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)

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.

3.
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.

De-Moivre formula identity is:
(1.1)
Which leads to:
(1.2)
Recalling the binomial expansion:
(1.3)
And:
(1.4)
We obtain:

Thus:
(1.5)
However, noting that
(1.6)
We can neglect all the odd terms and write:
(1.7)
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.

I have two problems with complexnumbers:
1) Use properties of moduli to sow that when |z3| does not equal |z4|,
Re(z1 + z2) / |z3 + z4| <= (is smaller or equals) (|z1| + |z2|) / (| |z3| -|z4| |)
2) Verify that sqrt(2) * |z| >= |Re z| + |Im z|
(suggestion: reduce this inequality to (|x| - |y|)^2 +. 0

Is the multiplication of complexnumbers similar to multiplication of polynomials? Is it possible to apply the FOIL method when multiplying complexnumbers? Explain your answers.

Design class Complex for working with complexnumbers of the form a + bi, where i is the square root of -1. Your class must have two overloaded operators for adding and subtracting the complexnumbers. The sum and the difference of two complexnumbers a + bi and c + di is defined as (a+c) + (b+d)i (respectively, (a-c) + (b-d)i).

I am having diffilculties understanding the whole concept of Real Numbers and Their Properties, Real Numbers, Fractions, Addition and Subtraction of Real Numbers, Exponential Expressions and the Order of Operations, Properties of the Real Numbers and Using the Properties to Simplify Expressions. What I need is a break-down of th

Decide whether each of the given sets is a group with respect to the indicated operation.
1- For a fixed positive integer n, the set of all complexnumbers x such that x^n=1(that is, the set of all nth roots of 1),with operation multiplication.
2-The set of all complexnumbers x that have absolute value 1, with operation m

1. Find the area bounded by the lines y=0, y=2 an y=sqrt(x)
2. Find the partial decomposition of:
x/(x^2+2x-3)
3. Find the critical points and the inflection points of the following function:
f(x) = x^4 - 4x^3 + 10
4. Simplify the following complex expressions, expressing each in the form (a + jb)
(i) (4+j|1)+(8+j6

1. show that for a 2x2 unitary matrix its determinant is a complex number of unit modulus
2. Verify that the Pi/2 rotation matrix is orthogonal.
3. Verify that the matrices:
A= 1/sqrt(2) * ( 1 i )
( i 1)
and:
B = 1/2 * ( 1+i 1-i )
( 1-i 1+i )
are unitary. Verify