Chap. 3:  A  Few  First  Steps

3.4.1 	Compute analytically (as far as possible) and numerically thefollowing quantities:  
	tg(π/4),  cot(π/3); log(1), log(I), log(I π).

3.4.2  	Divide  (x^9)-(a^9)   by   x-a;  get the quotient as a polynomial.

3.4.3  	Decompose   f=(x^4)+2 (x^3)-13 (x^2)-14 x+24   into factors;
	return to the original polynomial.

3.4.4	Get the values ot the binomial coefficients: 	(({{12}, {n}}))
	Hint: Compute (((x+y))^12.).

3.4.5  	z = 2 + i π. Compute the real and the imaginary part, the conjugate to z, |z|  and the corresponding
	numerical values.

3.5.1    Plot the function   f=(x^2)-5 x+1   with  the range of  x  chosen such 
	that both zeros of   f   are shown.
3.5.2    Plot the curve     x = cos(t),  y = cos(2t + a)    for  a = 0,   π/3,  π/2 .
3.5.3    Plot the hyperbolic paraboloid  z=((((x/2)))^2)-((((y/3)))^2) .
3.5.4    Plot the torus:  x =  r cos(p), y = r sin(p), z = sin(t)with   r = 2 + cos(t).

3.6.1   Derive the function   f=(x^2)-5 x+1   w.r.t.  x .
3.6.2   Derive the functions {(e^(2 a (x^2))) cos(c x),tan((e^(3 x))),sin(a x) cos(b y)} w.r.t.  x .
3.6.3   Derive  ((((x/2)))^2) ((((y/3)))^4)   w.r.t.  x   and  y .

3.7.1  	Compute the following definite integral in several ways and 
	compare the results:  ∫01 x20 ex  dx  
	a) Numeric integration.
	b) Analytic integration, go over to numeric values with default 
	     precision  and with 20 decimal places.

3.8.1	Expand  sin(sin x)   around   x = 0  up to the  7-th term.
3.8.2	Expand   tg(x)  around  x = π/2   up to the third term.

3.9.1	Compute the determinant, the inverse, the characteristic  values  and the 
	characteristic  equation of the following matrix:     {{1, 0, -4}, {0, 5, 4}, {-4,  4, 3}}. 
	Factorize the characteristic  polynomial.

3.10.1	Solve the following system of linear equations:
	x + y + z = 1,   x + 2y + 3z = 4,   x + 3y + 6z = 10.

3.11.1	Solve the followingalgebraic equation (analytically and numerically):
	(x^5) +  x  -  1   =  0. Verify by inserting that the solutions fulfil the equation.

3.12.1 Find the numeric values of two other roots of the function: f = sin(x) - cos(x).

3.13.1 Solve the 2-dimensional equation of motion for the motion of a projectile 
	experiencing the field of gravity and a linear friction force.

3.14.1  Go over to cylindrical coordinates  r, ϕ, z  (SetCoordinates Cylindrical[r,ϕ,z]).
	Compute the gradient and Laplacian of a scalar function  psi(r,ϕ,z),
	divergence and curl of a vector  ( ar(r,ϕ,z), ap(r,ϕ,z), az(r,ϕ,z) ).

Chapter 4: 	Defining Functions. Substitutions.  Delayed Assignments. 
		Transforming Expressions.

4.1          Define the function   f(x,n)=(x^n).    Evaluate f(2,1),f(3,2),f(4,7),f(y,k).

4.3	Decompose the following expressions into partial fractions; thereafter put them over common 
            denominator and expand completely. At the end  simplify all these expression as much as possible.

2 )    (((x^3)+3 (x^2)-4 x+3)/(((x^2)-1) ((((x^2)+1))^2)))

4.4           Plot the function f(x)=sin(x)/(1+x^2) in the intervall (0,π )  and determine the maximum  xm    and    f(xm).

4.5          Assuming that x and y are real, compute the real and imaginary parts of the following expression:

3)        (((x + i y))^2)sin(x + i y).

4.6	Transform the following expressions into Fourier sums and into pure powers of  sin x   and  cos x.   
	These results are not unique in view of the relation   (sin^2)x  +  (cos^2)x  = 1.  In addition some 
	transformations must be imposed by presenting some trigonometric relations as substitution rules.

3)  sin(3x) cos(5 x) (cos^2)x .

4.8	Spherical Bessel functions  z(m,x) are proportional to Bessel functions with half odd integer
	order:
			(Subscript[j, m])(x) := (Sqrt[(π/(2x))]) (Subscript[J, m + 1/2])(x),  ym(x) := (Sqrt[(π/(2x))]) (Subscript[Y, m + 1/2])(x) ;
	
	they fulfil the following recurrence relation:
	
	z(m+1,x) -  (2m + 1)/x z(m,x) + z(m-1,x)  =  0;	
	j(0,x) = y(-1,x) = sin(x)/x;  j(-1,x) = - y(0,x) = cos(x)/x
	
	Define a function, which computes  j(m,x)  or  y(m,x) for arbitrary natural   m. 
	So z is either j or y. Compute the first few  j's and  y's.

