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Integration over a non-flat region in 3D space
calculus, a surface integral is a generalization of multiple integrals to integration over surfaces. It can be thought of as the double integral analogue
Surface_integral
Operation in mathematical calculus
curve connecting two points in space. In a surface integral, the curve is replaced by a piece of a surface in three-dimensional space. The first documented
Integral
Mathematical concept applicable to physics
a scalar quantity, defined as the surface integral of the perpendicular component of a vector field over a surface. The word flux comes from Latin: fluxus
Flux
Definite integral of a scalar or vector field along a path
mathematics, a line integral is an integral where the function to be integrated is evaluated along a curve. The terms path integral, curve integral, and curvilinear
Line_integral
Theorem in calculus
closed surface to the divergence of the field in the volume enclosed. More precisely, the divergence theorem states that the surface integral of a vector
Divergence_theorem
Circulation density in a vector field
calculus is Stokes' theorem, which relates the surface integral of the curl of a vector field to the line integral of the vector field around the boundary curve
Curl_(mathematics)
Geometric model of the physical space
{r} .} A surface integral is a generalization of multiple integrals to integration over surfaces. It can be thought of as the double integral analog of
Three-dimensional_space
Branch of mathematics
differential calculus and integral calculus. Differential calculus studies instantaneous rates of change and slopes of curves; integral calculus studies accumulation
Calculus
Differentiation under the integral sign formula
Leibniz integral rule or the Leibniz rule for differentiation under the integral sign, named after Gottfried Wilhelm Leibniz, states that for an integral of
Leibniz_integral_rule
Theorem in vector calculus
relates the integral of the curl of the vector field over some surface, to the line integral of the vector field around the boundary of the surface. The classical
Stokes'_theorem
Integral over a 3-D domain
calculus), a volume integral (∭) is an integral over a 3-dimensional domain; that is, it is a special case of multiple integrals. Volume integrals are especially
Volume_integral
Mathematical identities
The following are important identities involving derivatives and integrals in vector calculus. For a function f ( x , y , z ) {\displaystyle f(x,y,z)}
Vector_calculus_identities
Method to solve scalar wave equation
The Kirchhoff integral theorem (sometimes referred to as the Fresnel–Kirchhoff integral theorem) is a surface integral to obtain the value of the solution
Kirchhoff_integral_theorem
Surface integral of the magnetic field
the magnetic flux through a surface is the surface integral of the normal component of the magnetic field B over that surface. It is usually denoted Φ or
Magnetic_flux
Concept of mass used in general relativity
b}\;dA} To make this demonstration, we need to express this surface integral as a volume integral. In flat space-time, we would use Stokes theorem and integrate
Komar_mass
Method of evaluating certain integrals along paths in the complex plane
To solve multivariable contour integrals (i.e. surface integrals, complex volume integrals, and higher order integrals), we must use the divergence theorem
Contour_integration
Mathematical symbol used to denote integrals and antiderivatives
The integral symbol (see below) is used to denote integrals and antiderivatives in mathematics, especially in calculus. ∫ (Unicode), ∫ {\displaystyle
Integral_symbol
Basic integral in elementary calculus
analysis, the Riemann integral is a rigorous definition of the integral of a function on an interval. It defines the integral by approximating the region
Riemann_integral
Conditions for switching order of integration in calculus
theorem gives the conditions under which a double integral can be computed as an iterated integral, i.e. by integrating in one variable at a time. Intuitively
