Maxwell's equations

Examples of Maxwell's equations in the following topics:

• Applications of Multiple Integrals

  • In electromagnetism, Maxwell's equations can be written using multiple integrals to calculate the total magnetic and electric fields.

• The Divergence Theorem

  • The first equation of the Maxwell's equations is often written as $\nabla \cdot \mathbf{E} = \frac {\rho} {\varepsilon_0}$ in a differential form, where $\rho$ is the electric density.

• Parametric Equations

  • Parametric equations are a set of equations in which the coordinates (e.g., $x$ and $y$) are expressed in terms of a single third parameter.
  • Converting a set of parametric equations to a single equation involves eliminating the variable from the simultaneous equations.
  • If one of these equations can be solved for $t$, the expression obtained can be substituted into the other equation to obtain an equation involving $x$ and $y$ only.
  • In some cases there is no single equation in closed form that is equivalent to the parametric equations.
  • One example of a sketch defined by parametric equations.

• Linear Equations

  • A linear equation is an algebraic equation in which each term is either a constant or the product of a constant and a single variable.
  • A common form of a linear equation in the two variables $x$ and $y$ is:
  • The parametric form of a linear equation involves two simultaneous equations in terms of a variable parameter $t$, with the following values:
  • Linear differential equations are differential equations that have solutions which can be added together to form other solutions.
  • Linear differential equations are of the form:

• Solving Differential Equations

  • Differential equations are solved by finding the function for which the equation holds true.
  • A differential equation is a mathematical equation for an unknown function of one or several variables that relates the values of the function itself to its derivatives of various orders.
  • As you can see, such an equation relates a function $f(x)$ to its derivative.
  • Solving the differential equation means solving for the function $f(x)$.
  • The "order" of a differential equation depends on the derivative of the highest order in the equation.

• Nonhomogeneous Linear Equations

  • When $f(t)=0$, the equations are called homogeneous second-order linear differential equations.
  • Otherwise, the equations are called nonhomogeneous equations.
  • Linear differential equations are differential equations that have solutions which can be added together to form other solutions.
  • This can be confirmed by substituting $y(x) = c_1y_1(t) + c_2 y_2(t)$ into the equation and using the fact that both $y_1(t)$ and $y_2(t)$ are solutions of the equation.
  • Identify when a second-order linear differential equation can be solved analytically

• Models Using Differential Equations

  • Differential equations can be used to model a variety of physical systems.
  • Differential equations are very important in the mathematical modeling of physical systems.
  • The mathematical theory of differential equations first developed together with the sciences where the equations had originated and where the results found application.
  • Conduction of heat is governed by another second-order partial differential equation, the heat equation .
  • Give examples of systems that can be modeled with differential equations

• Separable Equations

  • Separable differential equations are equations wherein the variables can be separated.
  • One of these forms is separable equations.
  • A separable equation is a differential equation of the following form:
  • The original equation is separable if this differential equation can be expressed as:
  • Integrating such an equation yields:

• Second-Order Linear Equations

  • For a function dependent on time, we may write the equation more expressly as $L y(t) = f(t)$ and, even more precisely, by bracketing $L [y(t)] = f(t)$.
  • It is convenient to rewrite this equation in an operator form:
  • When $f(t)=0$, the equations are called homogeneous second-order linear differential equations.
  • (Otherwise, the equations are called nonhomogeneous equations.)
  • A simple pendulum, under the conditions of no damping and small amplitude, is described by a equation of motion which is a second-order linear differential equation.

• Predator-Prey Systems

  • The relationship between predators and their prey can be modeled by a set of differential equations.
  • The predator–prey equations are a pair of first-order, non-linear, differential equations frequently used to describe the dynamics of biological systems in which two species interact, one a predator and one its prey.
  • As differential equations are used, the solution is deterministic and continuous.
  • The solutions to the equations are periodic.
  • Identify type of the equations used to model the predator-prey systems