Find the curl of the fields shown on the image below.

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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    ...

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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` .

 

 

 

 

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