Find whether the vector is solenoidal, E = yz i + xz j + xy k

To determine if the vector field ( mathbf{E} = yz , mathbf{i} + xz , mathbf{j} + xy , mathbf{k} ) is solenoidal, we need to check if its divergence is zero.

The divergence of a vector field ( mathbf{F} = P , mathbf{i} + Q , mathbf{j} + R , mathbf{k} ) is given by:

[

nabla cdot mathbf{F} = frac{partial P}{partial x} + frac{partial Q}{partial y} + frac{partial R}{partial z}

]

For the vector field ( mathbf{E} ), we identify:

– ( P = yz )

– ( Q = xz )

– ( R = xy )

We calculate each partial derivative:

1. ( frac{partial P}{partial x} = frac{partial}{partial x}(yz) = 0 )
2. ( frac{partial Q}{partial y} = frac{partial}{partial y}(xz) = 0 )
3. ( frac{partial R}{partial z} = frac{partial}{partial z}(xy) = 0 )

Now, substituting these results into the divergence formula:

[

nabla cdot mathbf{