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Properties of complex numbers

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Use De-Moivre's theorem to show that

cos(5x) = 16cos^5(x) - 20cos^3(x)+5cos(x)
sin(5x) = 5cos^4(x)sin(x)-10cos^2(x)sin^3(x)+sin^5(x)

given the complex numbers z1 = (3+4i)/(3-4i) and z2=[(1+2i)/(1-3i)]^2 find their
polar form, complex conjugate, moduli, product and quotients.

Prove for any complex numbers:

a. |z1 * z2| = |z1|*|z2|

b. |z1 / z2| = |z1|/|z2|

c. |z1 + z2| <= |z1|+|z2|

d. |z1 - z2| >= ||z1|-|z2||

Find all the complex values of (-32)^(1/5) and (1+i)^(1/3)



Find the unit vector normal to the surface x^2+y^2-z=1 at point (1,1,1)
Find the directional derivative of F(x,y,z)=yzx^2+4xz^3 at points (1,-2,-1) in the direction (1,-2,2)

Find constants a,b and c such that the vector field
A = (x+2y+az)i + (bx-3y-z)j +(4x+cy+2z)k is irrotational.

show that the resulting field can be expressed as a gradient of a scalar function.

show that the divergence of an inverse square force field in three dimensions is zero except in the origin.

show that the flux of the above force field through a sphere of radius a about the origin is 4Pi

use Gauss law to show that the divergence of F cannot vanish at the origin,

if f is differentiable function and A is a differentiable vector field then show that
div(fA) = A . grad(f) +fdiv(A)

Calculate the divergence of a radial field in three dimensions. What are the conditions in which the divergence vanishes.

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Solution Preview

See attached file for solutions

De-Moivre formula identity is:
Which leads to:
Recalling the binomial expansion:
We obtain:

However, noting that
We can neglect all the odd terms and write:
Where indicates the floor function (closets integer to m from below - inclusive).
In ...

Solution Summary

For the complex part ,The 13 pages file contain full derivations and explanations of the solutions to the problems described below.
for the vector calculus part the solution file is 15 pages long.