# Mathematics

Formal Sciences

### Canonical Form and Jordan Blocks

We discuss Jordan bases and the fact that an operator can be put into Jordan canonical form if its characteristic and minimal polynomials factor into

Formal Sciences

### Green’s Theorem

Green’s Theorem for Simple Connected Regions Theorem. (Green’s Theorem) Let $R$ be a simply connected region with a piecewise smooth boundary curve $C$ oriented counterclockwise

Formal Sciences

### Probability Density Functions (Applications of Integrals)

Applications of Integrals We will consider the following applications: average value of a function over a region, mass of a lamina, electric charge, moments and

Formal Sciences

### Invariant Subspaces and Generalized Eigenvectors

We let $V$ and $W$ denote real or complex vector spaces. Suppose $T \in \mathcal{L}(V)$ and $U$ a subspace of $V$, then we say $U$

Formal Sciences

### Conservative Vector Fields and Independence of Path

Independence of Path Definition. The line integral is called independent of path if in a region $D$, if for any two points $P$ and $Q$

Formal Sciences

### Diagonalization of a Matrix (with Examples)

An $n\times n$ matrix $A$ is called diagonalizable if $A$ is similar to some diagonal matrix $D.$ If the matrix of a linear transformation $T$

Formal Sciences

### Line Integrals

Definition of a Line Integral Let $C$ be a smooth curve, with parametric equations $x=x(t),$ $y=y(t),$ and $z=z(t)$ for $a\leq t\leq b,$ that lies within

Formal Sciences

### Finding Eigenvalues and Eigenvectors

Eigenvalues and Eigenvectors Let $\mathbb{F}$ be either the real numbers or the complex numbers. A nonzero vector $v$ in $\mathbb{F}^n$ is called an eigenvector of

Formal Sciences

### Divergence and Curl of a Vector Field

The Divergence and Curl Definition. Let $\mathbf{V}$ be a given vector field. The divergence of $\mathbf{V}$ is defined by div $\mathbf{V}=\nabla \cdot \mathbf{V}$ and the

Formal Sciences

### Determinant of a Matrix (Theory and Examples)

Determinants and Trace The determinant function can be defined by essentially two different methods. The advantage of the first definition, one which uses permutations, is

Formal Sciences

### Vector Fields and Gradient Fields

Introduction to Vector Fields Definition. A vector field in $\mathbb{R}^n$ is a function $\mathbf{V}$ that assigns a vector from each point in its domain. A

Recall that the norm of $x\in \mathbb{R}^n$ defined by $\left|\left |x\right|\right | = \sqrt{x_1^2+x_2^2}$ is not linear. To injective linearity into the discussion we introduce