Find Curl of vector F (x2y + xy2z + 3x2z2) at point (1,-1,1)

Interpretation: The printed components are taken as \(\vec F=x^2y\,\hat i+xy^2z\,\hat j+3x^2z^2\,\hat k\).

\[ \nabla\times\vec F=\begin{vmatrix}\hat i&\hat j&\hat k\\ \partial_x&\partial_y&\partial_z\\ x^2y&xy^2z&3x^2z^2\end{vmatrix} \]

• \(\hat i\): \(\partial_y(3x^2z^2)-\partial_z(xy^2z)=0-xy^2=-xy^2\)
• \(\hat j\): \(\partial_z(x^2y)-\partial_x(3x^2z^2)=0-6xz^2=-6xz^2\)
• \(\hat k\): \(\partial_x(xy^2z)-\partial_y(x^2y)=y^2z-x^2\)

\[ \nabla\times\vec F=-xy^2\,\hat i-6xz^2\,\hat j+(y^2z-x^2)\,\hat k \]

At (1, −1, 1):

• \(-1\cdot1=-1\)
• \(-6\cdot1\cdot1=-6\)
• \(1\cdot1-1=0\)

\(\nabla\times\vec F=-\hat i-6\,\hat j\) (magnitude \(\sqrt{37}\approx6.08\))