a) A curl is the vector derivative of a vector field. It can be denoted as

`vec grad xx vec F` , where `vec F` is the vector field. `

The curl is calculated as three-dimensional determinant:

i j k

d/dx ...

## Unlock

This Answer NowStart your **48-hour free trial** to unlock this answer and thousands more. Enjoy eNotes ad-free and cancel anytime.

Already a member? Log in here.

a) A curl is the vector derivative of a vector field. It can be denoted as

`vec grad xx vec F` , where `vec F` is the vector field. `

The curl is calculated as three-dimensional determinant:

i j k

d/dx d/dy d/dz

`F_x` `F_y` `F_z`

This determinant equals the vector quantity

`((dF_z)/dy - (dF_y)/(dz)) veci - ((dF_z)/(dx) - (dF_x)/(dz))vecj + ((dF_y)/(dx) - (dF_x)/(dy))veck` .

To find curl of the given field, let's first find all the required partial derivatives:

`(dF_y)/(dx) = y^2z`

`(dF_x)/(dy) = x^2z`

`(dF_z)/(dy) = xz^2`

`(dF_y)/(dz) = xy^2`

`(dF_z)/(dx) = yz^2`

`(dF_x)/(dz) = x^2y`

Substituting these into the expression above, we get

`vec grad xx vecF = x(z^2 - y^2) veci + y(x^2 - z^2) vecj + z(y^2 - x^2) veck` .

**This is the curl of the given vector field.**

b) Follow the same procedure to find the curl of this given vector field, as well.

These are the partial derivatives:

`(dF_y)/(dx) = y^2`

`(dF_x)/(dy) = -xsiny`

`(dF_z)/(dy) = 0`

`(dF_y)/(dz) = 0`

`(dF_z)/(dx) = 0`

`(dF_x)/(dz) = 0`

So the only component of the curl of this field that is non-zero is the z-component:

`vecgrad xx vecF = (y^2+xsiny) veck` .