Fubini's_theorem
Statement about integration on manifolds
to the line integral of the vector field over the surface boundary. The second fundamental theorem of calculus states that the integral of a function
Generalized_Stokes_theorem
Vector operator in vector calculus
point x0 is defined as the limit of the ratio of the surface integral of F out of the closed surface of a volume V enclosing x0 to the volume of V, as V
Divergence
Differential operator in mathematics
\textstyle \int _{{\text{shell}}_{R}}f({\vec {r}})dr^{n-1}} is the surface integral over an n-sphere of radius R {\displaystyle R} , and A n − 1 {\displaystyle
Laplace_operator
Equations describing classical electromagnetism
\iint _{\Sigma }} is a surface integral over the surface Σ, The total electric charge Q enclosed in Ω is the volume integral over Ω of the charge density
Maxwell's_equations
Equation describing the transport of some quantity
is an imaginary surface S, then the surface integral of flux over S is equal to the amount of q that is passing through the surface S per unit time:
Continuity_equation
Generalization of definite integrals to functions of multiple variables
the x-axis, the double integral of a positive function of two variables represents the volume of the region between the surface defined by the function
Multiple_integral
Relationship between derivatives and integrals
to curve and surface integrals in higher dimensions and on manifolds. One such generalization offered by the calculus of moving surfaces is the time evolution
Fundamental theorem of calculus
Fundamental_theorem_of_calculus
Calculus of functions of several variables
line integrals, surface integrals and volume integrals. Due to the non-uniqueness of these integrals, an antiderivative or indefinite integral cannot
Multivariable_calculus
Electric charge per unit length, area or volume
_{q}(\mathbf {r} )\,d\ell } similarly a surface integral of the surface charge density σq(r) over a surface S, Q = ∫ S σ q ( r ) d S {\displaystyle Q=\int
Charge_density
Multivariate derivative (mathematics)
(continuous) gradient field is always a conservative vector field: its line integral along any path depends only on the endpoints of the path, and can be evaluated
Gradient
Method of mathematical integration
In mathematics, the integral of a non-negative function of a single variable can be regarded, in the simplest case, as the area between the graph of that
Lebesgue_integral
Closed surface in three-dimensional space
simplify the calculation of the surface integral. If the Gaussian surface is chosen such that for every point on the surface the component of the electric
Gaussian_surface
Matrix of partial derivatives of a vector-valued function
determinant is fundamentally used for changes of variables in multiple integrals. Let f : R n → R m {\textstyle \mathbf {f} :\mathbb {R} ^{n}\to \mathbb
Jacobian matrix and determinant
Jacobian_matrix_and_determinant
Surface created by rotating a curve about an axis
{\displaystyle 0\leq \theta \leq 2\pi } . Then the surface area is given by the surface integral A x = ∬ S d S = ∬ [ a , b ] × [ 0 , 2 π ] ‖ ∂ r ∂ t
Surface_of_revolution
Indefinite integral
antiderivative, inverse derivative, primitive function, primitive integral or indefinite integral of a function f is a differentiable function F whose derivative
Antiderivative
Theorem in mathematics
theorem, in integral form, as an instant reflex but this use requires the continuity of the derivative. If one uses the Henstock–Kurzweil integral one can
Mean_value_theorem
Study of rates of change
calculus, the other being integral calculus—the study of accumulation or area beneath a curve.Differential calculus and integral calculus are connected by
Differential_calculus
Mathematical notion of infinitesimal difference
integrator in a Stieltjes integral is represented as the differential of a function. Formally, the differential appearing under the integral behaves exactly as
Differential_(mathematics)
Integrals not expressible in closed-form from elementary functions
antiderivative of a given elementary function is an antiderivative (or indefinite integral) that is, itself, not an elementary function. A theorem by Liouville in
Nonelementary_integral
Derivative of a function with multiple variables
{\displaystyle {\frac {\partial z}{\partial x}}=2x+y.} The so-called partial integral can be taken with respect to x (treating y as constant, in a similar manner
Partial_derivative
Vector calculus formulas relating the bulk with the boundary of a region
where A {\displaystyle {\mathcal {A}}} is the area of the surface S {\displaystyle S} . The integral can be simplified to ψ ( η ) = ⟨ ψ ⟩ S − ∮ ∂ U G ( y
Green's_identities
Restatement of Newton's law of universal gravitation
Friedrich Gauss. It states that the flux (surface integral) of the gravitational field over any closed surface is proportional to the mass enclosed. Gauss's
Gauss's_law_for_gravity
Integration is the basic operation in integral calculus. While differentiation has straightforward rules by which the derivative of a complicated function
Lists_of_integrals
Mapping involving integration between function spaces
In mathematics, an integral transform is a type of transformation that maps a function from its original function space into another function space via
Integral_transform
Instantaneous rate of change (mathematics)
Geometry of Curves and Surfaces with Mathematica, CRC Press, ISBN 978-1-58488-448-4 Guzman, Alberto (2003), Derivatives and Integrals of Multivariable Functions
Derivative
Mathematical theorem, used in calculus
In mathematics, integrals of inverse functions can be computed by means of a formula that expresses the antiderivatives of the inverse f − 1 {\displaystyle
Integral_of_inverse_functions
Theorem in calculus relating line and double integrals
Green's theorem relates a line integral around a simple closed curve C to a double integral over the plane region D (surface in R 2 {\displaystyle \mathbb
Green's_theorem
Branch of mathematical analysis
derivatives and integrals. Let f ( x ) {\displaystyle f(x)} be a function defined for x > 0 {\displaystyle x>0} . Form the definite integral from 0 to x {\displaystyle
Fractional_calculus
Calculus on stochastic processes
disciplines). The Stratonovich integral can readily be expressed in terms of the Itô integral, and vice versa. Stochastic integrals do NOT obey the usual chain
Stochastic_calculus
Concept in mathematical analysis
improper integral is an extension of the notion of a definite integral to cases that violate the usual assumptions for that kind of integral. In the context
Improper_integral
Matrix of second derivatives
Geometric Definitions Partial derivative Multiple integral Line integral Surface integral Volume integral Jacobian Hessian Theorems Clairaut's Fubini's Advanced
Hessian_matrix
Amount of charge flowing through a unit cross-sectional area per unit time
{\displaystyle \mathbf {j} =\rho \mathbf {v} .} The surface integral of j over a surface S, followed by an integral over the time duration t1 to t2, gives the
Current_density
Notation of differential calculus
second integral, f ( − 3 ) ( x ) {\displaystyle f^{(-3)}(x)} for the third integral, and f ( − n ) ( x ) {\displaystyle f^{(-n)}(x)} for the nth integral. Dxy
Notation_for_differentiation
Test for infinite series of monotonous terms for convergence
In mathematics, the integral test for convergence is a method used to test infinite series of monotonic terms for convergence. It was developed by Colin
Integral_test_for_convergence
Mathematical criterion about whether a series converges
the integral diverges, then the series does so as well. In other words, the series a n {\displaystyle {a_{n}}} converges if and only if the integral converges
Convergence_tests
Integral of sin(x)/x from 0 to infinity
several integrals known as the Dirichlet integral, after the German mathematician Peter Gustav Lejeune Dirichlet, one of which is the improper integral of
Dirichlet_integral
Formula for the derivative of a ratio of functions
rule – Formula in calculus Differentiation of integrals – Problem of the derivative of the mean value integral Differentiation rules – Rules for computing
Quotient_rule
