Question

find the rank of the matrix
3 0 2 2
-6 42 24 54
21 -21 0 -15

Answer 1

i.e., the vector potential is:
\[
\vec{F}=(y+z)\hat{i}+(z+x)\hat{j}+(x+y)\hat{k}
\]
fot the line integral to be path independent, we show that F is conservative.
i.e.,\[\nabla\times\vec{F}=0\\
\left|\begin{matrix}\hat{i}&\hat{j}&\hat{k}\\
{\partial\over\partial x}&{\partial\over\partial y}&{\partial\over\partial z}\\
y+z&z+x&x+y
\end{matrix}\right|=[1-1]\hat{i}-[1-1]\hat{j}+[1-1]\hat{k}=\vec{0}
\]
this, F is conservative and independent of path

Answer 2

thanks for solution.....