Question

Consider the two-dimensional vector field $\vec{F}(x,y)=x\,\hat{i}+y\,\hat{j}$, where $\hat{i}$ and $\hat{j}$ denote the unit vectors along the $x$-axis and the $y$-axis, respectively. A contour $C$ in the $xy$-plane, as shown in the figure, is composed of two horizontal lines connected at the two ends by two semicircular arcs of unit radius. The contour is traversed in the counter-clockwise sense. The value of the closed path integral \[ \oint_{C} \vec{F}(x,y)\cdot(dx\,\hat{i}+dy\,\hat{j}) \] is ________________.

• A. 0
• B. 1
• C. $8+2\pi$
• D. $-1$

Hint:

If the curl of a vector field is zero everywhere in a simply connected domain, the field is conservative and all closed-loop integrals are automatically zero.

Answer 1

The Correct Option is A

The given vector field is \[ \vec{F}(x,y)=x\,\hat{i}+y\,\hat{j}. \] This field is \textit{conservative} because its curl is zero.
Step 1: Compute the curl.
For a 2D field $\vec{F}=\langle x,\,y\rangle$, \[ \nabla \times \vec{F}=\frac{\partial y}{\partial x}-\frac{\partial x}{\partial y}=0-0=0. \] Step 2: Use the fact that the curl is zero.
A vector field with zero curl in a simply connected region is conservative, meaning it can be written as the gradient of a potential function. Thus, the line integral around any closed curve in such a field is zero: \[ \oint_C \vec{F}\cdot d\vec{r}=0. \] Step 3: Verify the domain is simply connected.
The contour consists of two horizontal lines and two semicircular arcs forming a closed loop with no holes inside the region. Hence, the region is simply connected. Therefore the integral must vanish.
Thus, \[ \oint_C (x\,dx + y\,dy) = 0. \] Final Answer: 0