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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
Calculus of vector-valued functions
Vector calculus or vector analysis is a branch of mathematics concerned with the differentiation and integration of vector fields, primarily in three-dimensional
Vector_calculus
Vector calculus formulas relating the bulk with the boundary of a region
In mathematics, Green's identities are a set of three identities in vector calculus relating the bulk with the boundary of a region on which differential
Green's_identities
such as dot product, cross product, etc. Vector calculus identities — regarding operations on vector fields such as divergence, gradient, curl, etc. This
Lists_of_vector_identities
Formulas about vectors in three-dimensional Euclidean space
The following are important identities in vector algebra. Identities that only involve the magnitude of a vector ‖ A ‖ {\displaystyle \|\mathbf {A} \|}
Vector_algebra_relations
trigonometric functions Logarithmic identities Summation identities Vector calculus identities List of inequalities List of set identities and relations – Equalities
List of mathematical identities
List_of_mathematical_identities
Specialized notation for multivariable calculus
matrix calculus into two separate groups. The two groups can be distinguished by whether they write the derivative of a scalar with respect to a vector as
Matrix_calculus
Rules for computing derivatives of functions
Matrix calculus – Specialized notation for multivariable calculus Trigonometric functions – Functions of an angle Vector calculus identities – Mathematical
Differentiation_rules
Vector differential operator
or nabla, is an operator used in mathematics (particularly in vector calculus) as a vector differential operator, usually represented by ∇ (the nabla symbol)
Del
Multivariate derivative (mathematics)
In vector calculus, the gradient of a scalar-valued differentiable function f {\displaystyle f} of several variables is the vector field (or vector-valued
Gradient
Assignment of a vector to each point in a subset of Euclidean space
In vector calculus and physics, a vector field is an assignment of a vector to each point in a space, most commonly Euclidean space R n {\displaystyle
Vector_field
Type of derivative in mathematics
one-variable calculus, this is the tangent line approximation. In multivariable calculus, the same property is generalized to define the derivative of a vector-valued
Derivative (multivariable calculus)
Derivative_(multivariable_calculus)
Circulation density in a vector field
In vector calculus, the curl, also known as rotor, is a vector operator that describes the infinitesimal circulation of a vector field in three-dimensional
Curl_(mathematics)
Vector field that is the gradient of some function
In vector calculus, a conservative vector field is a vector field that is the gradient of some function. A conservative vector field has the property
Conservative_vector_field
Tensor index notation for tensor-based calculations
manipulating indices, such as using index notation to verify vector calculus identities or identities of the Kronecker delta and Levi-Civita symbol (see also
Ricci_calculus
Calculus of functions of several variables
calculus in three dimensional space is often called vector calculus. In single-variable calculus, operations like differentiation and integration are
Multivariable_calculus
Formula for the derivative of a ratio of functions
descriptions of redirect targets Vector calculus identities – Mathematical identities Stewart, James (2008). Calculus: Early Transcendentals (6th ed.)
Quotient_rule
Infinitesimal calculus on functions defined on a geometric algebra
and can be shown to reproduce other mathematical theories including vector calculus, differential geometry, and differential forms. With a geometric algebra
Geometric_calculus
This article summarizes several identities in exterior calculus, a mathematical calculus used in differential geometry. The following notation is used
Exterior_calculus_identities
Instantaneous rate of change (mathematics)
variables, with the others held constant. Partial derivatives are used in vector calculus and differential geometry. As with ordinary derivatives, multiple notations
Derivative
Formula for the derivative of a product
displaying short descriptions of redirect targets Vector calculus identities – Mathematical identities Note: This is a usual image since the 17th century
Product_rule
Vector operator in vector calculus
In vector calculus, divergence is a vector operator that operates on a vector field, producing a scalar field giving the rate that the vector field alters
Divergence
Study of rates of change
