Question

Which of the following statements is/are true?
(I) The vector field \(\vec{F}(x,y,z)=(yz,\,zx,\,xy)\) is solenoidal.
(II) The vector field \(\vec{F}(x,y,z)=(x^2-y^2,\,2xy,\,z^3)\) is conservative.

• A. Only (I)
• B. Only (II)
• C. Both (I) and (II)
• D. Neither (I) nor (II)

Hint:

Use divergence for the solenoidal test and curl for the conservative test.

Answer 1

The Correct Option is A

Step 1: A vector field \(\vec F\) is solenoidal if \(\nabla\cdot\vec F=0\). For \(\vec F=(yz,zx,xy)\), \[\nabla\cdot\vec F=\frac{\partial}{\partial x}(yz)+\frac{\partial}{\partial y}(zx)+\frac{\partial}{\partial z}(xy)=0+0+0=0.\] So \(\vec F\) is solenoidal, and statement (I) is true.

Step 2: A vector field is conservative only if its curl vanishes everywhere. For \(\vec F=(x^2-y^2,2xy,z^3)\), \[\nabla\times\vec F=\left(\frac{\partial}{\partial y}(z^3)-\frac{\partial}{\partial z}(2xy),\ \frac{\partial}{\partial z}(x^2-y^2)-\frac{\partial}{\partial x}(z^3),\ \frac{\partial}{\partial x}(2xy)-\frac{\partial}{\partial y}(x^2-y^2)\right)\] \[=(0-0,\ 0-0,\ 2y-(-2y))=(0,0,4y).\] This is not the zero vector for \(y\neq0\), so \(\vec F\) is not conservative. Statement (II) is false.

Step 3: Only statement (I) is true. \[\boxed{\text{Only (I)}}\]