Use the Divergence Theorem to calculate the surface integral S
F · dS;
that is, calculate the flux of F across S.
F(x, y, z) = x2yi + xy2j + 3xyzk,

S is the surface of the tetrahedron bounded by the planes x = 0, y = 0, z = 0, and
x + 4y + z = 4.

Step-by-step explanation:

Given that

F(x, y, z) = x^2y \hat i + xy^2 \hat j + 3xyz \hat k

The flux of \vec F across the surface S is given by

Where the survare S is the surface of the tetrahedron bounded by the planes x = 0, y = 0, z = 0, and  x + 4y + z = 4.

Now, by using the Gauss Divergence theorem:

...(i)

where V is the volume enclosed by the surface S and

\bigtriangledown \cdot \vec F = \hat i \frac {\partial \vec F}{\partial x}+\hat j \frac {\partial \vec F}{\partial y}+\hat k \frac {\partial \vec F}{\partial z}

=2xy+2xy+3xy=7xy

Putting this value in equation (i), we have

\int \int \int _v \bigtriangledown \cdot \vec F dV = \int \int \int _v (7xy)dxdydz

=

Hence, the flux of F across S is .

3< x< 5

step-by-step explanation:

|x - 4| < 1

there are two solutions, one positive and one negative.   remember to flip the inequality when taking the negative solution

x-4 < 1       x-4 > -1

add 4 to each side

x-4+4 < 1+4       x-4+4 > -1+4

x< 5                 x> 3

3< x< 5

i found two solutions,   n = 2 • ± √3 = ± 3.4641

step-by-step explanation: