Question

State Green s theorem and verify for *(x^{2}+y^{2})dx+(3y+2x)dy ,whereC is triangle with vertices at (0,0),(1,0),(1,1)

Answer 1

According to Green's theorem, the line integral equals the double integral of this quantity across the contained region. We might begin by assuming that the region is both vertically and horizontally simple. As a result, the two line integrals across this line cancel each other out.

$∮_c(x²+y²)dx+(3y+2x)dy\\ \int_{x=0}^1\int_{y=0}^1\{ \frac{\partial}{\partial x}(3y+2x)- \frac{\partial}{\partial y}(x^2+y^2)\}dydx\\ =\int _0^1\int _0^12-\frac{\partial \:}{\partial \:y}\left(x^2+y^2\right)dydx\\ =\int _0^1\int _0^12-2ydydx\\ =\int _0^11dx\\ =1$