subjects such as real analysis, vector calculus, and multivariable calculus. The central idea of differential calculus is the derivative. For a real-valued
Differential_calculus
Certain vector fields are the sum of an irrotational and a solenoidal vector field
theorem of vector calculus states that certain differentiable vector fields can be resolved into the sum of an irrotational (curl-free) vector field and
Helmholtz_decomposition
Geometric object that has length and direction
physics, and engineering, a Euclidean vector or simply a vector (sometimes called a geometric vector or spatial vector) is a geometric object that has magnitude
Euclidean_vector
Differential calculus on function spaces
The calculus of variations (or variational calculus) is a field of mathematical analysis that uses variations, which are small changes in functions and
Calculus_of_variations
of multivariable calculus topics. See also multivariable calculus, vector calculus, list of real analysis topics, list of calculus topics. Closed and
List of multivariable calculus topics
List_of_multivariable_calculus_topics
Study of still or slow electric charges
is the negative gradient of the electric potential, as well as vector calculus identities in a way that resembles integration by parts. These two integrals
Electrostatics
Method for estimating new data within known data points
of field (scalar, vector, pseudo-vector or pseudo-scalar). A key feature of mimetic interpolation is that vector calculus identities are satisfied, including
Interpolation
Foundational law of classical magnetism
added onto A to get an alternative choice for A, by the identity (see Vector calculus identities): ∇ × A = ∇ × ( A + ∇ ϕ ) {\displaystyle \nabla \times
Gauss's_law_for_magnetism
In vector calculus, a Laplacian vector field is a vector field which is both irrotational and incompressible. If the field is denoted as v, then it is
Laplacian_vector_field
Theorem in vector calculus
theorem in vector calculus on three-dimensional Euclidean space and real coordinate space, R 3 {\displaystyle \mathbb {R} ^{3}} . Given a vector field, the
Stokes'_theorem
Operation in mathematical calculus
the gradient and curl of vector calculus, and Stokes' theorem simultaneously generalizes the three theorems of vector calculus: the divergence theorem
Integral
Vector field with zero divergence
In vector calculus a solenoidal vector field (also known as an incompressible vector field, a divergence-free vector field, or a transverse vector field)
Solenoidal_vector_field
Section of maths dealing with creating simulations
band-limited fractal signals. Other approaches developed later that use vector calculus identities to produce divergence free fields, such as "Curl-Noise" as suggested
Simulation_noise
Statement about integration on manifolds
In vector calculus and differential geometry the generalized Stokes theorem (sometimes with apostrophe as Stokes' theorem or Stokes's theorem), also called
Generalized_Stokes_theorem
Algebra associated to any vector space
calculus variously as the calculus of extension (Whitehead 1898; Forder 1941), or extensive algebra (Clifford 1878), and recently as extended vector algebra
Exterior_algebra
Branch of mathematical analysis
Fractional calculus is a branch of mathematical analysis that studies the several different possibilities of defining real number powers or complex number
Fractional_calculus
Method of differentiating single-term polynomials
differentiation Product rule Quotient rule Table of derivatives Vector calculus identities If r {\displaystyle r} is a rational number whose lowest terms
Power_rule
Algebraic manipulation of "true" and "false"
propositional calculus have an equivalent expression in Boolean algebra. Thus, Boolean logic is sometimes used to denote propositional calculus performed
Boolean_algebra
Mathematical gradient operator in certain coordinate systems
This is a list of some vector calculus formulae for working with common curvilinear coordinate systems. This article uses the standard notation ISO 80000-2
Del in cylindrical and spherical coordinates
Del_in_cylindrical_and_spherical_coordinates
Equation for the velocity of a body in viscous fluid
{\boldsymbol {\omega }}=\nabla \times \mathbf {u} .} By using some vector calculus identities, these equations can be shown to result in Laplace's equations
Stokes's_law
Vector calculus construction
dependent vector field is a construction in vector calculus which generalizes the concept of vector fields. It can be thought of as a vector field which
Time_dependent_vector_field
Differential operator in mathematics
B_{z}\end{bmatrix}}.} This identity is a coordinate dependent result, and is not general. An example of the usage of the vector Laplacian is the Navier-Stokes
Laplace_operator
Mathematical notion of infinitesimal difference
differential refers to several related notions derived from the early days of calculus, put on a rigorous footing, such as infinitesimal differences and the derivatives
Differential_(mathematics)
Branch of mathematics
infinitesimal calculus or the calculus of infinitesimals, it has two major branches, differential calculus and integral calculus. Differential calculus studies
Calculus
Mathematical operation in linear algebra
Matrix calculus, for the interaction of matrix multiplication with operations from calculus Nykamp, Duane. "Multiplying matrices and vectors". Math Insight
Matrix_multiplication
Equations of fluid dynamics
single dependent variable in 2D, or one vector equation in 3D. This is enabled by two vector calculus identities: ∇ × ( ∇ ϕ ) = 0 ∇ ⋅ ( ∇ × A ) = 0 {\displaystyle
Derivation of the Navier–Stokes equations
Derivation_of_the_Navier–Stokes_equations
Calculus on stochastic processes
Stochastic calculus is a branch of mathematics that operates on stochastic processes. It allows a consistent theory of integration to be defined for integrals
Stochastic_calculus
Topics referred to by the same term
Green formula may refer to: Green's theorem in integral calculus Green's identities in vector calculus Green's function in differential equations the Green
Green_formula
Vector behavior under coordinate changes
Ricci calculus (2 ed.). Springer. p. 6. Bowen, Ray; Wang, C.-C. (2008) [1976]. "§3.14 Reciprocal Basis and Change of Basis". Introduction to Vectors and
Covariance and contravariance of vectors
Covariance_and_contravariance_of_vectors
Theorem in calculus
In vector calculus, the divergence theorem, also known as Gauss's theorem or Ostrogradsky's theorem, is a theorem relating the flux of a vector field through
Divergence_theorem
Force acting on charged particles in electric and magnetic fields
\mathbf {B} \right)\mathrm {d} V.} Using Maxwell's equations and vector calculus identities, the force density can be reformulated to eliminate explicit reference
Lorentz_force
Vector logic is an algebraic model of elementary logic based on matrix algebra. Vector logic assumes that the truth values map on vectors, and that the
Vector_logic
Two Advanced Placement courses and exams
parametric equations, vector calculus, and polar coordinate functions, among other topics. AP Calculus AB is an Advanced Placement calculus course. It is traditionally
AP_Calculus
Ternary operation on vectors
simplifying vector calculations in physics. A related identity regarding gradients and useful in vector calculus is Lagrange's formula of vector cross-product
Triple_product
Generalized chain rule in calculus
displaying short descriptions of redirect targets Vector calculus identities – Mathematical identities (Arbogast 1800). According to Craik (2005, pp. 120–122):
Faà_di_Bruno's_formula
Formula for the derivative of an inverse function
displaying short descriptions of redirect targets Vector calculus identities – Mathematical identities "Derivatives of Inverse Functions". oregonstate.edu
Inverse_function_rule
Four-dimensional number system
Quaternions can be used to represent vectors in three-dimensional space, which provides a definition of the quotient of two vectors. Quaternions were first described
Quaternion
Historical term in mathematics
correspondence techniques of the calculus of finite differences. The method is a notational procedure used for deriving identities involving indexed sequences
Umbral_calculus
Type of motion of magnetic fields
derivative. This can be rearranged into a more useful form using vector calculus identities and ∇ ⋅ B = 0 {\displaystyle \nabla \cdot \mathbf {B} =0} : ∂
Magnetic_diffusion
Operation on differential forms
generalization of Stokes' theorem, Gauss's theorem, and Green's theorem from vector calculus. If a differential k {\displaystyle k} -form is thought of as measuring
Exterior_derivative
Representation of a tensor in Euclidean space
products, together with this identity, greatly facilitate the manipulation and derivation of other identities in vector calculus and algebra, which in turn
Cartesian_tensor
Graphical language for quantum processes
The ZX-calculus is a graphical language. It was conceived for reasoning about linear maps between qubits, which are represented as string diagrams called
ZX-calculus
Mathematical operation on vectors in 3D space
all true vectors, the magnetic field B is a pseudovector. In vector calculus, the cross product is used to define the formula for the vector operator
Cross_product
Mathematical techniques used in probability theory and related fields
