Final answer:
To find the values of the curl of the vector at the point (1, 1, 1), we need to calculate the partial derivatives of wx, wy, and wz with respect to x, y, and z, respectively, and substitute the values (1, 1, 1) into the expressions. The curl of the vector is -2j - 3k.
Step-by-step explanation:
The curl of a vector is given by the expression: curl(V) = (1/J) * [d(wz)/dy - d(wy)/dz]i + [d(wx)/dz - d(wz)/dx]j + [d(wy)/dx - d(wx)/dy]k where V is the given vector and J is the Jacobian determinant. In this case, the vector V = xz^2i - yj + 3xz^3. To find the values of the curl of the vector at the point (1, 1, 1), we need to calculate the partial derivatives of wx, wy, and wz with respect to x, y, and z, respectively, and substitute the values (1, 1, 1) into the expressions.
We have:
wx = xz^2
wy = -y
wz = 3xz^3
Calculating the partial derivatives:
d(wz)/dy = 0
d(wy)/dz = 0
d(wx)/dz = -2xz
d(wz)/dx = 3z^3
d(wy)/dx = 0
d(wx)/dy = 0
Substituting the values (1, 1, 1):
curl(V) = (1/J) * [0 - 0]i + [-2(1)(1) - 0]j + [0 - 3(1)(1)]k
curl(V) = -2j - 3k