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Instantaneous rate of change of the function
In multivariable calculus, the directional derivative measures the instantaneous rate at which a function changes along a specified vector through a given
Directional_derivative
Instantaneous rate of change (mathematics)
directional derivatives. Given a vector v = ( v 1 , … , v n ) {\displaystyle \mathbf {v} =(v_{1},\ldots ,v_{n})} , then the directional derivative
Derivative
Type of derivative in mathematics
0)\end{cases}}} whose directional derivatives are all 0 at (0,0), but which fails to be differentiable there. However, if all the partial derivatives of f {\displaystyle
Derivative (multivariable calculus)
Derivative_(multivariable_calculus)
Theorem in convex analysis
monograph provides a formula for the directional derivative of the maximum of a (not necessarily convex) directionally differentiable function. An extension
Danskin's_theorem
Calculus of functions of several variables
difference in the definition of the limits and continuity. Directional limits and derivatives define the limit and differential along a 1D parametrized
Multivariable_calculus
Multivariate derivative (mathematics)
is the rate of increase in that direction, the greatest absolute directional derivative. Further, a point where the gradient is the zero vector is known
Gradient
Specification of a derivative along a tangent vector of a manifold
Euclidean space, the covariant derivative can be viewed as the orthogonal projection of the Euclidean directional derivative onto the manifold's tangent
Covariant_derivative
Derivative of a function with multiple variables
In fact, the last equality shows that the partial derivative is just the directional derivative where the direction is the i {\displaystyle i} -th standard
Partial_derivative
Fundamental construction of differential calculus
covariant derivative makes a choice for taking directional derivatives of vector fields along curves. This extends the directional derivative of scalar
Generalizations of the derivative
Generalizations_of_the_derivative
Infinitesimal calculus on functions defined on a geometric algebra
{\displaystyle F} be a multivector-valued function of a vector. The directional derivative of F {\displaystyle F} along b {\displaystyle b} at a {\displaystyle
Geometric_calculus
Vector differential operator
gradient, divergence, curl, directional derivative, and Laplacian. The gradient of a scalar field f {\displaystyle f} is its derivative as a resultant vector
Del
Time rate of change of some physical quantity of a material element in a velocity field
{\displaystyle \mathbf {u} \cdot \nabla y} , or as involving the streamline directional derivative of the field ( u ⋅ ∇ ) y {\displaystyle (\mathbf {u} \cdot \nabla
Material_derivative
Generalization of the concept of directional derivative
mathematics, the Gateaux differential or Gateaux derivative is a generalization of the concept of directional derivative in differential calculus. Named after René
Gateaux_derivative
Manifold upon which it is possible to perform calculus
derivative of a function on a differentiable manifold, the most fundamental of which is the directional derivative. The definition of the directional
Differentiable_manifold
Mathematical operation
second derivative, or the second-order derivative, of a function f is the derivative of the derivative of f. Informally, the second derivative can be
Second_derivative
Type of derivative in differential geometry
derivative of a tensor field with respect to a vector field would be to take the components of the tensor field and take the directional derivative of
Lie_derivative
Mathematical operation in calculus
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. Intuitively
Logarithmic_derivative
Derivative defined on normed spaces
Fréchet derivative should be contrasted to the more general Gateaux derivative which is a generalization of the classical directional derivative. The Fréchet
Fréchet_derivative
simulations. The directional derivative provides a systematic way of finding these derivatives. The definitions of directional derivatives for various situations
Tensor derivative (continuum mechanics)
Tensor_derivative_(continuum_mechanics)
Assignment of vector fields to manifolds
as directional derivatives. Given a vector v {\displaystyle v} in R n {\displaystyle \mathbb {R} ^{n}} , one defines the corresponding directional derivative
Tangent_space
