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Topics referred to by the same term
Clairaut's formula may refer to: Clairaut's equation (mathematical analysis) Clairaut's relation (differential geometry) Clairaut's theorem (calculus)
Clairaut's_formula
Mathematical theorem
for the symmetry to hold are given by Schwarz's theorem, also called Clairaut's theorem or Young's theorem. In the context of partial differential equations
Symmetry of second derivatives
Symmetry_of_second_derivatives
International Gravity Formula 1967, the 1967 Geodetic Reference System Formula, Helmert's equation or Clairaut's formula. An alternative formula for g as a function
Gravity_of_Earth
Type of ordinary differential equation
In mathematical analysis, Clairaut's equation (or the Clairaut equation) is a differential equation of the form y ( x ) = x d y d x + f ( d y d x ) {\displaystyle
Clairaut's_equation
Length of a line segment
Descartes in 1637. The distance formula itself was first published in 1731 by Alexis Clairaut. Because of this formula, Euclidean distance is also sometimes
Euclidean_distance
Theorem about gravity
Clairaut's theorem characterizes the surface gravity on a viscous rotating ellipsoid in hydrostatic equilibrium under the action of its gravitational field
Clairaut's_theorem_(gravity)
French mathematician, astronomer, and geophysicist (1713–1765)
Alexis Claude Clairaut (/klɛərˈroʊ/; French: [alɛksi klod klɛʁo]; 13 May 1713 – 17 May 1765) was a French mathematician, astronomer, and physicist. He
Alexis_Clairaut
Approximation of a function by a polynomial
elementary tools in mathematical analysis. It gives simple arithmetic formulas to accurately compute values of many transcendental functions such as the
Taylor's_theorem
Mathematical approximation of a function
This series can be written by using sigma notation, as in the right side formula. The corresponding Taylor polynomial of degree n is T n ( x ) = ∑ k = 0
Taylor_series
Generalized chain rule in calculus
Faà di Bruno's formula is an identity in mathematics generalizing the chain rule to higher derivatives. It is named after Francesco Faà di Bruno (1855
Faà_di_Bruno's_formula
Formula in classical differential geometry
Clairaut's relation, named after Alexis Claude de Clairaut, is a formula that characterizes the great circle paths on the unit sphere. The formula states
Clairaut's relation (differential geometry)
Clairaut's_relation_(differential_geometry)
Identity relating to differential equations
In mathematics, Abel's identity (also called Abel's formula or Abel's differential equation identity) is an equation that expresses the Wronskian of two
Abel's_identity
Theorem in calculus
\end{aligned}}} In the last equality we used the Voss–Weyl coordinate formula for the divergence, although the preceding identity could be used to define
Divergence_theorem
How quickly an object undergoes movement in a circular path
Horrocks Halley Maupertuis Daniel Bernoulli Johann Bernoulli Euler d'Alembert Clairaut Lagrange Laplace Poisson Hamilton Jacobi Cauchy Routh Liouville Appell
Tangential_speed
Speed and direction of a motion
potential energy (which is always negative), is equal to zero. The general formula for the escape velocity of an object at a distance r from the center of
Velocity
Change of variable for integrals involving trigonometric functions
parametrized by angle measure onto the real line. The general transformation formula is: ∫ f ( sin x , cos x ) d x = ∫ f ( 2 t 1 + t 2 , 1 − t 2 1 + t 2
Tangent half-angle substitution
Tangent_half-angle_substitution
Operation on differential forms
}{\partial x^{i_{k}}}}\right)={\frac {1}{k!}}.} Alternatively, an explicit formula can be given for the exterior derivative of a k {\displaystyle k} -form
Exterior_derivative
Formula in calculus
In calculus, the chain rule is a formula that expresses the derivative of the composition of two differentiable functions z and y in terms of the derivatives
Chain_rule
Rules for computing derivatives of functions
