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8 months ago

1. Graph each point on the imaginary plane. The first number x is along thehorizontal axis, referred to as the real axis. The se

cond number y is along thevertical axis, referred to as the imaginary axis. Record the values of x and y. Then graph the points.Then graph the points.imaginary bais. Recordsnecoration(x + yi)ху(3 - 41)2(-5-21)O2(4 + 3i)2(-2+ 41)4

Mathematics

1 answer:

AlekseyPX8 months ago

4 0

$z=3-4i$

$z=-5-2i$ $z=4+3i$ $z=-2+4i$

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2 years ago

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$x=2,y=0\implies z=-8\implies(2,0,-8)$

Parameterize $S$ by

$\vec s(u,v)=(1-v)(2,0,-8)+v\bigg((1-u)(0,2,-2)+u(0,0,2)\bigg)=(2-2v,2v-2uv,-8+6v+4uv)$

with $0\le u\le1$ and $0\le v\le1$ . Take the normal vector to $S$ to be

$\vec s_v\times\vec s_u=(20v,8v,4v)$

Then the flux of $\vec F$ across $S$ is

$\displaystyle\iint_S\vec F(x,y,z)\cdot\mathrm d\vec S=\int_0^1\int_0^1\vec F(x(u,v),y(u,v),z(u,v))\cdot(\vec s_v\times\vec s_u)\,\mathrm du\,\mathrm dv$

$=\displaystyle\int_0^1\int_0^1(6-6v,2v-2uv,-8+6v+4uv)\cdot(20v,8v,4v)\,\mathrm du\,\mathrm dv$

$=\displaystyle8\int_0^1\int_0^1(11v-10v^2)\,\mathrm du\,\mathrm dv=\boxed{\frac{52}3}$

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3 years ago