This video from IMS Delhi provides a step-by-step mathematical solution to a vector analysis problem featured in the UPSC IAS Mathematics Optional PYQ 2021. The lecture focuses on evaluating the line integral of a specific vector field around a closed curve in the XY plane, applying Green’s Theorem.

Vector Field: F = (−y / (x² + y²))i + (x / (x² + y²))j

Case 1: Origin outside the curve (6:02)
The instructor demonstrates that when the origin (0,0) lies outside the closed curve, the partial derivatives satisfy the conditions for the vector field to be irrotational, resulting in an integral value of zero.

Case 2: Origin inside the curve (8:25)
When the origin is enclosed by the curve, the vector field is not defined at the origin. To solve this, the instructor uses the concept of a multiply connected region and creates a small circular cutout (C1) around the origin to transform it into a simply connected region (10:01-13:05).

Mathematical Derivation: The instructor utilizes polar coordinates (x = ε cosθ, y = ε sinθ) to evaluate the integral over the small circle (C1) (14:34-18:42).

Final Result: By relating the integral over the original curve to the integral over the circular cutout, the final solution is derived as -2π (20:32-20:44).