Calculus of vector-valued functions
Geometric Definitions Partial derivative Multiple integral Line integral Surface integral Volume integral Jacobian Hessian Theorems Clairaut's Fubini's Advanced
Vector_calculus
Commonly encountered and tricky integral
The integral of secant cubed is a frequent and challenging indefinite integral of elementary calculus. Integral of sec³x is as follows: ∫ sec 3 x d
Integral_of_secant_cubed
3D generalization of the Leibniz integral rule
Reynolds (1842–1912), is a three-dimensional generalization of the Leibniz integral rule. It is used to recast time derivatives of integrated quantities and
Reynolds_transport_theorem
Formula in calculus
Integration by substitution – Technique in integral evaluation Leibniz integral rule – Differentiation under the integral sign formula Product rule – Formula
Chain_rule
Divergent sum of positive unit fractions
can also be proven to diverge by comparing the sum to an integral, according to the integral test for convergence. Applications of the harmonic series
Harmonic_series_(mathematics)
the integral sign Trigonometric substitution Partial fractions in integration Quadratic integral Proof that 22/7 exceeds π Trapezium rule Integral of the
List_of_calculus_topics
Test for series convergence
non-negative monotonically decreasing function, then the integral of fg is a convergent improper integral. Démonstration d’un théorème d’Abel. Journal de mathématiques
Dirichlet's_test
Two Advanced Placement courses and exams
AP Calculus AB covers basic introductions to limits, derivatives, and integrals. AP Calculus BC covers all AP Calculus AB topics plus integration by parts
AP_Calculus
Approximation of a function by a polynomial
in the sense of Riemann integral provided the (k + 1)th derivative of f is continuous on the closed interval [a,x]. Integral form of the remainder—Let
Taylor's_theorem
Concept in classical electromagnetism
density only, ∮C is the closed line integral around the closed curve C, ∬S denotes a surface integral over the surface S bounded by the curve C, · is the
Ampère's_circuital_law
Evaluates a line integral through a gradient field using the original scalar field
also known as the fundamental theorem of calculus for line integrals, says that a line integral through a gradient field can be evaluated by evaluating the
Gradient_theorem
Change of time of the value of an integral
applications, one needs to calculate the rate of change of a volume or surface integral whose domain of integration, as well as the integrand, are functions
Time_evolution_of_integrals
On converting relations to functions of several real variables
Geometric Definitions Partial derivative Multiple integral Line integral Surface integral Volume integral Jacobian Hessian Theorems Clairaut's Fubini's Advanced
Implicit_function_theorem
Property of space that quantifies the magnetic influence at a given location
\mathbf {A} =-q_{\mathrm {m} }\,,} where the integral is a closed surface integral over the closed surface S and qM is the "magnetic charge" (in units
Magnetic_field
Method for partial-fraction expansion
In integral calculus we would want to write a fractional algebraic expression as the sum of its partial fractions in order to take the integral of each
Heaviside_cover-up_method
Concept in 3-dimensional geometry
bounded surface in three dimensions can be associated with a unique area vector called its vector area. It is equal to the surface integral of the surface normal
Vector_area
Rate at which electrical energy is transferred by an electric circuit
equation P = IV may be replaced by a more complex calculation. The closed surface integral of the cross-product of the electric field intensity and magnetic field
Electric_power
Buoyancy principle in fluid dynamics
surface of the body which is in contact with the fluid: B = ∮ σ d A . {\displaystyle \mathbf {B} =\oint \sigma \,d\mathbf {A} .} The surface integral
Archimedes'_principle
Theorem in magnetohydrodynamics
the surface integral over S3, the differential surface element dS3 = dl × v δt where dl is the line element around the boundary ∂S1 of the surface S1.
Alfvén's_theorem
Mathematical approximation of a function
series for arctan x, tan x, sec x, ln sec x (the integral of tan), ln tan 1/2(1/2π + x) (the integral of sec, the inverse Gudermannian function), arcsec(√2
Taylor_series
Theorem in classical electromagnetism
far away. In this case, if one integrates throughout space then the surface-integral terms cancel (see below) and one obtains: ∫ J 1 ⋅ E 2 d V = ∫ E 1 ⋅
Reciprocity (electromagnetism)
Reciprocity_(electromagnetism)
Mathematical operation in calculus
exp ( ∫ F ) {\displaystyle \exp \textstyle (\int F)} with any indefinite integral of F.[citation needed] The formula as given can be applied more widely;
Logarithmic_derivative
Rules for computing derivatives of functions
of integrals – Problem of the derivative of the mean value integral Differentiation under the integral sign – Differentiation under the integral sign
Differentiation_rules
Foundational law of electromagnetism relating electric field and charge distributions
resulting electric field. In its integral form, it states that the flux of the electric field out of an arbitrary closed surface is proportional to the electric
Gauss's_law
Technique in integral evaluation
reverse chain rule or change of variables, is a method for evaluating integrals and antiderivatives. It is the counterpart to the chain rule for differentiation
Integration_by_substitution
Antiderivative of the secant function
In calculus, the integral of the secant function can be evaluated using a variety of methods and there are multiple ways of expressing the antiderivative
Integral of the secant function
Integral_of_the_secant_function
Measure of electric field through surface
perpendicular to the field). The electric flux over a surface is therefore given by the surface integral: Φ E = ∬ S E ⋅ d A {\displaystyle \Phi _{\text{E}}=\iint
Electric_flux
Upward force that opposes the weight of an object immersed in fluid
surface of the body which is in contact with the fluid: B = ∮ σ d A . {\displaystyle \mathbf {B} =\oint \sigma \,d\mathbf {A} .} The surface integral
Buoyancy
In mathematics, the definite integral ∫ a b f ( x ) d x {\displaystyle \int _{a}^{b}f(x)\,dx} is the area of the region in the xy-plane bounded by the
List_of_definite_integrals
Type of derivative in mathematics
Ronald D. Kriz (2007) Envisioning total derivatives of scalar functions of two dimensions using raised surfaces and tangent planes from Virginia Tech
Derivative (multivariable calculus)
Derivative_(multivariable_calculus)
Convergence test for infinite series
→ 2 n f ( 2 n ) {\textstyle f(n)\rightarrow 2^{n}f(2^{n})} recalls the integral variable substitution x → e x {\textstyle x\rightarrow e^{x}} yielding
Cauchy_condensation_test
Course designed to prepare students for calculus
analysis and analytic geometry preliminary to the study of differential and integral calculus." He began with the fundamental concepts of variables and functions
Precalculus
Test for convergence of alternating series
Geometric Definitions Partial derivative Multiple integral Line integral Surface integral Volume integral Jacobian Hessian Theorems Clairaut's Fubini's Advanced
Alternating_series_test
integral (named after Hermann Weyl) is an operator defined, as an example of fractional calculus, on functions f on the unit circle having integral 0
Weyl_integral
Determining convergence in mathematics
whether an infinite series or an improper integral converges or diverges by comparing the series or integral to one whose convergence properties are known
Direct_comparison_test
Generalization of the concept of directional derivative
F(u+h)-F(u)=\int _{0}^{1}dF(u+th;h)\,dt} where the integral is the Gelfand–Pettis integral (the weak integral) (Vainberg (1964)). Many of the other familiar
Gateaux_derivative
Formula for the derivative of a product
therefore for all natural n. Differentiation of integrals – Problem of the derivative of the mean value integral Differentiation of trigonometric functions –
Product_rule
named after Vyacheslav Ivanovich Lebedev, is an approximation to the surface integral of a function over a three-dimensional sphere. The grid is constructed
Lebedev_quadrature
Certain vector fields are the sum of an irrotational and a solenoidal vector field
mathematically correct since the last integral diverges as ln R at R tends to infinity. This divergence of the integral is significant for the electromagnetic
Helmholtz_decomposition
To find the minimal surface with a given boundary
Sheldon Xu-Dong (1988), "Two-dimensional area minimizing integral currents are classical minimal surfaces", Journal of the American Mathematical Society, 1 (4):
Plateau's_problem
Partial differential equation describing the evolution of temperature in a region
divergence theorem, the previous surface integral for heat flow into V can be transformed into the volume integral q t ( V ) = − ∫ ∂ V H ( x ) ⋅ n (
Heat_equation
Historically important optical effect
This means that the field at a point P1 on the screen is given by a surface integral: U ( P 1 ) = A e i k r 0 r 0 ∫ S e i k r 1 r 1 K ( χ ) d S , {\displaystyle
Arago_spot
Scientific principles enabling the use of the calculus of variations
Kiyohisa Tokunaga, "Variational Principle for Electromagnetic Field". Total Integral for Electromagnetic Canonical Action, Part Two, Relativistic Canonical
Variational_principle
Operation on differential forms
_{V}} is locally the scalar triple product with V {\displaystyle V} .) The integral of ω V {\displaystyle \omega _{V}} over a hypersurface is the flux of V
Exterior_derivative
SURFACE INTEGRAL
SURFACE INTEGRAL
Boy/Male
Irish
Surname.