Notation of differential calculus
standard notation for differentiation. Instead, several notations for the derivative of a function or a dependent variable have been proposed by various mathematicians
Notation_for_differentiation
Evaluates a line integral through a gradient field using the original scalar field
path-independent. Let v be any nonzero vector in Rn. By the definition of the directional derivative, ∂ f ( x ) ∂ v = lim t → 0 f ( x + t v ) − f ( x ) t = lim t → 0
Gradient_theorem
Differentiation under the integral sign formula
the integrands are functions dependent on x , {\displaystyle x,} the derivative of this integral is expressible as d d x ( ∫ a ( x ) b ( x ) f ( x , t
Leibniz_integral_rule
Branch of mathematical analysis
Sonin–Letnikov derivative Liouville derivative Caputo derivative Hadamard derivative Marchaud derivative Riesz derivative Miller–Ross derivative Weyl derivative Erdélyi–Kober
Fractional_calculus
Indefinite integral
inverse derivative, primitive function, primitive integral or indefinite integral of a function f is a differentiable function F whose derivative is equal
Antiderivative
Specialized notation for multivariable calculus
notation just defined for the derivative of a scalar with respect to a vector we can re-write the directional derivative as ∇ u f = ∂ f ∂ x u . {\displaystyle
Matrix_calculus
Formula for the derivative of a ratio of functions
In calculus, the quotient rule is a method of finding the derivative of a function that is the ratio of two differentiable functions. Let h ( x ) = f
Quotient_rule
Operation on differential forms
{\displaystyle df(X)=d_{X}f} , where d X f {\displaystyle d_{X}f} is the directional derivative of f {\displaystyle f} in the direction of X {\displaystyle X}
Exterior_derivative
Mathematical theorem
of the proof is to show that, for any fixed unit vector v, the v-directional derivative of u exists almost everywhere. This is a consequence of a special
Rademacher's_theorem
Function in mathematics
typically given in the form of a covariant derivative, which gives a means for taking directional derivatives of vector fields, measuring the deviation
Connection_(mathematics)
In mathematics, the Hadamard derivative is a concept of directional derivative for maps between Banach spaces. It is particularly suited for applications
Hadamard_derivative
Formula for the derivative of a product
abstractly as an operator on real-valued functions which behaves like a directional derivative at p: that is, a linear functional v which is a derivation, v (
Product_rule
Formula in calculus
general case is to use the total derivative, which is a linear transformation that captures all directional derivatives in a single formula. Consider differentiable
Chain_rule
Matrix of partial derivatives of a vector-valued function
function of several variables is the matrix of all its first-order partial derivatives. If this matrix is square, that is, if the number of variables equals
Jacobian matrix and determinant
Jacobian_matrix_and_determinant
Numerical calculations carrying along derivatives
directional derivative operator. That is, if it is sufficient to compute y ′ = ∇ f ( x ) ⋅ x ′ {\textstyle y'=\nabla f(x)\cdot x'} , the directional derivative
Automatic_differentiation
Concept in calculus of variations
like the Gateaux derivative is preferred. In many practical cases, the functional differential is defined as the directional derivative δ F [ ρ , ϕ ] =
Functional_derivative
Theorem in mathematics
differentiable in an open interval, with a continuous derivative, then in a neighborhood of any point where the derivative is not zero, f has an inverse function. The
Inverse_function_theorem
Mathematical operation in calculus
In calculus, implicit differentiation is a method for finding the derivative of a function that is defined by an equation rather than by an explicit formula
Implicit_differentiation
Study of rates of change
variables, analogous ideas lead to partial derivatives, directional derivatives, and the total derivative. The derivative can also be understood as the coefficient
Differential_calculus
Types generalized of derivatives
f:\mathbb {R} ^{n}\rightarrow \mathbb {R} ,} the Clarke generalized directional derivative of f {\displaystyle f} at x ∈ R n {\displaystyle x\in \mathbb {R}
Clarke_generalized_derivative
Differential form of degree one or section of a cotangent bundle
differential of a function, d f ( v ) {\displaystyle df(v)} is the directional derivative, or the rate at which f {\displaystyle f} is changing, at p {\displaystyle