\mathbb {R} } ) that return real values, although, more generally, the formulas below apply wherever they are well defined, including the case of complex
Differentiation_rules
Mathematical operation in calculus
of a function that is defined by an equation rather than by an explicit formula. If an equation such as F ( x , y ) = 0 {\displaystyle F(x,y)=0} defines
Implicit_differentiation
Technique in integral evaluation
and derivatives. The formula is used to transform one integral into another integral that is easier to compute. Thus, the formula can be read from left
Integration_by_substitution
Operation in mathematical calculus
Hasan Ibn al-Haytham, Latinized as Alhazen (c. 965 – c. 1040 AD) derived a formula for the sum of fourth powers. Alhazen determined the equations to calculate
Integral
Generalization of definite integrals to functions of multiple variables
inequality from the formula of D (and then directly transforming x2 + y2 into ρ2). The new function is simply ρ2. Applying the integration formula ∭ T ρ 2 ρ d
Multiple_integral
Matrix of partial derivatives of a vector-valued function
Power Quotient L'Hôpital's rule Inverse General Leibniz Faà di Bruno's formula Reynolds Integral Lists of integrals Integral transform Leibniz integral
Jacobian matrix and determinant
Jacobian_matrix_and_determinant
Classical statement of gravity as force
is also called Newton's constant, Newton did not use this constant or formula, he only discussed proportionality. That was sufficient to show that the
Newton's law of universal gravitation
Newton's_law_of_universal_gravitation
Use of complex numbers to evaluate integrals
calculus, Euler's formula for complex numbers may be used to evaluate integrals involving trigonometric functions. Using Euler's formula, any trigonometric
Integration using Euler's formula
Integration_using_Euler's_formula
Instantaneous rate of change (mathematics)
{\displaystyle f} in the direction v {\displaystyle \mathbf {v} } by the formula: D v f ( x ) = ∑ j = 1 n v j ∂ f ∂ x j . {\displaystyle D_{\mathbf {v}
Derivative
Divergent sum of positive unit fractions
the Euler–Maclaurin formula. Using alternating signs with only odd unit fractions produces a related series, the Leibniz formula for π ∑ n = 0 ∞ ( − 1
Harmonic_series_(mathematics)
Differential operator in mathematics
Laplace operator. This can be seen to be a special case of Lagrange's formula; see Vector triple product. For expressions of the vector Laplacian in
Laplace_operator
Distance measured along the surface of the Earth
{\displaystyle D_{\textrm {t}}} , is calculated on Spherical Earth. This formula takes into account the variation in distance between meridians with latitude
Geographical_distance
Matrix of second derivatives
Power Quotient L'Hôpital's rule Inverse General Leibniz Faà di Bruno's formula Reynolds Integral Lists of integrals Integral transform Leibniz integral
Hessian_matrix
Mathematical method in calculus
it is indeed derived using the product rule. The integration by parts formula states: ∫ a b u ( x ) v ′ ( x ) d x = [ u ( x ) v ( x ) ] a b − ∫ a b u
Integration_by_parts
Differentiation under the integral sign formula
situation (for example, in the proof of Cauchy's repeated integration formula), the Leibniz integral rule becomes: d d x ( ∫ a x f ( x , t ) d t ) =
Leibniz_integral_rule
Mathematical technique for simplification
A ) ) {\displaystyle T^{*}\mu :=\mu (T(A))} . The change of variables formula for pullback measures is ∫ T ( Ω ) g d μ = ∫ Ω g ∘ T d T ∗ μ {\displaystyle
Change_of_variables
Theorem in calculus relating line and double integrals
closed boundary. Green's theorem also yields practical boundary-integral formulas for the area and centroid of a plane region. The following is a proof of
Green's_theorem
Vector operator in vector calculus
is used to refer to an arbitrary component, such as xi. The Voss-Weyl formula, which allows the divergence to be determined using simply partial coordinate
Divergence
Generalized function whose value is zero everywhere except at zero