Surname or Lastname
Probably an Americanized spelling of the Swiss German surname Bunz (see Bunce).English
Probably an Americanized spelling of the Swiss German surname Bunz (see Bunce).English : possibly a variant of Bunt.
Boy/Male
Irish American Welsh Scandinavian Scottish English
Surname.
Boy/Male
Irish
Surname.
Boy/Male
Irish
Surname.
Boy/Male
Irish American Biblical Hebrew
Surname.
Boy/Male
Irish
Surname.
Boy/Male
Irish American English
Surname.
Boy/Male
Irish
Surname.
Boy/Male
Irish
Surname.
Boy/Male
Irish Gaelic
Surname.
Boy/Male
Indian
Part of Sun
Surname or Lastname
English (Cumbria and Durham)
English (Cumbria and Durham) : variant spelling of Furness.
Boy/Male
Irish
Surname.
Boy/Male
Irish
Surname.
Boy/Male
Irish American Welsh
Surname.
Boy/Male
Irish Gaelic
Surname.
Boy/Male
Indian, Sanskrit
Surface of the Earth
Boy/Male
Scottish American English
Surname.
Boy/Male
Irish
Surname.
SURFACE INTEGRAL
SURFACE INTEGRAL
Girl/Female
British, English
God's Angel
Boy/Male
Australian, Gaelic, Irish
Dark One
Girl/Female
Welsh
Legendary daughter of Tuduathar.
Boy/Male
Hindu, Indian, Telugu
Name of Deity in Ahobilam
Girl/Female
Hindu
Girl/Female
Tamil
Boy/Male
Indian, Punjabi, Sikh
Lord of Ocean
Girl/Female
Assamese, Hindu, Indian, Kannada, Marathi, Sindhi, Telugu
Another Name for Saraswathi
Boy/Male
Biblical
A rejoicing; our proud lord.
Girl/Female
Indian
Desire
SURFACE INTEGRAL
SURFACE INTEGRAL
SURFACE INTEGRAL
SURFACE INTEGRAL
SURFACE INTEGRAL
n.
To throw out, or exhale, as from a furnace; also, to put into a furnace.
n.
That part of the side which is terminated by the flank prolonged, and the angle of the nearest bastion.
v. t.
To name or call by an appellation added to the original name; to give a surname to.
a.
meeting a curve or surface at a point and having at that point the same direction as the curve or surface; -- said of a straight line, curve, or surface; as, a line tangent to a curve; a curve tangent to a surface; tangent surfaces.
n.
Hence, outward or external appearance.
imp. & p. p.
of Surface
a.
Having the surface smooth and polished; -- said of leaves, the surfaces of shells, etc.
v. t.
To work over the surface or soil of, as ground, in hunting for gold.
n.
An instrument for gauging or testing a plane surface. See Surface gauge, under Surface.
v. t.
To give a surface to; especially, to cause to have a smooth or plain surface; to make smooth or plain.
n.
An inclosed place in which heat is produced by the combustion of fuel, as for reducing ores or melting metals, for warming a house, for baking pottery, etc.; as, an iron furnace; a hot-air furnace; a glass furnace; a boiler furnace, etc.
p. pr. & vb. n.
of Surface
n.
A form of machine for dressing the surface of wood, metal, stone, etc.
n.
A magnitude that has length and breadth without thickness; superficies; as, a plane surface; a spherical surface.
n.
Alt. of Serfdom
n.
Surface; superficies; externality.
n.
Surface; body; substance.
n.
The exterior part of anything that has length and breadth; one of the limits that bound a solid, esp. the upper face; superficies; the outside; as, the surface of the earth; the surface of a diamond; the surface of the body.