Math Problem Statement
Evaluate the surface integral ∫SF⋅ dS∫SF⋅ dS where F=⟨−x,5z,−5y⟩F=⟨−x,5z,−5y⟩ and SS is the part of the sphere x2+y2+z2=9x2+y2+z2=9 in the first octant, with orientation toward the origin.
∫∫SF⋅ dS=∫∫SF⋅ dS=
Solution
To evaluate the surface integral
where and is the part of the sphere in the first octant with orientation toward the origin, we use the Divergence Theorem.
Step 1: Verify Conditions for Divergence Theorem
The Divergence Theorem states: where is the volume enclosed by the surface .
The first octant portion of the sphere is not a closed surface. To apply the Divergence Theorem, we close the surface by adding the planar surface in the first octant (, , ), and we compute the total flux through the closed surface.
The orientation is toward the origin, so we reverse the sign of the flux.
Step 2: Compute the Divergence of
The divergence of is:
Step 3: Set up the Volume Integral
The volume is the first octant portion of the sphere . The volume element in spherical coordinates is: where:
- is the radial distance (),
- is the polar angle (),
- is the azimuthal angle ().
Thus, the volume integral becomes:
Step 4: Evaluate the Integral
Separate the integral into its components:
Combine these results:
Step 5: Account for Orientation
Since the orientation is toward the origin, we reverse the sign:
Final Answer:
Let me know if you have questions or would like detailed clarifications!
Follow-Up Questions:
- How does the Divergence Theorem simplify surface integrals?
- What is the significance of the divergence in vector fields?
- How is the orientation of the surface accounted for in flux integrals?
- Why do we need to close the surface when applying the Divergence Theorem?
- How can we evaluate surface integrals directly without using the Divergence Theorem?
Tip:
In spherical coordinates, always remember the volume element , which accounts for the geometry of the coordinate system.
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Math Problem Analysis
Mathematical Concepts
Surface integrals
Vector fields
Spherical coordinates
Divergence theorem
Formulas
∫∫S F ⋅ dS = ∫∫∫V (∇ ⋅ F) dV
∇ ⋅ F = ∂F₁/∂x + ∂F₂/∂y + ∂F₃/∂z
Volume element in spherical coordinates: dV = ρ² sinφ dρ dφ dθ
Theorems
Divergence Theorem
Suitable Grade Level
Undergraduate (Calculus III or Multivariable Calculus)
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