The curl of a vector field is a vector field that describes the circulation of a fluid around a point. It is calculated using the following formula:
$$\text{curl}(F) = \det \begin{pmatrix} \hat{\imath} & \hat{\jmath} & \hat{k} \\ \dfrac{\partial}{\partial x} & \dfrac{\partial}{\partial y} & \dfrac{\partial}{\partial z} \\ F_x & F_y & F_z \end{pmatrix}$$
In this case, the vector field is $F(x, y, z) = 2x^2y + 5z^2j – 4uyzk$. Substituting this into the formula for the curl, we get:
$$\text{curl}(F) = \det \begin{pmatrix} \hat{\imath} & \hat{\jmath} & \hat{k} \\ \dfrac{\partial}{\partial x} & \dfrac{\partial}{\partial y} & \dfrac{\partial}{\partial z} \\ 2x^2 & 5z^2 & -4uy \end{pmatrix} = -14zi – 2x^2k$$
Therefore, the curl of the vector field is $-14zi – 2x^2k$.
Here is a brief explanation of each option:
- Option A: $-14zi – 2x^2k$. This is the correct answer. It is calculated using the formula for the curl of a vector field, and it matches the result of the calculation.
- Option B: $6zi + 4x^2j – 2x^2k$. This option is incorrect because it does not include the term $-14zi$.
- Option C: $-14zi + 6yj + 2x^2k$. This option is incorrect because it does not include the term $-2x^2k$.
- Option D: $6zi – 8xyj + 2x^2yk$. This option is incorrect because it does not include the term $-14zi$.