GATE 2020 EE Q.44: solution and answer

GATE 2020 · EEQ.44 · 2 marks · NAT

Let $a_{x}$ and $a_{y}$ be unit vectors along x and y directions, respectively. A vector function is given by

F = a_{x} y - a_{y} x

The line integral of the above function

\int_{C} F \cdot d l

along the curve $C$ , which follows the parabola $y = x^{2}$ as shown below is ________ (rounded off to 2 decimal places).

GATE 2020 EE question 44: x-y axes with the parabola y = x^2 drawn from the point (-1, 1) down through the origin (0, 0) and up to the point (2, 4). — Figure from the GATE 2020 EE paper. It is part of the question.

Answer and solution

ANSWER

The line integral along the parabola from $(- 1, 1)$ to $(2, 4)$ is $- 3.00$ .

Official answer: -3.05 to -2.95Source: Official answer key, GATE 2020 EE (IIT Delhi)

Read the path from the figure: $C$ is the arc of $y = x^{2}$ from $(- 1, 1)$ through the origin to $(2, 4)$ . The arrowheads show that direction, so $x$ runs from $- 1$ to $2$ .
Write out the integrand using $d l = a_{x} d x + a_{y} d y$ : $F \cdot d l = y d x - x d y$
Parametrise by $x$ along the curve: $y = x^{2}$ , so $d y = 2 x d x$ .
Substitute: $y d x - x d y = x^{2} d x - x (2 x d x) = - x^{2} d x$
Integrate along the path: $\int_{C} F \cdot d l = \int_{- 1}^{2} - x^{2} d x = - [\frac{x ^{3}}{3}]_{- 1}^{2} = - (\frac{8}{3} + \frac{1}{3}) = - 3$
Check: $\nabla \times F = (\frac{\partial ( - x )}{\partial x} - \frac{\partial y}{\partial y}) a_{z} = - 2 a_{z} \neq = 0$ . The field is not conservative, so the integral depends on the path and has to be taken along the parabola itself. The answer is $- 3.00$ .

KEY IDEA

For a line integral along $y = x^{2}$ , write $F \cdot d l = F_{x} d x + F_{y} d y$ . Then replace $y$ and $d y$ with $x^{2}$ and $2 x d x$ and integrate in $x$ from the start point to the end point shown by the arrows. Here $\nabla \times F = - 2 a_{z} \neq = 0$ , so no potential function exists. The integral cannot be found from the endpoints alone and must be taken along the given curve.

WATCH OUT FOR

Integrating only over the right branch ( $x$ from $0$ to $2$ ), which gives $- 8/3 \approx - 2.67$ . Other common slips: reversing the direction of travel, which gives $+ 3$ ; dropping the minus sign on the $a_{y} x$ term, so the integrand becomes $3 x^{2} d x$ and the answer $9$ ; and treating the field as conservative by taking a difference between the endpoints.

From the GATE 2020 EE paper. Every question of this paper · Download the full paper · All 23 Vector calculus questions, with notes

GATE 2020 EE Q.44: solution and answer

Answer and solution

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