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)...
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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