\oint \oint _S \bar{f}\cdot d\bar{S}\\ \oint \oint _S \bar{f}\cdot d\bar{S} &= \oint \oint _S \left ( \frac{1}{r^2} \cdot \hat{r}\right ) \times r^2 \sin \theta \cdot d\theta \cdot d\phi \cdot \hat{r} \\ &= \oint \oint _S \sin \theta \cdot d\theta \cdot d\phi=4\pi \end{aligned}
 Question 7
The direction of vector A is radially outward from the origin, with $|A|=kr^{n}$, where $r^{2}=x^{2}+y^{2}+z^{2}$ and k is a constant. The value of n for which $\bigtriangledown \cdot A= 0$ is
 A -2 B 2 C 1 D 0
GATE EE 2012   Electromagnetic Theory
Question 7 Explanation:
\begin{aligned} \vec{A} &=kr^n\hat{i}_r \\ \triangledown \cdot \vec{A} &= \frac{1}{r^2}\frac{\partial }{\partial r}(r^2 k r^n)\\ &=\frac{1}{r^2}\frac{\partial }{\partial r}( k r^{n+2}) \\ &= (n+2)k\frac{r^{n+1}}{r^2}\\ &= (n+2)kr^{n-1}\\ \text{For}\;\; \triangledown \cdot \vec{A} &= 0\\ n+2 &=0\\ \Rightarrow \;\; n=-2 \end{aligned}
 Question 8
Divergence of the vector field
$V(x,y,z)=-(x \cos xy+y)\hat{i}+(y \cos xy)\hat{j}$ $+(\sin z^{2}+x^{2}+y^{2})\hat{k}$ is
 A $2z \cos z^{2}$ B $\sin xy+2z \cos z^{2}$ C $x sinxy - cos z$ D None of these
GATE EE 2007   Electromagnetic Theory
Question 8 Explanation:
\begin{aligned} V(x,y,z) &=-(x \cos xy+y)i +(y \cos xy)j \\ & \times[\sin (z^2)+(x^2)+(y^2)]k \\ \text{Divergence}&=\triangledown \cdot V \\ &=\frac{\partial V_x}{\partial x} +\frac{\partial V_y}{\partial y}+\frac{\partial V_z}{\partial z}\\ &=-\cos xy+x(\sin xy)y +\cos xy \\ &-y \sin (xy)x+2z \cos z^2 \\ &= 2z \cos z^2 \end{aligned}
 Question 9
Consider the following statements with reference to the equation $\frac{\delta p}{\delta t}$

(1) This is a point form of the continuity equation.
(2) Divergence of current density is equal to the decrease of charge per unit volume per unit at every point.
(3) This is Max well's divergence equation
(4) This represents the conservation of charge

If $\vec{E}$ is the electric intensity, $\triangledown \cdot (\triangledown \times \vec{E})$ is equal to
 A $\vec{E}$ B |$\vec{E}$| C null vector D Zero
i.e. $\triangledown \cdot (\triangledown \times \vec{E})=0$