One-form
Topics referred to by the same term
derivative, a generalization of the concept of directional derivative in differential calculus. Lie derivative, the change of a tensor field (including scalar
Derivative_(disambiguation)
Branch of mathematics
derivative of a function. The process of finding the derivative is called differentiation. Given a function and a point in the domain, the derivative
Calculus
Operation in calculus
Integrals also refer to the concept of an antiderivative, a function whose derivative is the given function; in this case, they are also called indefinite integrals
Integral
Mathematical rule for evaluating limits
functions, each of which tends to zero or infinity, by taking each function's derivative. The rule is named after the 17th-century French mathematician Guillaume
L'Hôpital's_rule
Mathematical techniques used in probability theory and related fields
and h ∈ H {\displaystyle h\in {\mathcal {H}}} one now defines the directional derivative ⟨ D F , h ⟩ = D h F = lim ε → 0 ( ρ ( ε h ) − I ) F ε . {\displaystyle
Malliavin_calculus
Image processing method
strength, usually a first-order derivative expression such as the gradient magnitude, and then searching for local directional maxima of the gradient magnitude
Edge_detection
Mathematical identities
} where A ⋅ ∇ {\displaystyle \mathbf {A} \cdot \nabla } is the directional derivative in the direction of A {\displaystyle \mathbf {A} } multiplied by
Vector_calculus_identities
Calculus of vector-valued functions
0)} tensor fields. Mathematics portal Conservative vector field Directional derivative Geometric calculus Helmholtz decomposition Laplacian vector field
Vector_calculus
Mathematical theorem
symmetry of second derivatives (also called the equality of mixed partials) is the fact that exchanging the order of partial derivatives of a multivariate
Symmetry of second derivatives
Symmetry_of_second_derivatives
Rate of change of the second derivative
a branch of mathematics, the third derivative or third-order derivative is the rate at which the second derivative, or the rate of change of the rate
Third_derivative
Vector operator in vector calculus
exterior derivative is usually easier than working with the vector field and divergence, because unlike the divergence, the exterior derivative commutes
Divergence
Mathematical approximation of a function
infinite sum of terms that are expressed in terms of the function's derivatives at a single point. For most common functions, the function and the sum
Taylor_series
Continuous function Derivative Notation Newton's notation for differentiation Leibniz's notation for differentiation Simplest rules Derivative of a constant
List_of_calculus_topics
Matrix of second derivatives
(less commonly) Hesse matrix is a square matrix of second-order partial derivatives of a scalar-valued function, or scalar field. It describes the local
Hessian_matrix
Mathematical notion of infinitesimal difference
put on a rigorous footing, such as infinitesimal differences and the derivatives of functions. The term is used in various branches of mathematics such
Differential_(mathematics)
Notion in calculus
point of view is to define the differential directly as a kind of directional derivative: d f ( x , h ) = lim t → 0 f ( x + t h ) − f ( x ) t = d d t f (
Differential_of_a_function
Technique in integral evaluation
{\displaystyle g:[a,b]\to I} be a differentiable function with a continuous derivative, where I ⊂ R {\displaystyle I\subset \mathbb {R} } is an interval. Suppose
Integration_by_substitution
Mathematical notation
_{1}}x_{2}^{\alpha _{2}}\ldots x_{n}^{\alpha _{n}}} . Higher-order partial derivative ∂ α = ∂ 1 α 1 ∂ 2 α 2 … ∂ n α n , {\displaystyle \partial ^{\alpha }=\partial
Multi-index_notation
Rules for computing derivatives of functions
a summary of differentiation rules, that is, rules for computing the derivative of a function in calculus. Unless otherwise stated, all functions are
Differentiation_rules
Course designed to prepare students for calculus
separate parts of the coursework. For students to succeed at finding the derivatives and antiderivatives with calculus, they will need facility with algebraic
Precalculus
Generalization of the product rule in calculus
after Gottfried Wilhelm Leibniz, generalizes the product rule for the derivative of the product of two functions (which is also known as "Leibniz's rule")
General_Leibniz_rule
