10 Lecture in calculus Projects Surfaces Flux Substitution Perpendicularity

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Projects Topics, ideas

Surfaces

Surfaces (continued)

(continued) Surfaces

Surface integral In mathematics, a surface integral is a generalization of multiple integrals to integration over surfaces. It can be thought of as the double integral analog of the line integral. Given a surface, one may integrate over its scalar fields (that is, functions which return scalars as values), and vector fields (that is, functions which return vectors as values).mathematicsmultiple integralssurfacesdouble integralline integralscalar fields functionsscalars vector fieldsvectors Surface integrals have applications in physics, particularly with the classical theory of electromagnetism.physicsclassical theory electromagnetism

Surface integral (continued)

Flux In the various subfields of physics, there exist two common usages of the term flux, each with rigorous mathematical frameworks. A simple and ubiquitous concept throughout physics and applied mathematics is the flow of a physical property in space, frequently also with time variation. It is the basis of the field concept in physics and mathematics, with two principal applications: in transport phenomena and surface integrals. The terms "flux", "current", "flux density", "current density", can sometimes be used interchangeably and ambiguously, though the terms used below match those of the contexts in the literature.physics applied mathematics physicstransport phenomenasurface integrals

Flux (continued)

Green's theorem In mathematics, Green's theorem gives the relationship between a line integral around a simple closed curve C and a double integral over the plane region D bounded by C.mathematicsline integraldouble integral

Green's theorem (continued)

Stokes' theorem Stokes' theorem (also called the generalized Stokes' theorem) is a statement about the integration of differential forms on manifolds, which both simplifies and generalizes several theorems from vector calculus. integrationdifferential formsmanifolds theoremsvector calculus

Stokes' theorem (continued)

Gauss's law Gauss's law, also known as Gauss's flux theorem, is a law relating the distribution of electric charge to the resulting electric field.electric chargeelectric field The law was formulated by Carl Friedrich Gauss in 1835, but was not published until It is one of the four Maxwell's equations which form the basis of classical electrodynamics, the other three being Gauss's law for magnetism, Faraday's law of induction, and Ampère's law with Maxwell's correction. Gauss's law can be used to derive Coulomb's law, and vice versa.Carl Friedrich GaussMaxwell's equationsclassical electrodynamics Gauss's law for magnetismFaraday's law of inductionAmpère's law with Maxwell's correction Coulomb's law

Gauss's law (continued)

Divergence theorem In vector calculus, the divergence theorem, also known as Gauss's theorem or Ostrogradsky's theorem, is a result that relates the flow (that is, flux) of a vector field through a surface to the behavior of the vector field inside the surface.vector calculusflux vector fieldsurface More precisely, the divergence theorem states that the outward flux of a vector field through a closed surface is equal to the volume integral of the divergence over the region inside the surface. Intuitively, it states that the sum of all sources minus the sum of all sinks gives the net flow out of a region.fluxvolume integral divergence The divergence theorem is an important result for the mathematics of engineering, in particular in electrostatics and fluid dynamics. engineeringelectrostaticsfluid dynamics In physics and engineering, the divergence theorem is usually applied in three dimensions. However, it generalizes to any number of dimensions. In one dimension, it is equivalent to the fundamental theorem of calculus. In two dimensions, it is equivalent to Green's theorem.fundamental theorem of calculusGreen's theorem The theorem is a special case of the more general Stokes' theorem.Stokes' theorem

Divergence theorem (continued)

Ampère's law Ampère's circuital law, discovered by André- Marie Ampère in 1826, relates the integrated magnetic field around a closed loop to the electric current passing through the loop. James Clerk Maxwell derived it again using hydrodynamics in his 1861 paper On Physical Lines of Force and it is now one of the Maxwell equations, which form the basis of classical electromagnetism.André- Marie Ampèreintegrated magnetic field electric currentJames Clerk Maxwell hydrodynamicsOn Physical Lines of ForceMaxwell equationsclassical electromagnetism

Ampère's law (continued)

