Laplace’s equation is a second-order elliptic partial differential equation. The solutions of Laplace’s equation are important in many fields of engineering, notably electromagnetism, astronomy, and fluid dynamics, because they can be used to accurately describe the behavior of electric, gravitational, and fluid potentials.
The Cauchy Momentum Equation is a vector partial differential equation that describes the non-relativistic momentum transport in any continuum. Essentially, the time derivative of the flow vector field is related to the divergence of the stress tensor and to the body forces per unit mass.
The Reynolds Number is an important dimensionless quantity in fluid mechanics that is used predict flow patterns in different fluid flow situations. It is the ratio of inertial forces to viscous forces.
The Lagrange Equations are a convenient form to model the motion of systems. They may be viewed as a different form of Newton’s second laws. The Lagrange equations are ideal for systems with conservative forces and for bypassing constraint forces in any coordinate system.
Newton’s Universal Law of Gravitation states that a particle attracts other particles in the universe through a force that is directly proportional to the product of their masses and inversely proportional to the square of the distance between their centers. This is a general physical law derived from empirical observations.
The Convection- Diffusion-Reaction Equation for mass transport is a combination of the diffusion and convection equations, and describes physical phenomena where particles, energy, or other physical quantities are transferred inside a physical system due to convection, diffusion and reactions.
The Wave Equation is a second-order linear hyperbolic partial differential equation for the description of waves as they occur in classical physics. The equation can be used to model sound waves, light waves, and water waves. It arises in fields like acoustics, electromagnetics, and fluid dynamics.
Einstein’s Mass-Energy Equivalence states that anything having mass has an equivalent amount of energy and vice versa, with these fundamental quantities directly relating to one another by Einstein’s famous formula. The formula arose from special relativity.
The Clausius Inequality is a fundamental statement of the Second Law of Thermodynamics. It recognizes that the cyclic integral of heat transfer (Q) between a system at temperature (T) and its surroundings is less than zero for all cases but the idealized case of reversibility. The inequality leads to the definition of the non-conserved property of entropy and the reality of entropy generation in real processes.
The Mechanical Equivalency of Heat states that the cyclic integral of heat transfer is equal to the cyclic integral of work transfer between a system and its surroundings. It was discovered by Joule in his famous experiments and created the basis upon which Conservation of Energy and the First Law of Thermodynamics were based. It leads to the definition of the property internal energy.
Newton’s Law of Cooling states that the rate of heat loss of a body is directly proportional to the difference in the temperatures between the body and its surroundings, provided the temperature difference is small and the nature of radiating surface remains same.
The Stefan-Boltzmann Law describes the power radiated from a black body in terms of its temperature. Specifically, the total energy radiated per unit surface area of a black body across all wavelengths per unit time is proportional to the fourth power of the black body’s thermodynamic temperature.
Archie’s Law relates the in-situ electrical conductivity of a sedimentary rock to its porosity and brine saturation. It is an empirical law attempting to describe ion flow in clean, consolidated sands with varying intergranular porosity.
Darcy’s Law is an equation that describes the flow of a fluid through a porous medium. The law was formulated on the results of experiments involving the flow of water through beds of sand, and it forms the basis of hydrogeology.
Amdahl’s Law is a formula that gives the theoretical speedup in latency of the execution of a task at fixed workload that can be expected of a system whose resources are improved. This law is commonly used in parallel computing to predict the theoretical speedup when using multiple processors.
An M/M/1 Queue represents the queue length in a system having a single server, where arrivals are determined by a Poisson process, and job service times have an exponential distribution. The model is the most elementary of queueing models.
Stirling’s approximation is an approximation for factorials. It is a good-quality approximation, leading to accurate results even for small values of n. It is commonly used in the analysis of algorithms.
Hooke’s Law is a principle of physics that states that the force (or stress) needed to extend or compress a spring (or continuum media) by some distance (or strain) scales linearly with respect to that distance.
Gauss’s law for magnetism is one of the four Maxwell’s equations that underlie classical electrodynamics. It states that the magnetic field B has divergence equal to zero. In other words, that it is a solenoidal vector field.
Faraday’s Law of Induction is a basic law of electromagnetism that predicts how a magnetic field will interact with an electric circuit to produce an electromotive force. It is the fundamental operating principle of transformers, inductors and many types of electrical motors, generators and solenoids.
Ampere’s Circuital Law relates the integrated magnetic field around a closed loop to the electric current passing through the loop. This law can be expressed in an integral form or differential form: the differential form is shown here.
Euler’s Identity is a mathematical expression containing three of the basic arithmetic operations: addition, multiplication, and exponentiation. The identity also links five fundamental mathematical constants.
The Gaussian Integral is encountered in physics and quantum field theory. This integral has a wide range of applications in engineering and science, such as computing the normalizing constant of the normal distribution.
The Tsiolkovsky Rocket Equation describes the motion of a vehicle that follows the basic principle of a rocket, which is a device that can apply acceleration to itself by expelling part of its mass with high velocity. The equation relates the change of vehicle velocity to the exhaust velocity and initial and final mass of the vehicle.
The Mohr-Coulomb Failure Criterion represents the linear envelope that is obtained from a plot of the shear strength of a material versus the applied normal stress. More generally, the Mohr–Coulomb theory is a mathematical model that describes the response of brittle materials, like concrete or rubble piles, to shear stress as well as normal stress. Most of the classical engineering materials somehow follow this rule in at least a portion of their shear failure envelope.
The Darcy-Weisbach Equation is a phenomenological equation that relates the head loss, or pressure loss, due to friction along a given length of pipe to the average velocity of the fluid flow for an incompressible fluid.
The nth Harmonic Number, which is the sum of the reciprocals of the first n natural numbers, can be approximated via the formula shown. This approximation is commonly used in the analysis of algorithms.