Consider the two-dimensional vector field \vec{F}(x,y)=x\vec{i}+y\vec{j}, where \vec{i} and \vec{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 _c \vec{F}(x,y)\cdot (dx\vec{i}+dy\vec{j})
\oint \vec{F} (x,y)\cdot [dx\vec{i}+dy\vec{j}] Given \vec{F} (x,y)=x\vec{i}+y\vec{j} \therefore \int_{c}xdx+ydy=0 Because here vector is conservative.
If the integral function is the total derivative
over the closed contoure then it will be zero
Question 2
Consider a system of linear equations Ax=b, where A=\begin{bmatrix}
1 & -\sqrt{2} & 3\\
-1& \sqrt{2}& -3
\end{bmatrix},b=\begin{bmatrix}
1\\
3
\end{bmatrix} This system of equations admits ______.