_{-\infty }^{\infty }dp\ \cos(px-p\alpha )\ .} Later, an infinitesimal formula for an infinitely tall, unit impulse delta function (infinitesimal version
Dirac_delta_function
Mathematical identities
dV} . Similar rules apply to algebraic and differentiation formulas. For algebraic formulas one may alternatively use the left-most vector position. Comparison
Vector_calculus_identities
Integration technique using recurrence relations
In-1 or In-2. This makes the reduction formula a type of recurrence relation. In other words, the reduction formula expresses the integral I n = ∫ f ( x
Integration by reduction formulae
Integration_by_reduction_formulae
Computation of an antiderivatives
the problem of finding a formula for the antiderivative, or indefinite integral, of a given function f(x), i.e. to find a formula for a differentiable function
Symbolic_integration
Mathematical rule for evaluating limits
_{x\to 0^{+}}-x=0.} Here is an example involving the mortgage repayment formula and 0 0 {\displaystyle {\frac {0}{0}}} . Let P {\displaystyle P} be the
L'Hôpital's_rule
Method for calculating the volume of a solid of revolution
the graph of the positive function f(x) on the interval [a, b]. Then the formula for the volume will be: 2 π ∫ a b x f ( x ) d x {\displaystyle 2\pi \int
Shell_integration
Surface created by rotating a curve about an axis
provided that x(t) is never negative between the endpoints a and b. This formula is the calculus equivalent of Pappus's centroid theorem. The quantity (
Surface_of_revolution
Integrals not expressible in closed-form from elementary functions
convergent Taylor series, its sequence of coefficients often has no elementary formula and must be evaluated term by term, with the same limitation for the integral
Nonelementary_integral
Method for partial-fraction expansion
undefined value to the expression since we do not divide by zero. General formula for a cubic denominator with three distinct roots: ℓ x 2 + m x + n ( x
Heaviside_cover-up_method
Multivariate derivative (mathematics)
particular example, under rotation of x-y coordinate system, the above formula for gradient fails to transform like a vector (gradient becomes dependent
Gradient
of integration Arbitrary constant of integration Cavalieri's quadrature formula Fundamental theorem of calculus Integration by parts Inverse chain rule
List_of_calculus_topics
{x^{n+1}}{n+1}}+C\qquad {\text{(for }}n\neq -1{\text{)}}} (Cavalieri's quadrature formula) ∫ ( a x + b ) n d x = ( a x + b ) n + 1 a ( n + 1 ) + C (for n ≠ − 1
Lists_of_integrals
Type of differential equation
solves the equation. However, it is often impossible to write down explicit formulas for solutions of partial differential equations. Hence there is a vast
Partial_differential_equation
To find the minimal surface with a given boundary
Power Quotient L'Hôpital's rule Inverse General Leibniz Faà di Bruno's formula Reynolds Integral Lists of integrals Integral transform Leibniz integral
Plateau's_problem
Branch of mathematics
find the area of a shape whose boundary is described by a complicated formula. In elementary algebra, one can calculate the distance traveled over time
Calculus
Point to which functions converge in analysis
Power Quotient L'Hôpital's rule Inverse General Leibniz Faà di Bruno's formula Reynolds Integral Lists of integrals Integral transform Leibniz integral
Limit_of_a_function
Relationship between derivatives and integrals
Conversely, if f is any integrable function, then F as given in the first formula will be absolutely continuous with F′ = f almost everywhere. The conditions
Fundamental theorem of calculus
Fundamental_theorem_of_calculus
Finite difference method for numerically solving parabolic differential equations
\alpha \in (0,1)} , may be better behaved. In expanded form, the update formula is x i + 1 = α x i + ( 1 − α ) [ x 0 + h 2 ( f ( x 0 ) + f ( x i ) ) ]
Crank–Nicolson_method
Statement relating differentiable symmetries to conserved quantities
\varphi }{\partial \varepsilon }}.} To avoid excessive complication of the formulas, this derivation assumed that the flow does not change as time passes.
