Please solve the screenshot (handwritten preferred) and explain your work, thanks! Use the Divergence Theorem to calculate the surface integralF• dS; that is, calculate the flux of F across S.F(x, y, z) = x²yi + xy²j + 2xyzk,S is the surface of the tetrahedron bounded by the planes x = 0, y = 0, z = 0, and x + 3y + z = 3.

Answered: Use the Divergence Theorem to calculate…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

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Please solve the screenshot (handwritten preferred) and explain your work, thanks!

Use the Divergence Theorem to calculate the surface integral
F• dS; that is, calculate the flux of F across S.
F(x, y, z) = x²yi + xy²j + 2xyzk,
S is the surface of the tetrahedron bounded by the planes x = 0, y = 0, z = 0, and x + 3y + z = 3.

Expert Solution

Step 1

According to divergence theorem Surface integral of normal component of flux $F$ is equal to volume integral of divergence of flux $F$

i.e. $\int \int_{S} F . d s = \iint \int_{V} d i v F d v$

Given $F (x, y, z) = x^{2} y i + x y^{2} j + 2 x y z k$

div $F = \nabla \cdot F = (\frac{\partial}{\partial x} i + \frac{\partial}{\partial y} j + \frac{\partial}{\partial z} k) . (x^{2} y i + x y^{2} j + 2 x y z k) = 2 x y + 2 x y + 2 x y = 6 x y$

Now surface $S$ is tetrahedron bounded by the planes $x = 0$ , $y = 0$ , $z = 0$ and $x + 3 y + z = 3$

$\Rightarrow 0 \leq z \leq 3 - x - 3 y$ , $0 \leq y \leq \frac{3 - x}{3}$ and $0 \leq x \leq 3$

$\iint \int_{V} d i v F d v = \int_{0}^{3} \int_{0}^{\frac{3 - x}{3}} \int_{0}^{3 - x - 3 y} 6 x y d z d y d x$

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