Verla
MathCalculus

Divergence Theorem: Outward Flux Through a Capped Paraboloid

Use the divergence theorem to compute outward flux through a paraboloid and its base, then check the boundary orientation.

Question

For the vector field F(x,y,z)=⟨x2,xy,z+1⟩\mathbf F(x,y,z)=\langle x^2,xy,z+1\rangle, calculate the outward flux through the entire boundary of the solid between z=4−x2−y2z=4-x^2-y^2 and z=0z=0. Apply the divergence theorem.

This question has been answered.

Explore the full answer and supporting references to better understand this question.

Verla BasicBest value

$19.99/month

  • Everything in Study Pass
  • Assignments with writing output
  • AI detection and humanizer
  • Basic quotas across every tool
Study Pass

$0.99/month

  • Entire question library
  • Knowledge guides and sources
  • Detailed answers and explanations
  • No AI tools or quotas

Evidence boundary

The problem data and outward orientation come from OpenStax Calculus Volume 3, Section 6.8, Exercise 394. The wording is normalized; the calculations here are an independent solution, not an official answer key. Flux includes both the curved surface and its bottom disk.