Question

If f⃗ =grad(2x2+3y3), find ∫c where c is the quarter of the circle x2+y2=1 in the first quadrant, oriented counterclockwise.

Answer 1

Presumably, the path starts on the horizontal axis at (1, 0) and ends on the vertical axis at (0, 1). Since we know the vector field is the gradient of a scalar function, we can apply the fundamental theorem of calculus (aka gradient theorem):

$\displaystyle\int_{\mathcal C}\mathbf f\cdot\mathrm d\mathbf r=f(0,1)-f(1,0)=3-2=1$