Question

for a certain field, , the two-dimensional curl is . what is the counterclockwise circulation of around the circle of radius 5 centered on the origin? group of answer choices

Answer 1

The two-dimensional curl of F is x(18y - 5). The two-dimensional divergence of F is 5y + 18xy + 15x⁴. F is not irrotational on R, and F is not source free on R.

(a) Calculate the two-dimensional curl of F.

The two-dimensional curl of F is given by:

curl(F) = ∂Qy/∂x - ∂Qx/∂y

where Q is the z-component of the vector field F.

In this case, Q = 9xy², so:

curl(F) = ∂(9xy²)/∂x - ∂(5xy)/∂y

= 18xy - 5x

= x(18y - 5)

(b) Calculate the two-dimensional divergence of F.

The two-dimensional divergence of F is given by:

div(F) = ∂Fx/∂x + ∂Fy/∂y

where Fx and Fy are the x- and y-components of the vector field F.

In this case, Fx = 5xy and Fy = 9xy² + 3x⁵, so:

div(F) = ∂(5xy)/∂x + ∂(9xy² + 3x⁵)/∂y

= 5y + 18xy + 15x⁴

(c) Is F irrotational on R?

A vector field is irrotational if its curl is zero. In this case, the curl of F is x(18y - 5), which is not zero for all x and y. Therefore, F is not irrotational on R.

(d) Is F source free on

A vector field is source free if its divergence is zero. In this case, the divergence of F is 5y + 18xy + 15x⁴, which is not zero for all x and y. Therefore, F is not source free on R.

Question