Information about Partial Derivative
In mathematics, a partial derivative of a function of several variables is its derivative with respect to one of those variables with the others held constant (as opposed to the total derivative, in which all variables are allowed to vary). Partial derivatives are useful in vector calculus and differential geometry.
The partial derivative of a function f with respect to the variable x is written as fx or ∂f/∂x. The partial-derivative symbol ∂ is a rounded letter, distinguished from the straight d of total-derivative notation. The notation was introduced by Legendre and gained general acceptance after its reintroduction by Jacobi.
The partial derivative of V with respect to r is
It describes the rate with which a cone's volume changes if its radius is varied and its height is kept constant. The partial derivative with respect to h is
and represents the rate with which the volume changes if its height is varied and its radius is kept constant.
Equations involving an unknown function's partial derivatives are called partial differential equations and are common in physics, engineering, and other sciences and applied disciplines.
First-order partial derivatives:
Second-order partial derivatives:
Second-order mixed derivatives:
Higher-order partial and mixed derivatives:
When dealing with functions of multiple variables, some of these variables may be related to each other, and it may be necessary to specify explicitly which variables are being held constant. In fields such as statistical mechanics, the partial derivative of f with respect to x, holding y and z constant, is often expressed as
Even if all partial derivatives ∂f/∂xi(a) exist at a given point a, the function need not be continuous there. However, if all partial derivatives exist in a neighborhood of a and are continuous there, then f is totally differentiable in that neighborhood and the total derivative is continuous. In this case, we say that f is a C1 function. We can use this fact to generalize for vector valued functions (f : U → R'm) by carefully using a componentwise argument.
The partial derivative
can be seen as another function defined on U and can again be partially differentiated. If all mixed second order partial derivatives are continuous at a point (or on a set), we call f a C2 function at that point (or on that set); in this case, the partial derivatives can be exchanged by Clairaut's theorem:
The partial derivative of a function f with respect to the variable x is written as fx or ∂f/∂x. The partial-derivative symbol ∂ is a rounded letter, distinguished from the straight d of total-derivative notation. The notation was introduced by Legendre and gained general acceptance after its reintroduction by Jacobi.
Examples
Consider the volume V of a cone; it depends on the cone's height h and its radius r according to the formula
The partial derivative of V with respect to r is
It describes the rate with which a cone's volume changes if its radius is varied and its height is kept constant. The partial derivative with respect to h is
and represents the rate with which the volume changes if its height is varied and its radius is kept constant.
Equations involving an unknown function's partial derivatives are called partial differential equations and are common in physics, engineering, and other sciences and applied disciplines.
Notation
For the following examples, let f be a function in x, y and z.First-order partial derivatives:
Second-order partial derivatives:
Second-order mixed derivatives:
Higher-order partial and mixed derivatives:
When dealing with functions of multiple variables, some of these variables may be related to each other, and it may be necessary to specify explicitly which variables are being held constant. In fields such as statistical mechanics, the partial derivative of f with respect to x, holding y and z constant, is often expressed as
Formal definition and properties
Like ordinary derivatives, the partial derivative is defined as a limit. Let U be an open subset of Rn and f : U → R a function. We define the partial derivative of f at the point a = (a1, ..., an) ∈ U with respect to the i-th variable xi as
Even if all partial derivatives ∂f/∂xi(a) exist at a given point a, the function need not be continuous there. However, if all partial derivatives exist in a neighborhood of a and are continuous there, then f is totally differentiable in that neighborhood and the total derivative is continuous. In this case, we say that f is a C1 function. We can use this fact to generalize for vector valued functions (f : U → R'm) by carefully using a componentwise argument.
The partial derivative
can be seen as another function defined on U and can again be partially differentiated. If all mixed second order partial derivatives are continuous at a point (or on a set), we call f a C2 function at that point (or on that set); in this case, the partial derivatives can be exchanged by Clairaut's theorem:
See also
- d'Alembertian operator
- Curl
- Directional derivative
- Divergence
- Exterior derivative
- Gradient
- Jacobian
- Laplacian
- Symmetry of second derivatives
- Triple product rule, also known as the cyclic chain rule.
Mathematics (colloquially, maths or math) is the body of knowledge centered on such concepts as quantity, structure, space, and change, and also the academic discipline that studies them. Benjamin Peirce called it "the science that draws necessary conclusions".
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function expresses dependence between two quantities, one of which is given (the independent variable, argument of the function, or its "input") and the other produced (the dependent variable, value of the function, or "output").
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derivative is a measurement of how a function changes when the values of its inputs change. Loosely speaking, a derivative can be thought of as how much a quantity is changing at some given point.
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In mathematics (more precisely in differential calculus), the term total derivative has a number of closely related meanings.
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- The total derivative of a function of several variables, with respect to one of its variables, is, in contrast to the partial derivative, a
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Vector calculus (also called vector analysis) is a field of mathematics concerned with multivariate real analysis of vectors in a metric space with two or more dimensions (some results can only be applied to three dimensions[1]).
