Math Problem Statement

answer

Use the divergence theorem to calculate the flux of F = (x − 4y)i + (y − 6z)j + (z − 4x)k out of the unit sphere.

Solution

Solution:

We use the Divergence Theorem to compute the flux of the vector field F out of the unit sphere.

The Divergence Theorem states: ∬∂VF⋅n dS=∭V(∇⋅F) dV,\iint_{\partial V} \mathbf{F} \cdot \mathbf{n} \, dS = \iiint_{V} (\nabla \cdot \mathbf{F}) \, dV, where:

  • F=(x−4y)i+(y−6z)j+(z−4x)k\mathbf{F} = (x - 4y) \mathbf{i} + (y - 6z) \mathbf{j} + (z - 4x) \mathbf{k},
  • VV is the volume enclosed by the surface (here, the unit sphere).

Step 1: Compute the Divergence of F\mathbf{F}

The divergence is given by: ∇⋅F=∂∂x(x−4y)+∂∂y(y−6z)+∂∂z(z−4x).\nabla \cdot \mathbf{F} = \frac{\partial}{\partial x}(x - 4y) + \frac{\partial}{\partial y}(y - 6z) + \frac{\partial}{\partial z}(z - 4x). Compute term by term:

\frac{\partial}{\partial y}(y - 6z) = 1, \quad \frac{\partial}{\partial z}(z - 4x) = 1.$$ Thus: $$\nabla \cdot \mathbf{F} = 1 + 1 + 1 = 3.$$ #### Step 2: Apply the Divergence Theorem We now compute the flux: $$\iint_{\partial V} \mathbf{F} \cdot \mathbf{n} \, dS = \iiint_{V} (\nabla \cdot \mathbf{F}) \, dV.$$ Since $$\nabla \cdot \mathbf{F} = 3$$ is constant, the integral becomes: $$\iiint_{V} 3 \, dV = 3 \iiint_{V} 1 \, dV = 3 \cdot (\text{Volume of the unit sphere}).$$ The volume of the unit sphere is: $$\text{Volume of the unit sphere} = \frac{4}{3} \pi (1)^3 = \frac{4}{3} \pi.$$ Therefore: $$\iiint_{V} 3 \, dV = 3 \cdot \frac{4}{3} \pi = 4\pi.$$ #### Final Answer: The flux of $$\mathbf{F}$$ out of the unit sphere is: $$\boxed{4\pi}.$$ Would you like a breakdown of a specific step or related concepts? ### Related Questions: 1. What is the Divergence Theorem, and how is it applied in general? 2. How would the answer change if the sphere radius were $$r > 1$$? 3. How is the flux computed directly without the Divergence Theorem? 4. Can the Divergence Theorem be applied to non-spherical surfaces? 5. What physical interpretations can be drawn from flux in vector fields? ### Tip: Always verify the divergence calculation carefully, as errors here propagate through the entire solution.

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Math Problem Analysis

Mathematical Concepts

Vector Calculus
Divergence
Flux Calculation
Volume Integration

Formulas

∇⋅F = ∂/∂x(F_x) + ∂/∂y(F_y) + ∂/∂z(F_z)
Flux = ∫∫_∂V F⋅n dS = ∫∫∫_V (∇⋅F) dV
Volume of a sphere = (4/3)πr^3

Theorems

Divergence Theorem

Suitable Grade Level

Undergraduate Level

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