related fields, Malliavin calculus is a set of mathematical techniques and ideas that extend the mathematical field of calculus of variations from deterministic
Malliavin_calculus
Kinetic energy per unit volume of a fluid
\nu \,\nabla ^{2}\mathbf {u} =-\nabla p+\rho \mathbf {g} } By a vector calculus identity ( u = | u | {\displaystyle u=|\mathbf {u} |} ) ∇ ( u 2 / 2 ) =
Dynamic_pressure
Relationship between derivatives and integrals
The fundamental theorem of calculus is a theorem that links the concept of differentiating a function (calculating its slopes, or rate of change at every
Fundamental theorem of calculus
Fundamental_theorem_of_calculus
Calculus of functions generalization
finite-dimensional real vector space. This calculus is also known as advanced calculus, especially in the United States. It is similar to multivariable calculus but is
Calculus_on_Euclidean_space
these are identities involving certain functions of one or more angles. They are distinct from triangle identities, which are identities potentially
List of trigonometric identities
List_of_trigonometric_identities
Conversion of a matrix or a tensor to a vector
allows vectorization and function vech() implemented in both packages 'ks' and 'sn' allows half-vectorization. Vectorization is used in matrix calculus and
Vectorization_(mathematics)
identities . trigonometric integral . trigonometric substitution . trigonometry . triple integral . upper bound . variable . vector . vector calculus
Glossary_of_calculus
Function for incompressible divergence-free flows in two dimensions
=\nabla \psi \times {\hat {\mathbf {z} }}} where we've used the vector calculus identity ∇ × ( ψ z ^ ) = ψ ∇ × z ^ + ∇ ψ × z ^ . {\displaystyle \nabla \times
Stream_function
On products on sums of squares
This identity is a generalisation of the Brahmagupta–Fibonacci identity and a special form of the Binet–Cauchy identity. In a more compact vector notation
Lagrange's_identity
Algebraic structure in linear algebra
operations of vector addition and scalar multiplication must satisfy certain requirements, called vector axioms. Real vector spaces and complex vector spaces
Vector_space
Notation of differential calculus
settings—such as partial derivatives in multivariable calculus, tensor analysis, or vector calculus—other notations, such as subscript notation or the ∇
Notation_for_differentiation
Collection of proofs of equations involving trigonometric functions
defining trigonometric functions, and the proofs of the trigonometric identities between them depend on the chosen definition. The oldest and most elementary
Proofs of trigonometric identities
Proofs_of_trigonometric_identities
Integration over a non-flat region in 3D space
In mathematics, particularly multivariable calculus, a surface integral is a generalization of multiple integrals to integration over surfaces. It can
Surface_integral
Definite integral of a scalar or vector field along a path
path L {\displaystyle L} . In qualitative terms, a line integral in vector calculus can be thought of as a measure of the total effect of a given tensor
Line_integral
In vector calculus, a Beltrami vector field, named after Eugenio Beltrami, is a vector field in three dimensions that is parallel to its own curl. That
Beltrami_vector_field
Fluid instability that causes turbulence in accretion disks
divergence-free displacement, then our equation reduces to because of the vector calculus identity ∇ × ( ξ × B ) = ξ ( ∇ ⋅ B ) − B ( ∇ ⋅ ξ ) + ( B ⋅ ∇ ) ξ − ( ξ ⋅
Magnetorotational_instability
Addition of several numbers or other values
Beside numbers, other types of values can be summed as well: functions, vectors, matrices, polynomials and, in general, elements of any type of mathematical
Summation
Algebraic operation on coordinate vectors
numbers (usually coordinate vectors), and returns a single number. In Euclidean geometry, the scalar product of two vectors is the dot product of their
Dot_product
Equation
_{m}v_{i}=\left[(\mathbf {u} \cdot \nabla )\mathbf {u} \right]_{i}\,.} The vector calculus identity of the cross product of a curl holds: v × ( ∇ × a ) = ∇ a ( v ⋅
Cauchy_momentum_equation
Mathematical method in calculus
In calculus, and more generally in mathematical analysis, integration by parts or partial integration is a process that finds the integral of a product
Integration_by_parts
Discrete (i.e., incremental) version of infinitesimal calculus
Discrete calculus or the calculus of discrete functions, is the mathematical study of incremental change, in the same way that geometry is the study of
Discrete_calculus
Shorthand notation for tensor operations
Summation Convention and Vector Identities". Oxford University. Archived from the original on 2017-01-06. Retrieved 2008-07-02. "Vector Calculation in Index
Einstein_notation