Noether's_theorem
Swiss mathematician (1707–1783)
which was called "the most remarkable formula in mathematics" by Richard Feynman. A special case of the above formula is known as Euler's identity, e i π
Leonhard_Euler
Force directed to the center of rotation
local path of the object, if the path is not circular). The speed in the formula is squared, so twice the speed needs four times the force, at a given radius
Centripetal_force
On converting relations to functions of several real variables
by explicit formulas. It guarantees that g1(x) and g2(x) are differentiable, and it even works in situations where we do not have a formula for f(x, y)
Implicit_function_theorem
In mathematics, the exponential response formula (ERF), also known as exponential response and complex replacement, is a method used to find a particular
Exponential_response_formula
Circulation density in a vector field
{u}} } as their normal. C is oriented via the right-hand rule. The above formula means that the component of the curl of a vector field along a certain
Curl_(mathematics)
Theorem in mathematics
Power Quotient L'Hôpital's rule Inverse General Leibniz Faà di Bruno's formula Reynolds Integral Lists of integrals Integral transform Leibniz integral
Mean_value_theorem
Physics problem related to laws of motion and gravity
gravitating bodies are not integrable and cannot be solved to give explicit formulas for the positions of the bodies as a function of time. For most initial
Three-body_problem
Derivative of a function with multiple variables
that set); in this case, the partial derivatives can be exchanged by Clairaut's theorem: ∂ 2 f ∂ x i ∂ x j = ∂ 2 f ∂ x j ∂ x i . {\displaystyle {\frac
Partial_derivative
Conditions for switching order of integration in calculus
x=\int _{Y}\left(\int _{X}f(x,y)\,\mathrm {d} x\right)\mathrm {d} y.} This formula is generally not true for the Riemann integral (however, it is true if
Fubini's_theorem
Mathematical techniques used in probability theory and related fields
F\in L^{\infty -0}(\Omega ,{\mathcal {F}},P)} the integration by parts formula E [ D h F ] = E [ M W ( h ) F ] = E [ W ( h ) F ] {\displaystyle \mathbb
Malliavin_calculus
Integration over a non-flat region in 3D space
scalar, vector, or tensor field defined on a surface S. To find an explicit formula for the surface integral of f over S, we need to parameterize S by defining
Surface_integral
Vector calculus formulas relating the bulk with the boundary of a region
expression of the Huygens principle, and leads to Kirchhoff's diffraction formula and other approximations. Green's identities hold on a Riemannian manifold
Green's_identities
Special case of the Euler-Lagrange equations
two points of equal height and at distance D {\displaystyle D} . By the formula for arc length, l = ∫ S d S = ∫ s 1 s 2 1 + y ′ 2 d x , {\displaystyle
Beltrami_identity
Differential calculus on function spaces
Power Quotient L'Hôpital's rule Inverse General Leibniz Faà di Bruno's formula Reynolds Integral Lists of integrals Integral transform Leibniz integral
Calculus_of_variations
Method of evaluating certain integrals along paths in the complex plane
function in the form for direct application of the formula. Then, by using Cauchy's integral formula, ∮ C f ( z ) d z = ∮ C 1 ( z + i ) 2 ( z − i ) 2 d
Contour_integration
Scottish mathematician (1698–1746)
Clairaut, Maupertuis, and d'Ortous de Mairan. Independently from Euler and using the same methods, Maclaurin discovered the Euler–Maclaurin formula.
Colin_Maclaurin
specific names, area by area. Ablowitz-Kaup-Newell-Segur (AKNS) system Clairaut's equation Hypergeometric differential equation Jimbo–Miwa–Ueno isomonodromy
List of named differential equations
List_of_named_differential_equations
Theorem in vector calculus
machinery of geometric measure theory; for that approach see the coarea formula. In this article, we instead use a more elementary definition, based on
Stokes'_theorem
Branch of mathematics
Fourier transform were used by Alexis Clairaut in 1754 to compute an orbit, which has been described as the first formula for the DFT, and in 1759 by Joseph
Fourier_analysis
Rate of change of the second derivative
Power Quotient L'Hôpital's rule Inverse General Leibniz Faà di Bruno's formula Reynolds Integral Lists of integrals Integral transform Leibniz integral
Third_derivative
Mathematical theorem, used in calculus
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
Mathematical operation in calculus
analysis, the logarithmic derivative of a function f is defined by the formula f ′ f {\displaystyle {\frac {f'}{f}}} where f′ is the derivative of f.
