Calculate the circulation of a specific vector field along a counterclockwise unit circle using a line integral.

Vault Of Euclid

Circulation of a Vector Field Along a Unit Circle

Multivariable calculus core

Calculate the circulation of \( \mathbf{F}(x, y) = \left( -\frac{y}{x^2 + y^2}, \frac{x}{x^2 + y^2} \right) \) along a unit circle oriented counterclockwise.

The solution and a full step-by-step explanation are here. Sign in to read them.

What you need first

Parametric equations for a circle
Evaluating line integrals
Dot product of vectors
Basic trigonometric identities

What this problem tests

Tests the ability to parametrize a curve, substitute it into a vector field, compute a dot product, and evaluate a definite integral.

This is a classic multivariable calculus problem demonstrating a vector field with zero curl but non-zero circulation around a singularity.

Circulation of a Vector Field Along a Unit Circle

What you need first

What this problem tests

More on Multivariable calculus