Matrix of partial derivatives of a vector-valued function
In vector calculus, the Jacobian matrix (/dʒəˈkoʊbiən/, /dʒɪ-, jɪ-/) of a vector-valued function of several variables is the matrix of all its first-order
Jacobian matrix and determinant
Jacobian_matrix_and_determinant
Instantaneous rate of change of the function
multivariable calculus, the directional derivative measures the instantaneous rate at which a function changes along a specified vector through a given
Directional_derivative
matrix Curvature Green's theorem Divergence theorem Stokes' theorem Vector Calculus Infinite series Maclaurin series, Taylor series Fourier series Euler–Maclaurin
List_of_calculus_topics
Field theory coupling of charge but not higher moments
\mathbf {B} \end{aligned}}} The above derivation makes use of the vector calculus identity: 1 2 ∇ ( A ⋅ A ) = A ⋅ J A = A ⋅ ( ∇ A ) = ( A ⋅ ∇
Minimal_coupling
Algebraic structure designed for geometry
geometric algebras went somewhat ignored, greatly eclipsed by the vector calculus then newly developed to describe electromagnetism. The term "geometric
Geometric_algebra
Mathematical object used in fluid dynamics
Lamb vector is the cross product of vorticity vector and velocity vector of the flow field, named after the physicist Horace Lamb. The Lamb vector is defined
Lamb_vector
Derivative defined on normed spaces
to the case of a vector-valued function of multiple real variables, and to define the functional derivative used widely in the calculus of variations. Generally
Fréchet_derivative
Coordinate system using perpendicular axes
calculus by Isaac Newton and Gottfried Wilhelm Leibniz. The two-coordinate description of the plane was later generalized into the concept of vector spaces
Cartesian_coordinate_system
Course designed to prepare students for calculus
might spend more time on conic sections, Euclidean vectors, and other topics needed for calculus, used in fields such as medicine or engineering. A college
Precalculus
Force in which the work done in moving an object depends only on its displacement
Finally, assume that the third statement is true. A well-known vector calculus identity states that the curl of the gradient of any function is 0. (See
Conservative_force
derivative from vector calculus. As with the directional derivative, the covariant derivative is a rule, which takes as its inputs: (1) a vector, u, (along
Introduction to the mathematics of general relativity
Introduction_to_the_mathematics_of_general_relativity
Algebraic object with geometric applications
of algebraic objects associated with a vector space. Tensors may map between different objects such as vectors, scalars, and even other tensors. There
Tensor
Set of coordinates where the coordinate hypersurfaces all meet at right angles
are a special but extremely common case of curvilinear coordinates. While vector operations and physical laws are normally easiest to derive in Cartesian
Orthogonal_coordinates
VECTOR CALCULUS-IDENTITIES
VECTOR CALCULUS-IDENTITIES
Male
Portuguese
Galician-Portuguese form of Roman Latin Victor, VITOR means "conqueror."
Boy/Male
American, British, Christian, Danish, Dutch, English, Finnish, French, German, Greek, Hindu, Indian, Irish, Jamaican, Latin, Romanian, Slovenia, Spanish, Swedish, Swiss, Tamil, Ukrainian
Victorious; Conqueror; Winner; Champion; One who Conquers; Victory
Boy/Male
English American
Doctor; teacher.
Boy/Male
American, Australian, British, Chinese, Christian, Danish, Dutch, English, French, German, Greek, Italian, Latin, Portuguese, Shakespearean, Spanish
Steadfast; Anchor; Holds Fast; Star; Coined from Esther Vanhomrigh; Tenacious; Defend; Hold Fast; Coined from Esther Vanho
Boy/Male
Spanish American Shakespearean Greek Latin
Tenacious.
Male
English
Short form of English Sylvester, VESTER means "from the forest."
Male
Arthurian
, sir Hector de Maris; (defender).
Surname or Lastname
Scottish
Scottish : Anglicized form of the Gaelic personal name Eachann (earlier Eachdonn, already confused with Norse Haakon), composed of the elements each ‘horse’ + donn ‘brown’.English : found in Yorkshire and Scotland, where it may derive directly from the medieval personal name. According to medieval legend, Britain derived its name from being founded by Brutus, a Trojan exile, and Hector was occasionally chosen as a personal name, as it was the name of the Trojan king’s eldest son. The classical Greek name, HektÅr, is probably an agent derivative of Greek ekhein ‘to hold back’, ‘hold in check’, hence ‘protector of the city’.German, French, and Dutch : from the personal name (see 2 above). In medieval Germany, this was a fairly popular personal name among the nobility, derived from classical literature. It is a comparatively rare surname in France.