Theorem in calculus relating line and double integrals
point of R {\displaystyle R} . This implies the existence of all directional derivatives, in particular D e i A =: D i A , D e i B =: D i B , i = 1 , 2
Green's_theorem
Point on a curve where motion must move backwards
\end{aligned}}} a cusp is a point where both derivatives of f and g are zero, and the directional derivative, in the direction of the tangent, changes sign
Cusp_(singularity)
3D generalization of the Leibniz integral rule
generalization of the Leibniz integral rule. It is used to recast time derivatives of integrated quantities and is useful in formulating the basic equations
Reynolds_transport_theorem
Mathematical relation consisting of a multi-variable function equal to zero
pathologies. In calculus, implicit differentiation is a method for finding the derivative of a function that is defined by an equation rather than by an explicit
Implicit_function
Integral of sin(x)/x from 0 to infinity
condensation Dirichlet Abel Vector Gradient Divergence Curl Laplacian Directional derivative Identities Theorems Gradient Green's Stokes' Divergence Generalized
Dirichlet_integral
Method for partial-fraction expansion
condensation Dirichlet Abel Vector Gradient Divergence Curl Laplacian Directional derivative Identities Theorems Gradient Green's Stokes' Divergence Generalized
Heaviside_cover-up_method
Property of a mathematical function
Rn or in a Banach space. Directional derivative Partial derivative Gradient Gateaux derivative Fréchet derivative Derivative (generalizations) Phase space
Semi-differentiability
Differential operator in mathematics
coordinate system, the Laplacian is given by the sum of second partial derivatives of the function with respect to each independent variable. In other coordinate
Laplace_operator
Method of differentiating single-term polynomials
underlies the Taylor series as it relates a power series with a function's derivatives. Let f {\displaystyle f} be a function satisfying f ( x ) = x r {\displaystyle
Power_rule
Relationship between derivatives and integrals
original function. Thus, the derivative of the integral of a function (the area) is the original function, so that derivative and integral are inverse operations
Fundamental theorem of calculus
Fundamental_theorem_of_calculus
Calculus on stochastic processes
condensation Dirichlet Abel Vector Gradient Divergence Curl Laplacian Directional derivative Identities Theorems Gradient Green's Stokes' Divergence Generalized
Stochastic_calculus
Integral transform
derivative of integer order rather than the derivatives of fractional order as in the Riemann–Liouville derivative. The Caputo fractional derivative with
Riemann–Liouville_integral
greater than the values at nearby points inside the domain, then the directional derivative of the function in the direction of the outward pointing normal
Hopf_lemma
Type of function
pseudoconvex if it is increasing in any direction where it has a positive directional derivative. The property must hold in all of the function domain, and not only
Pseudoconvex_function
Integral over a 3-D domain
condensation Dirichlet Abel Vector Gradient Divergence Curl Laplacian Directional derivative Identities Theorems Gradient Green's Stokes' Divergence Generalized
Volume_integral
On converting relations to functions of several real variables
= 0), the theorem states that, under a mild condition on the partial derivatives (with respect to each yi ) at a point, the m variables yi are differentiable
Implicit_function_theorem
Historical mathematical concept; form of derivative
were introduced by Isaac Newton to describe his form of a time derivative (a derivative with respect to time). Newton introduced the concept in 1665 and
Fluxion
Matrix operation generalizing exponentiation of scalar numbers
{\displaystyle n\times n} Hermitian matrix V {\displaystyle V} , the directional derivative of exp : X → e X {\displaystyle \exp :X\to e^{X}} at X {\displaystyle
Matrix_exponential
Circulation density in a vector field
Christoffel symbols participating in the covariant derivative, this expression reduces to the partial derivative: ( ∇ × F ) = 1 g R k ε k ℓ m ∂ ℓ F m {\displaystyle
Curl_(mathematics)
Two Advanced Placement courses and exams
College Board. AP Calculus AB covers basic introductions to limits, derivatives, and integrals. AP Calculus BC covers all AP Calculus AB topics plus
AP_Calculus
Scientific principles enabling the use of the calculus of variations
