differential

Examples of differential in the following topics:

• Bibliography

  • Elementary differential equations and boundary value problems.

• Introduction to Waves and Modes in One and Two Spatial Dimensions

  • where the prime denotes differentiation with respect to the argument ( $x+ct$ for instance).
  • As a result, any differentiable function evaluated at $x \pm ct$ is a solution of the 1-D wave equation.

• Complex numbers and constant coefficient differential equations

  • This is an example of a constant coefficient differential equation.
  • ($x$ is the zeroth derivative of $x$. ) The most general linear n-th order constant coefficient differential equation is
  • These constant coefficient differential equations have a very special property: they reduce to polynomials for exponential $x$.

• Ampere's Law: Magnetic Field Due to a Long Straight Wire

  • In this equation, dl represents the differential of length of wire in the curved wire, and μ0 is the permeability of free space.
  • Integrating the previous differential equation, we find:

• Scattering

  • The evolution of the intensity of a particular ray depends not only the intensity of the ray and the local properties of the material but also on the intensity of all other rays passing through the same point—we have an integro-differential equation.
  • Even if one neglects scattering, one often has to solve an integro-differential equation.

• Damped Harmonic Motion

  • We can describe this situation using Newton's second law, which leads to a second order, linear, homogeneous, ordinary differential equation.
  • We solve this differential equation for our equation of motion of the system, x(t).
  • Plugging this into the differential equation we find that there are three results for a, which will dictate the motion of our system.

• Simple Harmonic Motion

  • For one-dimensional simple harmonic motion, the equation of motion (which is a second-order linear ordinary differential equation with constant coefficients) can be obtained by means of Newton's second law and Hooke's law.
  • Solving the differential equation above, a solution which is a sinusoidal function is obtained.
  • We can use differential calculus and find the velocity and acceleration as a function of time:
  • Using Newton's Second Law, Hooke's Law, and some differential Calculus, we were able to derive the period and frequency of the mass oscillating on a spring that we encountered in the last section!

• Introduction to Spherical and Cylindrical Harmonics

  • As an exercise, carry out the differentiations of $\frac{1}{r} = \left(x^2 +y^2+z^2\right)^{-1/2}$ and show that $\nabla ^2 \psi(x,y,z)$ is identically zero for $r>0$, where the charge is located.
  • He went on to make profound advances in differential equations and celestial mechanics.
  • When solving boundary value problems for differential equations like Laplace's equation, it is extremely handy if the boundary on which you want to specify the boundary con- ditions can be represented by holding one of the coordinates constant.
  • The disadvantage to using coordinate systems other than Cartesian is that the differential operators are more complicated.

• Changing Magnetic Flux Produces an Electric Field

  • Therefore, we get an alternative form of the Faraday's law of induction: $\nabla \times \vec E = - \frac{\partial \vec B}{\partial t}$.This is also called a differential form of the Faraday's law.

• Relation Between Electric Potential and Field

  • ∆P can be substituted for its definition as the product of charge (q) and the differential of potential (dV).
  • We can then replace W with its definition as the product of q, electric field (E), and the differential of distance in the x direction (dx):