Logarithmic_derivative
Infinite sum
}{\frac {\left(-1\right)^{n}}{2n-1}}=-{\frac {\pi }{4}},} the Leibniz formula for π . {\displaystyle \pi .} A telescoping series ∑ n = 1 ∞ ( b n − b
Series_(mathematics)
Geometric figure which approximates the Earth's shape
Gravimetry is another technique for determining Earth's flattening, as per Clairaut's theorem. Modern geodesy no longer uses simple meridian arcs or ground
Earth_ellipsoid
Approach to finding numerical solutions of ordinary differential equations
differential equation. Here, a differential equation can be thought of as a formula by which the slope of the tangent line to the curve can be computed at
Euler_method
Type of ordinary differential equation
Inspection Method of characteristics Ansatz Euler Exponential response formula Finite difference Crank–Nicolson Finite element Infinite element Finite
Bernoulli differential equation
Bernoulli_differential_equation
Mathematical operation
{\frac {d^{2}y}{dx^{2}}}.} This notation is derived from the following formula: d 2 y d x 2 = d d x ( d y d x ) . {\displaystyle {\frac {d^{2}y}{dx^{2}}}\
Second_derivative
Energy of a moving physical body
physical situation. For objects and processes in common human experience, the formula 1/2mv2 given by classical mechanics is suitable. However, if the speed
Kinetic_energy
Formula for the derivative of an inverse function
In calculus, the inverse function rule is a formula that expresses the derivative of the inverse of a bijective and differentiable function f in terms
Inverse_function_rule
Turning force around an axis
(lbf⋅ft) for torque and rpm for rotational speed. This results in the formula changing to: P h p = τ l b f ⋅ f t ⋅ 2 π r a d / r e v ⋅ ν r e v / m i
Torque
Study of rates of change
Power Quotient L'Hôpital's rule Inverse General Leibniz Faà di Bruno's formula Reynolds Integral Lists of integrals Integral transform Leibniz integral
Differential_calculus
the major axis length: 1/(2a). Also, geodesics on an unduloid obey the Clairaut relation, and their behavior is therefore predictable. Unduloids are not
Unduloid
State of balance between external forces on a fluid and internal pressure gradient
cuboid is equal to the area of the top or bottom, times the height – the formula for finding the volume of a cube. F weight = − ρ g A h {\displaystyle
Hydrostatic_equilibrium
Existence and uniqueness of solutions to initial value problems
Inspection Method of characteristics Ansatz Euler Exponential response formula Finite difference Crank–Nicolson Finite element Infinite element Finite
Picard–Lindelöf_theorem
Mathematical notation
Multi-index notation is a mathematical notation that simplifies formulas used in multivariable calculus, partial differential equations and the theory
Multi-index_notation
Gradient conjecture Recurrence plot Limit cycle Initial value problem Clairaut's equation Singular solution Poincaré–Bendixson theorem Riccati equations
List of dynamical systems and differential equations topics
List_of_dynamical_systems_and_differential_equations_topics
Family of implicit and explicit iterative methods
all and not passed to function f {\displaystyle f} , with only the final formula for t n + 1 {\displaystyle t_{n+1}} used. The family of explicit Runge–Kutta
Runge–Kutta_methods
Formula for the derivative of a ratio of functions
h''=\left({\frac {f}{g}}\right)''={\frac {f''-g''h-2g'h'}{g}}.} Chain rule – Formula in calculus Differentiation of integrals – Problem of the derivative of
Quotient_rule
Differential equations involving stochastic processes
Inspection Method of characteristics Ansatz Euler Exponential response formula Finite difference Crank–Nicolson Finite element Infinite element Finite
Stochastic differential equation
Stochastic_differential_equation
Formula for the derivative of a product