Faraday's law Faraday's law of induction is a basic law of electromagnetism predicting how a magnetic field will interact with an electric circuit to produce an electromotive force (EMF)—a phenomenon called electromagnetic induction. It is the fundamental operating principle of transformers, inductors, and many types of electrical motors, generators and solenoids. electromagnetismmagnetic fieldelectric circuit electromotive force (EMF) electromagnetic inductiontransformersinductorselectricalmotorsgenerators solenoids The Maxwell–Faraday equation is a generalization of Faraday's law, and forms one of Maxwell's equations.Maxwell's equations

Faraday's law (continued)

Maxwell's equations Maxwell's equations are a set of partial differential equations that, together with the Lorentz force law, form the foundation of classical electrodynamics, classical optics, and electric circuits. These fields in turn underlie modern electrical and communications technologies. Maxwell's equations describe how electric and magnetic fields are generated and altered by each other and by charges and currents. They are named after the Scottish physicist and mathematician James Clerk Maxwell, who published an early form of those equations between 1861 and 1862.partial differential equationsLorentz forceclassical electrodynamicsopticselectric circuitselectricmagnetic fieldschargescurrentsJames Clerk Maxwell

Maxwell's equations (continued) The equations have two major variants. The "microscopic" set of Maxwell's equations uses total charge and total current, including the complicated charges and currents in materials at the atomic scale; it has universal applicability but may be unfeasible to calculate. The "macroscopic" set of Maxwell's equations defines two new auxiliary fields that describe large-scale behavior without having to consider these atomic scale details, but it requires the use of parameters characterizing the electromagnetic properties of the relevant materials.atomic

(continued) Maxwell's equations The term "Maxwell's equations" is often used for other forms of Maxwell's equations. For example, space-time formulations are commonly used in high energy and gravitational physics. These formulations, defined on space-time rather than space and time separately, are manifestly compatible with special and general relativity. In quantum mechanics and analytical mechanics, versions of Maxwell's equations based on the electric and magnetic potentials are preferred. other forms space-time formulationsspace-timemanifestlyspecialgeneral relativity quantum mechanicsanalytical mechanics electricmagnetic potentials

Maxwell's equations (continued) Since the mid-20th century, it has been understood that Maxwell's equations are not exact laws of the universe, but are a classical approximation to the more accurate and fundamental theory of quantum electrodynamics. In most cases, though, quantum deviations from Maxwell's equations are immeasurably small. Exceptions occur when the particle nature of light is important or for very strong electric fields.quantum electrodynamicsparticle

(continued) Maxwell's equations

Integration by substitution In calculus, integration by substitution, also known as u-substitution, is a method for finding integrals. Using the fundamental theorem of calculus often requires finding an antiderivative. For this and other reasons, integration by substitution is an important tool for mathematicians. It is the counterpart to the chain rule of differentiation.calculus integralsfundamental theorem of calculus chain ruledifferentiation

Integration by substitution (continued)

Simpson's rule Simpson's rule is a method for numerical integration, the numerical approximation of definite integralsnumerical integration definite integrals

Perpendicular In elementary geometry, the property of being perpendicular (perpendicularity) is the relationship between two lines which meet at a right angle (90 degrees). The property extends to other related geometric objects.geometrylinesright angledegreesgeometric objects A line is said to be perpendicular to another line if the two lines intersect at a right angle. Explicitly, a first line is perpendicular to a second line if (1) the two lines meet; and (2) at the point of intersection the straight angle on one side of the first line is cut by the second line into two congruent angles. Perpendicularity can be shown to be symmetric, meaning if a first line is perpendicular to a second line, then the second line is also perpendicular to the first. For this reason, we may speak of two lines as being perpendicular (to each other) without specifying an order.intersectstraight anglecongruent anglessymmetric

Translation, rotation, stretch

Absolute value

Inequalities

Tasks 1. Prove that the lines y = sx + i and y = gx + I are perpendicular if sg = Calculate the Riemann sums and prove the result for the integral of xdx from 0 to 1.

Tasks (continued) 3. Find curvature (K) of each of these curves. a. x 2 + y 2 = 4b. x 2 = yc. xy = 1 d. Link the curvature radius (R) with the curvature (K).

(continued) Tasks 4. R is the radius-vector on a circumference. Calculate the dot-products and the cross- product. a. R. R' =... b. R'. R'' =... c. R×R'' =...

Tasks (continued)

(continued) Tasks

Tasks (continued) 7. Write the general equations of linear surface and quadratic surface.