Final answer:
To calculate the line integral of a vector field over a closed curve, parameterize the curve as a function of a single variable or parameter, calculate the dot product between the vector field and the derivative of the parameterization, and integrate over the range of the parameter corresponding to the curve.
Step-by-step explanation:
To calculate the line integral of the vector field F(x, y, z) = ⟨z^2, x^2, y^2⟩ over the closed curve C, we first parameterize the curve C as a function of a single variable or parameter. Let's say the parameterization is r(t) = ⟨x(t), y(t), z(t)⟩. We then calculate the dot product of F and the derivative of r with respect to the parameter, and integrate this dot product over the range of the parameter that corresponds to the closed curve C.
So, the line integral of the vector field F over the closed curve C can be expressed as:
∫CF(x, y, z) · dr = ∫abF(x(t), y(t), z(t)) · r'(t) dt
where t varies from a to b, depending on the parameterization of the curve C.