Consider the two-dimensional vector field $$\overrightarrow F (x,y) - x\overrightarrow i + y\overrightarrow j $$, where $$\overrightarrow i $$ and $$\widehat j$$ denote the unit vectors along the x-axis and the y-axis, respectively. A contour C in the x-y plane, as shown in the figure, is composed of two horizontal lines connected at the two ends by two semicircular arcs of unit radius. The contour is traversed in the counter-clockwise sense. The value of the closed path integral

$$\oint\limits_C {\overrightarrow F (x,y)\,.\,(dx\overrightarrow i + dy\overrightarrow j )} $$

is ___________.

The families of curves represented by the solution of the equation

$${{dy} \over {dx}} = - {\left( {{x \over y}} \right)^n}$$

for n = –1 and n = 1 respectively, are

Let $$f\left( {x,y} \right) = {{a{x^2} + b{y^2}} \over {xy}}$$, where $$a$$ and $$b$$ are constants. If $${{\partial f} \over {\partial x}} = {{\partial f} \over {\partial y}}$$ at x = 1 and y = 2, then the relation between $$a$$ and $$b$$ is

Taylor series expansion of $$f\left( x \right) = \int\limits_0^x {{e^{ - \left( {{{{t^2}} \over 2}} \right)}}} dt$$ around 𝑥 = 0 has the form

f(x) = $${a_0} + {a_1}x + {a_2}{x^2} + ...$$

The coefficient $${a_2}$$ (correct to two decimal places) is equal to _______.