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

SFdS\iint_S \mathbf{F} \cdot \mathbf{dS}

where F=x,5z,5y\mathbf{F} = \langle -x, 5z, -5y \rangle and SS is the part of the sphere x2+y2+z2=9x^2 + y^2 + z^2 = 9 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: SFdS=V(F)dV\iint_S \mathbf{F} \cdot \mathbf{dS} = \iiint_V (\nabla \cdot \mathbf{F}) \, dV where VV is the volume enclosed by the surface SS.

The first octant portion of the sphere x2+y2+z2=9x^2 + y^2 + z^2 = 9 is not a closed surface. To apply the Divergence Theorem, we close the surface by adding the planar surface in the first octant (z=0z = 0, x0x \geq 0, y0y \geq 0), 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 F\mathbf{F}

The divergence of F\mathbf{F} is: F=(x)x+(5z)z+(5y)y\nabla \cdot \mathbf{F} = \frac{\partial (-x)}{\partial x} + \frac{\partial (5z)}{\partial z} + \frac{\partial (-5y)}{\partial y} F=1+55=1.\nabla \cdot \mathbf{F} = -1 + 5 - 5 = -1.

Step 3: Set up the Volume Integral

The volume VV is the first octant portion of the sphere x2+y2+z2=9x^2 + y^2 + z^2 = 9. The volume element in spherical coordinates is: dV=ρ2sinϕdρdϕdθ,dV = \rho^2 \sin\phi \, d\rho \, d\phi \, d\theta, where:

  • ρ\rho is the radial distance (0ρ30 \leq \rho \leq 3),
  • ϕ\phi is the polar angle (0ϕπ/20 \leq \phi \leq \pi/2),
  • θ\theta is the azimuthal angle (0θπ/20 \leq \theta \leq \pi/2).

Thus, the volume integral becomes: V(F)dV=V(1)dV=0π/20π/203ρ2sinϕdρdϕdθ.\iiint_V (\nabla \cdot \mathbf{F}) \, dV = \iiint_V (-1) \, dV = - \int_0^{\pi/2} \int_0^{\pi/2} \int_0^3 \rho^2 \sin\phi \, d\rho \, d\phi \, d\theta.

Step 4: Evaluate the Integral

Separate the integral into its components: 03ρ2dρ=[ρ33]03=273=9,\int_0^3 \rho^2 \, d\rho = \left[\frac{\rho^3}{3}\right]_0^3 = \frac{27}{3} = 9, 0π/2sinϕdϕ=[cosϕ]0π/2=cos(π/2)+cos(0)=0+1=1,\int_0^{\pi/2} \sin\phi \, d\phi = \left[-\cos\phi\right]_0^{\pi/2} = -\cos(\pi/2) + \cos(0) = 0 + 1 = 1, 0π/2dθ=θ0π/2=π2.\int_0^{\pi/2} d\theta = \theta \big|_0^{\pi/2} = \frac{\pi}{2}.

Combine these results: V(1)dV=(9)(1)(π2)=9π2.\iiint_V (-1) \, dV = - (9)(1)\left(\frac{\pi}{2}\right) = -\frac{9\pi}{2}.

Step 5: Account for Orientation

Since the orientation is toward the origin, we reverse the sign: SFdS=9π2.\iint_S \mathbf{F} \cdot \mathbf{dS} = \frac{9\pi}{2}.

Final Answer:

9π2\boxed{\frac{9\pi}{2}}

Let me know if you have questions or would like detailed clarifications!


Follow-Up Questions:

  1. How does the Divergence Theorem simplify surface integrals?
  2. What is the significance of the divergence in vector fields?
  3. How is the orientation of the surface accounted for in flux integrals?
  4. Why do we need to close the surface when applying the Divergence Theorem?
  5. How can we evaluate surface integrals directly without using the Divergence Theorem?

Tip:

In spherical coordinates, always remember the volume element ρ2sinϕdρdϕdθ\rho^2 \sin\phi \, d\rho \, d\phi \, d\theta, which accounts for the geometry of the coordinate system.

Ask a new question for Free

By Image

Drop file here or Click Here to upload

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)