Consider the two-dimensional vector field F(x,y)=x î+y ĵ, where î and ĵ denote the unit vectors along the x-axis and the y-axis, respectively. A contour C in the x-y plane, as shown in the figure, is composed of two horizontal lines connected at the ends by two semicircular arcs of uni

Question

Consider the two-dimensional vector field F(x,y)=x î+y ĵ, where î and ĵ denote the unit vectors along the x-axis and the y-axis, respectively. A contour C in the x-y plane, as shown in the figure, is composed of two horizontal lines connected at the ends by two semicircular arcs of unit radius. The contour is traversed in the counter-clockwise sense. The value of the closed path integral ∮_C F(x,y)· (dx î+dy ĵ) is _________.

Accepted Answer

0. The vector field ( ext{F}(x,y) = x ext{hat{i}} + y ext{hat{j}}) is conservative, meaning that the line integral over any closed path is zero. This is because the field can be expressed as the gradient of a scalar potential function, which implies that the integral around a closed contour yields no net work done.

Solution

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