Consider the two-dimensional vector field ๐น⃗(๐ฅ,๐ฆ)=๐ฅ ๐ค⃗+๐ฆ ๐ฅ⃗, where ๐ค⃗ and ๐ฅ⃗ denote the unit vectors along the ๐ฅ-axis and the ๐ฆ-axis, respectively.
Question
Consider the two-dimensional vector field
F(x, y) = x i + y j,
where i and j denote the unit vectors along the x-axis and y-axis respectively. The contour C in the x-y plane is composed of two horizontal line segments connected at the two ends by semicircular arcs of unit radius. The contour is traversed in the counter-clockwise direction. Evaluate
∮C F(x,y) · (dx i + dy j)
Contour (Approximate)
Solution
The given vector field is
F(x,y) = x i + y j.
The required line integral is
∮C (x dx + y dy).
Observe that
x dx + y dy = d((x² + y²)/2).
Hence the vector field is conservative with potential function
ฯ(x,y) = (x² + y²)/2.
For any conservative vector field, the line integral over a closed curve is equal to the change in the potential between the initial and final points. Since a closed curve starts and ends at the same point,
∮C F · dr = ฯ(final) − ฯ(initial) = 0.
Alternative Verification (Green's Theorem)
Let
P = x, Q = y.
Green's theorem gives
∮C (P dx + Q dy) = ∬R (∂Q/∂x − ∂P/∂y) dA.
Since
∂Q/∂x = 0, ∂P/∂y = 0,
we obtain
∮C (x dx + y dy) = ∬R 0 dA = 0.