Male
English
Roman Latin name VICTOR means "conqueror."Â
Male
Portuguese
Portuguese form of Latin Hector, HEITOR means "defend; hold fast."
Male
English
 Anglicized form of Scottish Gaelic Eachann, HECTOR means "brown horse." Compare with another form of Hector.
Male
Scandinavian
 Scandinavian form of Roman Latin Victor, VIKTOR means "conqueror." Compare with another form of Viktor.
Boy/Male
Latin American Spanish
Conqueror.
Boy/Male
Australian, Basque, Czech, Czechoslovakian, Danish, Finnish, French, German, Hungarian, Latin, Polish, Slovenia, Swedish, Swiss, Ukrainian
The Conqueror; Victory; Victorious; Conquer
Boy/Male
Christian & English(British/American/Australian)
Conqueror
Boy/Male
Spanish
Victor.
Male
Russian
(Cyrillic Виктор): Slavic form of Roman Latin Victor, VIKTOR means "conqueror." In use by the Bulgarians, Russians and Serbians. Compare with another form of Viktor.
Male
Greek
(á¼ÎºÏ„ωÏ) Variant spelling of Greek Hektor, EKTOR means "defend; hold fast."
Male
Celtic
, Mars, the divinity.
Boy/Male
Christian & English(British/American/Australian)
Steadfast
VECTOR CALCULUS-IDENTITIES
VECTOR CALCULUS-IDENTITIES
Boy/Male
Bengali, Indian, Sanskrit
The Son of Asvini
Girl/Female
Indian
Blessing of God
Girl/Female
Tamil
Karnas wifes name in mahabharata, Success
Boy/Male
Hindu, Indian, Marathi, Tamil, Telugu
Born of the Mind
Boy/Male
Indian, Tamil
No One Equal
Boy/Male
Tamil
Panchjanya | பஂசஜநà¯à®¯
Male
English
English name derived from Latin Adrianus, ADRIAN means "from Hadria."Â
Female
Portuguese
Portuguese form of Roman Latin Lucia, LUZIA means "light."
Boy/Male
Tamil
One who does good deeds
Girl/Female
Tamil
Yashwini | யஷà¯à®µà¯€à®¨à¯€Â
Successful lady, Yash, Victory
VECTOR CALCULUS-IDENTITIES
VECTOR CALCULUS-IDENTITIES
VECTOR CALCULUS-IDENTITIES
VECTOR CALCULUS-IDENTITIES
VECTOR CALCULUS-IDENTITIES
n.
A calculous concretion, especially one in the kidneys or bladder; the disease arising from a calculus.
n.
Any solid concretion, formed in any part of the body, but most frequent in the organs that act as reservoirs, and in the passages connected with them; as, biliary calculi; urinary calculi, etc.
a.
Pertaining to a rector or a rectory; rectoral.
n.
The chief elective officer of some universities, as in France and Scotland; sometimes, the head of a college; as, the Rector of Exeter College, or of Lincoln College, at Oxford.
n.
A woman who wins a victory; a female victor.
a.
Of the nature of a calculus; like stone; gritty; as, a calculous concretion.
n.
The calculus; fluxions.
n. pl.
See Calculus.
n.
A directed quantity, as a straight line, a force, or a velocity. Vectors are said to be equal when their directions are the same their magnitudes equal. Cf. Scalar.
n.
A urinary calculus.
v. t.
To tamper with and arrange for one's own purposes; to falsify; to adulterate; as, to doctor election returns; to doctor whisky.
n.
Same as Radius vector.
n.
A term made up of the two parts / + /1 /-1, where / and /1 are vectors.
pl.
of Calculus
n.
The turning factor of a quaternion.
n.
An African weaver bird (Textor alector).
v. t.
To confer a doctorate upon; to make a doctor.
n.
A belly, or protuberant part; a broad surface; as, the venter of a muscle; the venter, or anterior surface, of the scapula.
n.
The ratio of one vector to another in length, no regard being had to the direction of the two vectors; -- so called because considered as a stretching factor in changing one vector into another. See Versor.
a.
Caused, or characterized, by the presence of a calculus or calculi; a, a calculous disorder; affected with gravel or stone; as, a calculous person.