engineering recuitment GATE ECE 2023-24 Test Series Engineering Mathematics Calculus Vector Calculus
If \(\vec F\left( {x,\;y,\;z} \right) = \left( {2x + 3yz} \right)\hat i + \left( {3xz + 2y} \right)\hat j + \left( {3xy + 2z} \right)\hat k\) for (x, y, z) ∈ R3, then which among the following is (are) TRUE?
1
\(\nabla \times \vec F = 0\)
2
\(\mathop \oint \nolimits_C \vec F \cdot d\vec r = 0\) along any simple closed curve C.
3
There exists a scalar function such that \(\nabla \cdot \vec F = {\phi _{xx}} + {\phi _{yy}} + {\phi _{zz}}\)
4
\(\nabla \cdot \vec F = 0\)