Question

(,,)=^3−^3+^3, where is the sphere ^2 + ^2 + ^2=^
Use divergence theorem to find flux of F. is in outward orientation.

Answer 1

Answer:

The flux of F in the outward orientation is equal to 0.

General Formulas and Concepts:
Calculus

Differentiation

• Derivatives
• Derivative Notation

Integration

• Integrals

Multivariable Calculus

Partial Derivatives

Vector Calculus (Line Integrals)

Del (Operator):
$\displaystyle \nabla = \hat{\i} \frac{\partial}{\partial x} + \hat{\j} \frac{\partial}{\partial y} + \hat{\text{k}} \frac{\partial}{\partial z}$

Div

• $\displaystyle \text{div \bf{F}} = \nabla \cdot \textbf{F}$

Divergence Theorem:
$\displaystyle \iint_S {\big( \nabla \times \textbf{F} \big) \cdot \textbf{n}} \, d\sigma = \iiint_D {\nabla \cdot \textbf{F}} \, dV$

Explanation:

*Note:

Your question is incomplete, but I have defined the portions of the questions that were missing below.

Step 1: Define

Identify given.

$\displaystyle \textbf{F}(x, y, z) = z^3 \hat{\i} - x^3 \hat{\j} + y^3 \hat{\text{k}}$

$\displaystyle \text{Region:} \{ \ \text{Sphere: } x^2 + y^2 + z^2 = a^2 \ \}$

Step 2: Integrate Pt. 1

1. Find div F:
   $\displaystyle \text{div \bf{F}} = \frac{\partial}{\partial x}z^3 - \frac{\partial}{\partial y}x^3 + \frac{\partial}{\partial z}y^3$
2. [div F] Differentiate [Partial Derivatives]:
   $\displaystyle \text{div \bf{F}} = 0$
3. [Divergence Theorem] Substitute in div F:
   $\displaystyle \iint_S {\big( \nabla \times \textbf{F} \big) \cdot \textbf{n}} \, d\sigma = \iiint_D {0} \, dV$

Step 3: Integrate Pt. 2

We can evaluate the Divergence Theorem integral pretty easily:

$\displaystyle \begin{aligned}\iiint_D {0} \, dV & = \iint_D {0} \, dV \\& = \int_D {0} \, dV \\& = \boxed{0}\end{aligned}$

∴ $\displaystyle \Phi = 0$

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Learn more about Divergence Theorem: https://brainly.com/question/14040950
Learn more about multivariable calculus: https://brainly.com/question/13933633

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Topic: Multivariable Calculus

Unit: Stokes' Theorem and Divergence Theorem