\( (1-x)y^{\prime \prime }+xy^{\prime }-y=(1-x)^{2},y_{1}=e^{x} \)

Label each expression as a scalar quantity, a vector quantity or undefined, if \( f \) is a scalar function and \( F \) is a vector field. \( \nabla \times(\nabla f) \) scalar vector

Compute the divergence of \( F(x, y)=1.2 x i+1.75 y j \) \( \operatorname{div} F=\square \)

Determine whether or not the vector field \( (x-46 x y) i+\left(y^{2}-23 x^{2}\right) j \) is conservative. If it is, find a potential function. \( f(x, y)=\frac{x^{2}}{2}-23 x^{2} y+y^{3}+c \). The field is not conservative. \( f(x, y)=x^{2}-23 x^{2} y+\frac{y^{3}}{3}+c \). The field is not conservative. \( f(x, y)=\frac{x^{2}}{2}-x^{2} y+\frac{y^{3}}{3}+c \). The field is conservative. \( f(x, y)=\frac{x^{2}}{2}-23 x^{2} y+\frac{y^{3}}{3}+c \). The field is conservative. \( f \)

Compute the work done by the force field \( F=\langle y, 0, z\rangle \) along the triangle from \( (0,0,0) \) to \( (2,3,4) \) then to \( (2,3,0) \) and then back to \( (0,0,0) \). \( W=\square \)

Evaluate the line integral \( \int_{C} 5 x d s \), where \( C \) is the line segment from \( (1,2) \) to \( (8,7) \). \( \int_{C} 5 x d s=\square \)