Divergence & Curl — Practice Problems and Worked Solutions

17 worked problems

Every step is generated and checked by a computer algebra system.

1. 1.

   Find the divergence of the vector field.

   $$F=\left[\begin{array}{c}x\\ y\\ z\end{array}\right],\nabla \cdot F=?$$

   Answer: $3$

   Step-by-step solution

   1. $$\nabla \cdot F=P_x+Q_y+R_z$$
   2. $$=3$$
2. 2.

   Find the curl of the vector field.

   $$F=\left[\begin{array}{c}xy\\ yz\\ xz\end{array}\right],\nabla \times F=?$$

   Answer: $\left[\begin{array}{c}-y\\ -z\\ -x\end{array}\right]$

   Step-by-step solution

   1. $$\nabla \times F=(R_y-Q_z,P_z-R_x,Q_x-P_y)$$
   2. $$=\left[\begin{array}{c}-y\\ -z\\ -x\end{array}\right]$$
3. 3.

   Find the divergence of the vector field.

   $$F=\left[\begin{array}{c}y\\ -x\\ z\end{array}\right],\nabla \cdot F=?$$

   Answer: $1$

   Step-by-step solution

   1. $$\nabla \cdot F=P_x+Q_y+R_z$$
   2. $$=1$$
4. 4.

   Find the curl of the vector field.

   $$F=\left[\begin{array}{c}{x}^2\\ y^2\\ z^2\end{array}\right],\nabla \times F=?$$

   Answer: $\left[\begin{array}{c}0\\ 0\\ 0\end{array}\right]$

   Step-by-step solution

   1. $$\nabla \times F=(R_y-Q_z,P_z-R_x,Q_x-P_y)$$
   2. $$=\left[\begin{array}{c}0\\ 0\\ 0\end{array}\right]$$
5. 5.

   Find the divergence of the vector field.

   $$F=\left[\begin{array}{c}yz\\ xz\\ xy\end{array}\right],\nabla \cdot F=?$$

   Answer: $0$

   Step-by-step solution

   1. $$\nabla \cdot F=P_x+Q_y+R_z$$
   2. $$=0$$
6. 6.

   Find the curl of the vector field.

   $$F=\left[\begin{array}{c}x+y\\ y+z\\ x+z\end{array}\right],\nabla \times F=?$$

   Answer: $\left[\begin{array}{c}-1\\ -1\\ -1\end{array}\right]$

   Step-by-step solution

   1. $$\nabla \times F=(R_y-Q_z,P_z-R_x,Q_x-P_y)$$
   2. $$=\left[\begin{array}{c}-1\\ -1\\ -1\end{array}\right]$$
7. 7.

   Find the divergence of the vector field.

   $$F=\left[\begin{array}{c}2x\\ 3y\\ z\end{array}\right],\nabla \cdot F=?$$

   Answer: $6$

   Step-by-step solution

   1. $$\nabla \cdot F=P_x+Q_y+R_z$$
   2. $$=6$$
8. 8.

   Find the curl of the vector field.

   $$F=\left[\begin{array}{c}z\\ x\\ y\end{array}\right],\nabla \times F=?$$

   Answer: $\left[\begin{array}{c}1\\ 1\\ 1\end{array}\right]$

   Step-by-step solution

   1. $$\nabla \times F=(R_y-Q_z,P_z-R_x,Q_x-P_y)$$
   2. $$=\left[\begin{array}{c}1\\ 1\\ 1\end{array}\right]$$
9. 9.

   Find the divergence of the vector field.

   $$F=\left[\begin{array}{c}xz\\ xy\\ yz\end{array}\right],\nabla \cdot F=?$$

   Answer: $x+y+z$

   Step-by-step solution

   1. $$\nabla \cdot F=P_x+Q_y+R_z$$
   2. $$=x+y+z$$
10. 10.

    Find the curl of the vector field.

    $$F=\left[\begin{array}{c}{x}^3\\ y\\ z^2\end{array}\right],\nabla \times F=?$$

    Answer: $\left[\begin{array}{c}0\\ 0\\ 0\end{array}\right]$

    Step-by-step solution

    1. $$\nabla \times F=(R_y-Q_z,P_z-R_x,Q_x-P_y)$$
    2. $$=\left[\begin{array}{c}0\\ 0\\ 0\end{array}\right]$$
11. 11.

    Find the divergence of the vector field.

    $$F=\left[\begin{array}{c}-y\\ x\\ 0\end{array}\right],\nabla \cdot F=?$$

    Answer: $0$

    Step-by-step solution

    1. $$\nabla \cdot F=P_x+Q_y+R_z$$
    2. $$=0$$
12. 12.

    Find the curl of the vector field.

    $$F=\left[\begin{array}{c}{y}^2\\ z^2\\ x^2\end{array}\right],\nabla \times F=?$$

    Answer: $\left[\begin{array}{c}-2z\\ -2x\\ -2y\end{array}\right]$

    Step-by-step solution

    1. $$\nabla \times F=(R_y-Q_z,P_z-R_x,Q_x-P_y)$$
    2. $$=\left[\begin{array}{c}-2z\\ -2x\\ -2y\end{array}\right]$$
13. 13.

    Find the divergence of the vector field.

    $$F=\left[\begin{array}{c}x+z\\ y\\ xy\end{array}\right],\nabla \cdot F=?$$

    Answer: $2$

    Step-by-step solution

    1. $$\nabla \cdot F=P_x+Q_y+R_z$$
    2. $$=2$$
14. 14.

    Find the curl of the vector field.

    $$F=\left[\begin{array}{c}0\\ xz\\ y\end{array}\right],\nabla \times F=?$$

    Answer: $\left[\begin{array}{c}1-x\\ 0\\ z\end{array}\right]$

    Step-by-step solution

    1. $$\nabla \times F=(R_y-Q_z,P_z-R_x,Q_x-P_y)$$
    2. $$=\left[\begin{array}{c}1-x\\ 0\\ z\end{array}\right]$$
15. 15.

    Find the divergence of the vector field.

    $$F=\left[\begin{array}{c}xyz\\ x\\ y\end{array}\right],\nabla \cdot F=?$$

    Answer: $yz$

    Step-by-step solution

    1. $$\nabla \cdot F=P_x+Q_y+R_z$$
    2. $$=yz$$
16. 16.

    Find the curl of the vector field.

    $$F=\left[\begin{array}{c}{x}^{2}y\\ z\\ yz\end{array}\right],\nabla \times F=?$$

    Answer: $\left[\begin{array}{c}z-1\\ 0\\ -x^2\end{array}\right]$

    Step-by-step solution

    1. $$\nabla \times F=(R_y-Q_z,P_z-R_x,Q_x-P_y)$$
    2. $$=\left[\begin{array}{c}z-1\\ 0\\ -x^2\end{array}\right]$$
17. 17.

    Find the divergence of the vector field.

    $$F=\left[\begin{array}{c}{z}^2\\ x^2\\ y^2\end{array}\right],\nabla \cdot F=?$$

    Answer: $0$

    Step-by-step solution

    1. $$\nabla \cdot F=P_x+Q_y+R_z$$
    2. $$=0$$

More from Multivariable Calculus

• Vectors in 3D
• Lines & Planes
• Partial Derivatives
• Gradients
• Directional Derivatives
• Tangent Planes
• Critical Points
• Double Integrals
• Polar Integrals
• Triple Integrals