calculus, the product rule (or Leibniz rule or Leibniz product rule) is a formula used to find the derivatives of products of two or more functions. For
Product_rule
Two Advanced Placement courses and exams
Power Quotient L'Hôpital's rule Inverse General Leibniz Faà di Bruno's formula Reynolds Integral Lists of integrals Integral transform Leibniz integral
AP_Calculus
Object movement along a circular path
.} Circular motion can be described using complex numbers and Euler's formula. Let the x axis be the real axis and the y {\displaystyle y} axis be the
Circular_motion
Book by Joseph-Louis Lagrange
harmonious system, the scattered developments of contributors such as Alexis Clairaut, Jean le Rond d'Alembert, Pierre-Simon Laplace, Leonhard Euler, and Johann
Mécanique_analytique
Time-varying quantity or variable
Power Quotient L'Hôpital's rule Inverse General Leibniz Faà di Bruno's formula Reynolds Integral Lists of integrals Integral transform Leibniz integral
Fluent_(mathematics)
German mathematician (1843–1921)
Schwarz minimal surface Schwarz theorem (also known as Clairaut's theorem) Schwarz integral formula Schwarz–Christoffel mapping Schwarz–Ahlfors–Pick theorem
Hermann_Schwarz
CLAIRAUTS FORMULA
CLAIRAUTS FORMULA
Boy/Male
Hindu, Indian
King of Enchanting Formulas
CLAIRAUTS FORMULA
CLAIRAUTS FORMULA
Boy/Male
Hindu, Indian, Marathi
Lord of Tolerance
Boy/Male
English
Falconer; one who trains falcons.
Girl/Female
English
A well-established compound of Jo-.
Boy/Male
Indian
Female sheep name of a Saha
Girl/Female
Tamil
River
Surname or Lastname
English
English : variant spelling of Edsall.
Surname or Lastname
English
English : from Middle English riggewey, hence a topographic name for someone who lived by such a route or a habitational name from any of various places so named, for example in Cheshire, Derbyshire, Dorset, and Staffordshire.
Girl/Female
Muslim/Islamic
Royal lady Princess
Boy/Male
British, English
From the Field Estate
Boy/Male
Indian, Punjabi, Sikh
Love of Glory
CLAIRAUTS FORMULA
CLAIRAUTS FORMULA
CLAIRAUTS FORMULA
CLAIRAUTS FORMULA
CLAIRAUTS FORMULA
n.
A rule or principle expressed in algebraic language; as, the binominal formula.
v. t.
To reduce to a forula; to formulate.
pl.
of Formulary
imp. & p. p.
of Formulate
v. t.
To reduce to, or express in, a formula; to put in a clear and definite form of statement or expression.
n.
The doctrine, as formulated by Luther, that Christ's glorified body is omnipresent.
n.
Prescribed form or model; formula.
n.
The act of formularizing; a formularized or formulated statement or exhibition.
p. pr. & vb. n.
of Formulate
n.
Accumulated and established knowledge, which has been systematized and formulated with reference to the discovery of general truths or the operation of general laws; knowledge classified and made available in work, life, or the search for truth; comprehensive, profound, or philosophical knowledge.
n.
A prayer; an invocation; a religious formula; a charm.
pl.
of Formula
v. i.
To plead against each other, or go to trial between themselves, as the claimants in an in an interpleader. See Interpleader.
a.
Pertaining to, or illustrating, the hypothetical space relations of atoms in the molecule; as, a stereo-chemic formula.
n.
The right of taking a profit in the land of another, in common either with the owner or with other persons; -- so called from the community of interest which arises between the claimant of the right and the owner of the soil, or between the claimants and other commoners entitled to the same right.
n.
The act, process, or result of formulating or reducing to a formula.
n.
A book containing stated and prescribed forms, as of oaths, declarations, prayers, medical formulaae, etc.; a book of precedents.
v. t.
To formulate into a theorem.
pl.
of Formula
a.
Pertaining to, or exhibiting, formularization.