condensation Dirichlet Abel Vector Gradient Divergence Curl Laplacian Directional derivative Identities Theorems Gradient Green's Stokes' Divergence Generalized
Variational_principle
Differential calculus on function spaces
often expressed as definite integrals involving functions and their derivatives. Functions that maximize or minimize functionals may be found using the
Calculus_of_variations
Commonly encountered and tricky integral
condensation Dirichlet Abel Vector Gradient Divergence Curl Laplacian Directional derivative Identities Theorems Gradient Green's Stokes' Divergence Generalized
Integral_of_secant_cubed
Definite integral of a scalar or vector field along a path
{\displaystyle \mathbf {F} =\nabla G,} then by the multivariable chain rule the derivative of the composition of G and r(t) is d G ( r ( t ) ) d t = ∇ G ( r ) ⋅
Line_integral
Use of complex numbers to evaluate integrals
condensation Dirichlet Abel Vector Gradient Divergence Curl Laplacian Directional derivative Identities Theorems Gradient Green's Stokes' Divergence Generalized
Integration using Euler's formula
Integration_using_Euler's_formula
Method of evaluating certain integrals along paths in the complex plane
and we require that each piece have a finite (non-vanishing) continuous derivative. These requirements correspond to requiring that we consider only curves
Contour_integration
Formula for the derivative of an inverse function
formula that expresses the derivative of the inverse of a bijective and differentiable function f in terms of the derivative of f. More precisely, if the
Inverse_function_rule
Mapping involving integration between function spaces
condensation Dirichlet Abel Vector Gradient Divergence Curl Laplacian Directional derivative Identities Theorems Gradient Green's Stokes' Divergence Generalized
Integral_transform
Statement relating differentiable symmetries to conserved quantities
motion – it is an invariant. Mathematically, the rate of change of X (its derivative with respect to time) is zero, d X d t = X ˙ = 0 . {\displaystyle {\frac
Noether's_theorem
Theorem in mathematics
there is at least one point in ( a , b ) {\displaystyle (a,b)} where the derivative equals the function's average rate of change over the whole interval.
Mean_value_theorem
Conditions for switching order of integration in calculus
_{0}^{u}w(x)\,\mathrm {d} x\right]=V(u)W(u)} By the product rule, the derivative of the right-hand side is d d x [ V ( x ) W ( x ) ] = V ( x ) w ( x )
Fubini's_theorem
Mathematical function with no sudden changes
be discontinuous in a restricted way, giving rise to the concept of directional continuity (or right- and left-continuous functions) and semi-continuity
Continuous_function
Vector calculus formulas relating the bulk with the boundary of a region
\psi \right)\cdot d\mathbf {S} .\end{array}}} Here, ∂φ/∂n is the directional derivative of φ in the direction of the outward pointing surface unit normal
Green's_identities
Mathematical technique for simplification
differentiation. For example, consider the problem of calculating the derivative d d x sin ( x 2 ) . {\displaystyle {\frac {d}{dx}}\sin(x^{2}).} Let
Change_of_variables
Change of variable for integrals involving trigonometric functions
condensation Dirichlet Abel Vector Gradient Divergence Curl Laplacian Directional derivative Identities Theorems Gradient Green's Stokes' Divergence Generalized
Tangent half-angle substitution
Tangent_half-angle_substitution
Mathematical criterion about whether a series converges
{\displaystyle f(1/n)=a_{n}} for all positive integers n and the second derivative f ″ {\displaystyle f''} exists at x = 0 {\displaystyle x=0} . Then ∑ n
Convergence_tests
Method of mathematical differentiation
a method used to differentiate functions by employing the logarithmic derivative of a function f, ( ln f ) ′ = f ′ f ⟹ f ′ = f ⋅ ( ln f ) ′ . {\displaystyle
Logarithmic_differentiation
Integrals not expressible in closed-form from elementary functions
expression – Mathematical formula involving a given set of operations Derivative – Instantaneous rate of change (mathematics) Differential algebra – Algebraic
Nonelementary_integral
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DIRECTIONAL DERIVATIVE
DIRECTIONAL DERIVATIVE
DIRECTIONAL DERIVATIVE
DIRECTIONAL DERIVATIVE
DIRECTIONAL DERIVATIVE
DIRECTIONAL DERIVATIVE
DIRECTIONAL DERIVATIVE
DIRECTIONAL DERIVATIVE
DIRECTIONAL DERIVATIVE
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