Question

Use the divergence theorem to find the outward flux of the vector field F(xyz)= 4x^2 i + 4y^2 j + 3z^2 k across the boundary of the rectangular prism: 0

Answer 1

$\vec F(x,y,z)=4x^2\,\vec\imath+4y^2\,\vec\jmath+3z^2\,\vec k\implies\nabla\cdot\vec F=8x+8y+6z$

Let $S$ be the surface of the rectangular prism bounded by the planes $x=0$, $x=a$, $y=0$, $y=b$, $z=0$, and $z=c$. By the divergence theorem, the integral of $\vec F$ over $S$ is given by the integral of $\nabla\cdot\vec F$ over the interior of $S$ (call it $R$):

$\displaystyle\iint_S\vec F\cdot\mathrm d\vec S=\iiint_R(\nabla\cdot\vec F)\,\mathrm dV$

$=\displaystyle\int_0^c\int_0^b\int_0^a(8x+8y+6z)\,\mathrm dx\,\mathrm dy\,\mathrm dz=\boxed{abc(4a+4b+3c)}$

Answer 2

Answer:

$\displaystyle \iiint_D {\nabla \cdot \textbf{F}} \, dV = \boxed{6875}$

General Formulas and Concepts:

Calculus

Differentiation

• Derivatives
• Derivative Notation

Derivative Property [Multiplied Constant]:
$\displaystyle (cu)' = cu'$

Derivative Rule [Basic Power Rule]:

1. f(x) = cxⁿ
2. f’(x) = c·nxⁿ⁻¹

Integration

• Integrals

Integration Rule [Reverse Power Rule]:
$\displaystyle \int {x^n} \, dx = \frac{x^{n + 1}}{n + 1} + C$

Integration Rule [Fundamental Theorem of Calculus 1]:
$\displaystyle \int\limits^b_a {f(x)} \, dx = F(b) - F(a)$

Integration Property [Multiplied Constant]:
$\displaystyle \int {cf(x)} \, dx = c \int {f(x)} \, dx$

Integration Property [Addition/Subtraction]:
$\displaystyle \int {[f(x) \pm g(x)]} \, dx = \int {f(x)} \, dx \pm \int {g(x)} \, dx$

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 and Curl:

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

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

Step-by-step explanation:

*Note:
Your question is incomplete, but I have defined the missing portions of the question below.

Step 1: Define

Identify given.

$\displaystyle \textbf{F} (x, y, z) = 4x^2 \hat{\i} + 4y^2 \hat{\j} + 3z^2 \hat{\text{k}}$

$\displaystyle \text{Region (Boundary de} \text{fined by rectangular prism):} \left\{ \begin{array}{ccc} 0 \leq x \leq 5 \\ 0 \leq y \leq 5 \\ 0 \leq z \leq 5 \end{array}$

Step 2: Find Flux Pt. 1

1. [Vector Field] Find div F:
   $\displaystyle \text{div } \textbf{F} = \frac{\partial}{\partial x}4x^2 + \frac{\partial}{\partial y}4y^2 + \frac{\partial}{\partial z}3z^2$
2. [div F] Differentiate [Partial Derivatives and Basic Differentiation Techniques]:
   $\displaystyle \text{div } \textbf{F} = 8x + 8y + 6z$

Step 3: Find Flux Pt. 2

1. [Flux] Define:
   $\displaystyle \Phi = \iiint_D {\nabla \cdot \textbf{F}} \, dV$
2. [Divergence Theorem] Substitute in div F:
   $\displaystyle \Phi = \iiint_D {\big( 8x + 8y + 6z \big)} \, dx \, dy \, dz$
3. [Flux] Substitute in region D:
   $\displaystyle \Phi = \int\limits^5_0 \int\limits^5_0 \int\limits^5_0 {\big( 8x + 8y + 6z \big)} \, dx \, dy \, dz$

We can evaluate the Flux integral (Divergence Theorem integral) using basic integration techniques listed under "Calculus":

$\displaystyle \begin{aligned}\int\limits^5_0 \int\limits^5_0 \int\limits^5_0 {\big( 8x + 8y + 6z \big)} \, dx \, dy \, dz & = \int\limits^5_0 \int\limits^5_0 {4x^2 + x \big( 8y + 6z \big) \bigg| \limits^{x = 5}_{x = 0}} \, dy \, dz \\& = \int\limits^5_0 \int\limits^5_0 {\big( 40y + 30z + 100 \big)} \, dy \, dz \\& = \int\limits^5_0 {20y^2 + y \big( 30z + 100 \big) \bigg| \limits^{y = 5}_{y = 0}} \, dz \\\end{aligned}$

$\displaystyle\begin{aligned}\int\limits^5_0 \int\limits^5_0 \int\limits^5_0 {\big( 8x + 8y + 6z \big)} \, dx \, dy \, dz & = \int\limits^5_0 {\big( 150z + 1000 \big)} \, dz \\& = \big( 75z^2 + 1000z \big) \bigg| \limits^{z = 5}_{z = 0} \\& = \boxed{6875} \\\end{aligned}$

∴  $\displaystyle \Phi = \boxed{6875}$

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Learn more about Divergence Theorem: https://brainly.com/question/2407209

Learn more about multivariable calculus: https://brainly.com/question/7964566

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Topic: Multivariable Calculus

Unit: Stokes' Theorem and Divergence Theorem