Math Problem Statement
Let ω be the upper unit hemisphere,* ω = {x ↓ R3 | ↘x↘ = 1, x3 ↔ 0} with the outward orientation (as in an example in the lectures). Let F : R3 → R3, F(x, y, z) = (0, x, 0) . (a) Compute I = ↓ ϖω F(x) · dx directly.
Solution
To compute the flux integral , where is the upper hemisphere of the unit sphere in with outward orientation and , we can proceed as follows:
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Understand the Geometry of : The region is defined as the upper hemisphere of the unit sphere, which is the set Since the outward orientation is specified, the surface normal will point away from the surface of the hemisphere.
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Express the Surface Integral: We want to compute the surface integral: where , with being the outward-pointing unit normal vector on , and representing the surface area element.
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Parameterize : Since is the upper unit hemisphere, we can use spherical coordinates: where (as it is the upper hemisphere), and .
The surface area element on the unit sphere is , and the outward unit normal vector is simply because lies on the unit sphere.
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Compute : Given , we have So,
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Set Up the Integral: Substitute into the integral :
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Evaluate the Integral: Separate the integrals over and : Notice that . Then, because the integral of over a full period is zero.
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Conclusion: Since the -integral evaluates to zero, the entire surface integral is zero:
Thus, the answer is:
Would you like a breakdown of each step or have any questions?
Related Questions:
- How would the result change if instead?
- What is the physical interpretation of the flux integral in this context?
- How does the divergence theorem apply to surface integrals like this?
- How would you set up this problem using a different coordinate system?
- What is the significance of choosing an outward versus inward orientation?
Tip: For integrals over symmetric regions, look for terms that may integrate to zero due to symmetry—it often simplifies the problem greatly.
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Math Problem Analysis
Mathematical Concepts
Vector Calculus
Surface Integrals
Flux
Spherical Coordinates
Formulas
Surface Integral Formula: I = ∬ₓω F · dS
Spherical Coordinate Transformation: x = sin(θ)cos(φ), y = sin(θ)sin(φ), z = cos(θ)
Surface Area Element: dS = sin(θ) dθ dφ
Theorems
Divergence Theorem
Surface Integral Theorem
Suitable Grade Level
Undergraduate (Calculus III/Vector Calculus)
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