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 Fijk=𝑦+π‘₯βˆ’π‘§ across the boundary of the region 𝐷: the solid cylinder π‘₯+𝑦≀4 between the plane 𝑧=0 and the paraboloid 𝑧=π‘₯+π‘¦οŠ¨οŠ¨.

  • A4πœ‹
  • B0
  • C8πœ‹
  • Dβˆ’4πœ‹
  • Eβˆ’8πœ‹

Q3:

Find the flux of the vector field Fijk(π‘₯,𝑦,𝑧)=𝑧+𝑦+π‘₯ over the sphere π‘₯+𝑦+𝑧=1.

  • A4πœ‹
  • B8πœ‹3
  • Cβˆ’4πœ‹3
  • D0
  • E4πœ‹3

Q4:

Use the divergence theorem to find the outward flux of Fijk=π‘₯βˆ’2π‘₯𝑦+3π‘₯π‘§οŠ¨ across the boundary of the region 𝐷, the region cut from the first octant by the sphere π‘₯+𝑦+𝑧=4.

  • A6πœ‹
  • Bβˆ’6πœ‹
  • Cβˆ’3πœ‹
  • D0
  • E3πœ‹

Q5:

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

  • A⋅𝑆=0curldF
  • B⋅𝑆=ο„½β‹…π‘†οŒ²οŒ€curldcurldFF
  • C⋅𝑆=ο…‚β‹…π‘†οŒ²οŒ€curldcurldFF
  • D⋅𝑆=ο…‚β‹…π‘†οŒ²οŒ€curlddFF
  • E⋅𝑆=curldFF

Q6:

Use the divergence theorem to find the outward flux of Fijk=(π‘¦βˆ’π‘₯)+(π‘§βˆ’π‘¦)+(π‘¦βˆ’π‘₯) across the boundary of the region 𝐷: the cube bounded by the planes π‘₯=Β±1, 𝑦=Β±1, and 𝑧=Β±1.

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