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 .

77

step-by-step explanation:

a

step-by-step explanation:

when a number is between two straight lines it means the absolute value.

to find absolute value you need to know how many numbers is that number away from 0.

when finding absolute value the number will always come out positive

so absolute value of -3 is 3

absolute value of 3/2 is 3/2

absolute value of -2 is 2

absolute value of 3.5 is 3.5

absolute value of -1 is 1

now put these numbers in order:

a)   |-1|, |3/2|, |-2|, |-3|, |3.5|