Paper 2, Section I, B

Vector Calculus | Part IA, 2020

(a) Evaluate the line integral

(0,1)(1,2)(x2y)dx+(y2+x)dy\int_{(0,1)}^{(1,2)}\left(x^{2}-y\right) d x+\left(y^{2}+x\right) d y


(i) a straight line from (0,1)(0,1) to (1,2)(1,2),

(ii) the parabola x=t,y=1+t2x=t, y=1+t^{2}.

(b) State Green's theorem. The curve C1C_{1} is the circle of radius aa centred on the origin and traversed anticlockwise and C2C_{2} is another circle of radius b<ab<a traversed clockwise and completely contained within C1C_{1} but may or may not be centred on the origin. Find

C1C2y(xyλ)dx+x2ydy\int_{C_{1} \cup C_{2}} y(x y-\lambda) d x+x^{2} y d y

as a function of λ\lambda.

Typos? Please submit corrections to this page on GitHub.