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In mathematics, differential topology is the field dealing with differentiable functions on differentiable manifolds. It arises naturally from the study of the theory of differential equations. Differential geometry is the study of geometry using differential calculus (cf.
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Adrien-Marie Legendre
Adrien-Marie Legendre
Born September 18 1752
Paris, France
Died January 10 1833 (aged 82)
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Adrien-Marie Legendre
Born September 18 1752
Paris, France
Died January 10 1833 (aged 82)
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Carl Jacobi
Carl Gustav Jacob Jacobi
Born November 10 1804
Potsdam, Germany
Died January 18 1851 (aged 48)
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Carl Gustav Jacob Jacobi
Born November 10 1804
Potsdam, Germany
Died January 18 1851 (aged 48)
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The volume of a solid object is the three-dimensional concept of how much space it occupies, often quantified numerically. One-dimensional figures (such as lines) and two-dimensional shapes (such as squares) are assigned zero volume in the three-dimensional space.
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cone is a three-dimensional geometric shape consisting of all line segments joining a single point (the apex or vertex) to every point of a two-dimensional figure (the base).
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Height is the measurement of vertical distance, but has two meanings in common use. It can either indicate how "tall" something is, or how "high up" it is. For example one could say "That is a tall building", or "That airplane is high up in the sky".
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In classical geometry, a radius (plural: radii) of a circle or sphere is any line segment from its center to its perimeter. By extension, the radius of a circle or sphere is the length of any such segment. The radius is half the diameter.
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In mathematics, a partial differential equation (PDE) is a type of differential equation, i. e. a relation involving an unknown function of several independent variables and its partial derivatives with respect to those variables.
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Physics is the science of matter[1] and its motion[2][3], as well as space and time[4][5] —the science that deals with concepts such as force, energy, mass, and charge.
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Engineering is the applied science of acquiring and applying knowledge to design, analysis, and/or construction of works for practical purposes. The American Engineers' Council for Professional Development, also known as ECPD,[1] (later ABET [2]
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Science (from the Latin scientia, 'knowledge'), in the broadest sense, refers to any systematic knowledge or practice.[1] Examples of the broader use included political science and computer science, which are not incorrectly named, but rather named according to
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In mathematics, the Hessian matrix is the square matrix of second-order partial derivatives of a function. Given the real-valued function
if all second partial derivatives of f exist, then the Hessian matrix of f is the matrix
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if all second partial derivatives of f exist, then the Hessian matrix of f is the matrix
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Statistical mechanics is the application of probability theory, which includes mathematical tools for dealing with large populations, to the field of mechanics, which is concerned with the motion of particles or objects when subjected to a force.
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In mathematics, the concept of a "limit" is used to describe the behavior of a function as its argument either "gets close" to some point, or as it becomes arbitrarily large; or the behavior of a sequence's elements as their index increases indefinitely.
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In topology and related fields of mathematics, a set U is called open if, intuitively speaking, starting from any point x in U one can move by a small amount in any direction and still be in the set U.
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In mathematics, a continuous function is a function for which, intuitively, small changes in the input result in small changes in the output. Otherwise, a function is said to be discontinuous.
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neighbourhood (or neighborhood) is one of the basic concepts in a topological space. Intuitively speaking, a neighbourhood of a point is a set containing the point where you can wiggle the point a bit without leaving the set.
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In mathematics, the symmetry of second derivatives refers to the possibility of interchanging the order of taking partial derivatives of a function
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- f(x1, x2, ...
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In special relativity, electromagnetism and wave theory, the d'Alembert operator , also called the d'Alembertian or the wave operator, is the Laplace operator of Minkowski space and other solutions of the Einstein equation.
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cURL is a command line tool for transferring files with URL syntax, supporting FTP, FTPS, HTTP, HTTPS, TFTP, SCP, SFTP, Telnet, DICT, and LDAP. cURL supports HTTPS certificates, HTTP POST, HTTP PUT, FTP uploading, Kerberos, HTTP form based upload, proxies, cookies, user+password
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In mathematics, the directional derivative of a multivariate differentiable function along a given vector V at a given point P intuitively represents the instantaneous rate of change of the function, moving through P, in the direction of V.
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In vector calculus, the divergence is an operator that measures the magnitude of a vector field's source or sink at a given point; the divergence of a vector field is a (signed) scalar.
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In mathematics, the exterior derivative operator of differential geometry extends the concept of the differential of a function to differential forms of higher degree. It is important in the theory of integration on manifolds, and is the differential (coboundary) used to define de
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gradient of a scalar field is a vector field which points in the direction of the greatest rate of increase of the scalar field, and whose magnitude is the greatest rate of change.
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Jacobian is shorthand for either the Jacobian matrix or its determinant, the Jacobian determinant.
In algebraic geometry the Jacobian of a curve means the Jacobian variety: a group variety associated to the curve, in which the curve can be embedded.
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In algebraic geometry the Jacobian of a curve means the Jacobian variety: a group variety associated to the curve, in which the curve can be embedded.
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