# 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 .