L'affirmation suivante est-elle vraie ou fausse ? Justifier. La fonction \( f \) suivante est continue sur \( \mathbb{R} \). \( \left\{\begin{array}{ll}f(x)=-x^{2}-4 x-2 & \text { si } x \leqslant-1 \\ f(x)=\frac{x+1}{x+2} & \text { si } x>-1\end{array}\right. \)

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

\( \sum _ { y = 1 } ^ { n } \sum _ { x = 1 } ^ { n } 4 x + 7 y + x ^ { 2 } \)

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 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 \( (0,0,0) \). \( W=\square \)

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. \( f \)