4.12	Find the roots of the following polynomial and verify that they fulfil the corresponding 
	equation p(u) = 0.		p(u) = -2 u+2 (u^3)+3 ε-2 (u^2) ε

Chapter 5: Lists

5.1	Set up a list containing the roots of the polynomial
	24 - 14 x - 13 x^2 + 2 x^3 + x^4. 
	a)  Sort it according to the numeric values of the roots.
	b)  Set up a list containing only the positive roots.
		
5.3	Generate the list of natural numbers  between 0 and  10.
	Insert  Pi  at its appropiate place.
	
5.4	Generate a Table for Plots of  Cos[n x]  for  n = 1,3,5,7  in the intervall  [0  <= x <= π].
	
5.6	Write a function that works on a list such as {a1, a2, a3, a4, ...} 
	and returns a list  {a2/a1, a3/a2, a4/a3, ...}.
	
5.7	Reduce the following list  {a,b,c,e,b,e,f,f,a,d,b,c} such that each letter
	occurs only once and order it in inverse alphabetical order.

5.9	There are two lists, each containing 10 elements:
	xl  = { x[1], x[2], ..., x[10] },  yl = { y[1],...,y[10] }
	Combine them into one list  by one simple command:
               xyl =    { { x[1], y[1] }, ..., { x[10], y[10]} }

Chap.8. Linear Algebra: Matrices, Linear Equations, Eigenvalues and -vectors, Jordan Decomposition, Singular Value Decomposition, Pseudoinverse Matrices

8.10      Exercises

8.1.   	Solve the following systems of linear equations;   c) is a homogeneous system, the matrix gives the coefficients.  	
	Use at least 3 methods when the system is regular,  otherwise at  least  2 methods.

b)  	x + y + z = 1,    x + 2y + 3z = 4,    x + 3y + 6z = 10,       x + 4y + 10z = 19.

	c)        (({{0, 1, -3, -1}, {1, 0, 1, 1}, {3, 1, 0, 2}, {1, 1, -2, 0}})).

8.2.   	Replace  number  19   in  the system  8.1.b)  above by  an unknown   a. Without using the results obtained in  8.1.b)   
	find  the  value of   a   such  that the system of  4   equations has  a unique solution.

8.3.   	Compute the square, the transpose, the trace, the determinant, the inverse,  the characteristic polynomial, the latent roots 
	(= eigenvalues) and eigenvectors of the following  matrixs and  transform  it into  an equivalent  diagonal  matrix. 

	a)       (({{10, -14, -10}, {-14, 7, -4}, {-10, -4, 19}})) .

8. 4   Some powers of the matrix given below coincide with the indentity matrix; find the lowest positive power.   

	  e)   (({{0, 1, 0}, {0, 0, 1}, {1, 0, 0}}))

8.7.  	A zero divisor of a given matrix  A  is a (square or rectangular)  matrix, which differs from the null matrix, but for which   
	LA  =  0  (then L  is a left zero divisor)  or  AR = 0  (then  R  is a right zero divisor).  R, L  respectively, consist of the 	null space of  A, of the transpose of A  respectively.  Find the left and right  zero divisor of the following matrix:

a)             (({{2, 0, -1}, {3, 4, 2}, {0, -8, -7}}))       ,

8.8.  	Determine the shape (determined by the eigenvalues) and the orientation (given by the eigenvectors) 
	of the following conic:

	1)	   x 2 +   5 y 2   +   3 z 2  +  8 y z  -    8 x z 	= 1;	
	Hint: The equation above may be written as: (x,y,z).A.(x,y,z) with appropriate elements of A.

8.12  	Is the following matrix exactly equivalent to a diagonal matrix  ?

(({{1, 2}, {-2, -3}}))

Give  also the JordanDecomposition and the SingularValueDecomposition of this matrix.

9.   Non-linear Algebra and Non-Linear Equations

9.1.   Divide  the polynomial   f   by the polynomial  g.  Find the quotient and  the remainder.  

	 1)	f   =  x6  -  5 x5  +  4 x4  -  2 x3  +  3 x2  -  x  -  1, 	g  =  x2 -  x  +  1.

9.4	For which real value(s) of  the parameter  a  occur multiple roots of the following equation :
		(a^4)+3 (a^2) x+a (x^4)+(x^5)  =  0  
	These value(s) should be given with an accuracy of 30 decimal places.