Question

Consider the given vector field. F(x, y, z) = 8x^2 + y^2 + z^2 (x i + y j + z k)(a) Find the curl of the vector field. curl F = _______ (b) Find the divergence of the vector field. div F = _________

Answer 1

a) Curl F=0

b) div F=16x+2y+2z

Explanation:

Assuming the given vector field is
`\vec F=8x^2i +y^2j+z^2k`, then the curl of F is

`Curl F=\\abla * F`

`Curl F=\left|\begin{array}{ccc}i&j&k\\(\partial)/(\partial x) &(\partial)/(\partial y) &(\partial)/(\partial z) \\8x^2&y^2&z^2\end{array}\right|`

`\implies Curl F=((\partial (z^2))/(\partial y) -(\partial (y^2))/(\partial z) )i-((\partial (z^2))/(\partial x) -(\partial (8x^2))/(\partial z) )j+((\partial (y^2))/(\partial y) -(\partial (8x^2))/(\partial y) )k`

`\implies Curl F=0i+0j+0k`=0

b) The divergence of F is:

`div F=\\abla \cdot \vec F`

`div F=((\partial)/(\partial x)i+(\partial)/(\partial y)j+(\partial)/(\partial z)k)\cdot(8x^2i+y^2j+z^2k)`

`\implies div F=(\partial)/(\partial x) (8x^2)+(\partial)/(\partial y) (y^2)+(\partial)/(\partial z) (z^2)=16x+2y+2z`