# Worksheet: The Divergence Theorem

In this worksheet, we will practice using the divergence theorem to find the flux of a vector field over a surface by transforming the surface integral to a triple integral.

**Q1: **

Which of the following is true about the divergence theorem?

- AThe divergence theorem calculates the flux across the boundary surface of a simple solid region using the double integration of the divergence of the vector field.
- BThe divergence theorem calculates the flux across the boundary surface of a simple solid region using the triple integration of the divergence of the vector field.
- CThe divergence theorem calculates the flux across the boundary surface of a simple solid region using the triple integration of the vector field.
- DThe divergence theorem calculates the flux across the boundary surface of a simple solid region using the double integration of the vector field.
- EThe divergence theorem calculates the flux across the boundary surface of a simple solid region using the line integration of the divergence of the vector field.

**Q2: **

Use the divergence theorem to find the outward flux of across the boundary of the region : the solid cylinder between the plane and the paraboloid .

- A
- B0
- C
- D
- E

**Q3: **

Find the flux of the vector field over the sphere .

- A
- B
- C
- D0
- E

**Q4: **

Use the divergence theorem to find the outward flux of across the boundary of the region , the region cut from the first octant by the sphere .

- A
- B
- C
- D0
- E

**Q5: **

Assuming that and satisfy the conditions of the divergence theorem and that the vector field has a continuous second-order partial derivative, which of the following statements is true?

- A
- B
- C
- D
- E

**Q6: **

Use the divergence theorem to find the outward flux of across the boundary of the region : the cube bounded by the planes , , and .