Mathematics

Scientific look at Scalars and Vectors



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Introduction:
In vector analysis and matrix mathematics you'll find a manifold of expressions like scalar field, scalar product, scalar multiplication, vector field, vector product, row vector, column vector, Eigenvector etc.. However, the basic concept is easy to understand and starts mathematically spoken with something simple as a n-tuple i.e. an array of n numbers. The 3-dimensional space R^3 (say:'R-three') is defined in set theory as the set of all 3-tuple with 3 real numbers.
R^3={(x,y,z)|x.element_of.R,y.element_of.R,z.element_of.R}
A vector is a mathematical object that points from the origin to a specified point in space, specified by its coordinates. This mathematical object 'vector' can be shifted, stretched, compressed or rotated. A vector is characterized by its lengths and its direction. To distinguish a vector from a simple real number as mathematical object the term 'scalar' resp. the prefix 'scalar-' is used. Typical vectorial entities in physics are forces and streams, whereas temperature scales or energy scales are measuring scalar values.
Vector Calculus:
A vector is usually given by a matrix with a single line, a row vector. If you want a column vector, simply trans-pone it. The components are given by Cartesian coordinates in 3D space.
a=(a[1],a[2],a[3])
The basis vectors in 3D space in a Cartesian coordinate system are defined as unit vectors along the x,y and z axis. Any vector is composed out of a linear combination of basis vectors. The basis vectors are denoted by a small 'e' from the German word 'Einheitsvektor' for unit vector.
e[x]=(1,0,0);e[y]=(0,1,0);e[z]=(0,0,1)
In vector calculus two formalisms are used, a vector formalism and a matrix formalism. The scalar product is written in two ways:
a.dot.b vector formalism (that's why it's also called the 'dot-product')
ab^T matrix formalism (vector b is transponed)
The dot product is defined as the product of the lengths of the vectors times the cosine of the angle between them. Another vector multiplication is defined as vector product. The resulting vector c=a.x.b is perpendicular to both vector a and b. The length of the resulting vector is defined as the product of the lengths of the vectors times the sinus of the angle between them. A scalar multiplication of a vector with a scalar is defined as multiplication of each component with the scalar.
Scalar Fields, Vector Fields and Trajectories:
The basic mathematical objects in vector analysis are best understood by the conception of input/output. The most simple case is the ordinary function of one variable. Input and output is a scalar. See the following table for definition of scalar field, vector field and trajectories: r=(x,y,z)
Scalar field .phi.(r): input:vector output:scalar
Vector field V(r): input:vector output:vector
Trajectory C(t): input:scalar output:vector
Function f(x): input:scalar output:scalar
A 2D scalar field is displayed as a 3D surface, the 3rd axis representing the functional value or scalar. A function of more than one variable is synonymous to a scalar field. Examples for a scalar field is the electron density map of a molecule, showing contour lines or an energy hyper-surface. A vector field is usually displayed by its field lines. The electric force field in a condenser shows parallel lines between the condenser plates. The density of the field lines is a measure of the strength of a force field.
V(r)=V(x,y,z)=(V[x],V[y],V[z])
A trajectory is a parametrized vector used to follow the position of an object in space during time.
C(t)=(C[x](t),C[y](t),C[z](t))
Basics of Vector Analysis:
Operators in vector analysis are gradient(grad), divergence(div) and curl. The Nabla formalism uses the Nabla operator, symbolized by a triangle standing on the top with an arrow above, to define the operators.
.nabla.=(d/dx,d/dy,d/dz)
grad.phi.=.nabla..phi.
div V(r)=.nabla..dot.V(r)=.nabla.V(r)^T
curl V(r)=.nabla..x.V(r)
The gradient vector field of a potential (=scalar field) is a measure of the steepest slope and used to find extrema in a scalar field, the so called 'gradient method'. The divergence scalar field of a vector field shows regions and points of sources resp. negative sources.
The curve integral is given as:
.curve_integral.[t start,t end] V(r)ds^T with ds=(dx(t),dy(t),dz(t)), C(t)
If the curve integral is path independent - the curve integral is zero for a closed path - we call the vector field conservative i.e. a gradient field of a scalar field called potential. The curl of such a conservative vector field is zero.
To evaluate the potential the inverse Nabla operator is used.
.phi.=.nabla.^-1 V
.phi.=.integral. V[x]dx + C(y,z)
.phi.=.integral. V[y]dy + C(x,z)
.phi.=.integral. V[z]dz + C(x,y)
Eigenvectors, Eigenvalues - The Eigen Equation Reformulated
The well known Eigen equation is solved by certain Eigenfunctions, Eigenvectors respectively. Operator is acting on the Eigenfunction or Eigenvector giving the same Eigenfunction, Eigenvector respectively scalar multiplied by the Eigenvalue .lambda. (O operator, M matrix)
O f[i](x)=.lambda.[i]f[i](x)
M v[i]^T=.lambda.[i]v[i]^T
But how can this formula be interpreted in terms of fields?. The Eigenvector is transformed by the matrix multiplication, stretched or compressed. Other vectors do not fullfill the Eigen equation. My colleague at FU Berlin, the physicist Dr. rer. nat. Wilhelm Uebach put forward a new formulation of the Eigen equation, seeing it as a search mask or shablone for all radial parts of a vector field, the Uebach-Eigen vector field. A partial annihilating operator, annihilating all non radial parts, is acting on a vector field to transform it into a Uebach-Eigen vector field. Only the radial parts of the vector field solve the reformulated Eigen equation.
V[Uebach-Eigen](v[i])=.lambda.[i]v[i]
Vector Constructed Circular Matrices:
A circular matrix is defined as a matrix constructed by a vector. This vector is shifted and scrolled one place to the right by going one row below. The circular matrices have firstly been mentioned by Bronstein and Semendjajew. Circular matrices play an important role in matrix mathematics. The unit matrix itself is a circular matrix. The unit matrix is given by Kronecker delta matrix elements.
E=(.delta.[i,j])
The unit matrix can be constructed by circularizing the basis vector of the first dimension. Defining a circular operator circ, we get the unit matrix constructive equation.
E=circ e[1]
The convolution product of two vectors is given by:
a*b=a circ b
For deconvolution matrix multiplication with the inverse circular matrix is used. The inverse matrix of a circular matrix is itself circular and constructed by the convolution inverse vector.

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