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In mathematics, a critical point (or critical number) is a point on the domain of a function where: Euclid, Greek mathematician, 3rd century BC, as imagined by by Raphael in this detail from The School of Athens. ...
A spatial point is an entity with a location in space but no extent (volume, area or length). ...
In mathematics, the domain of a function is the set of all input values to the function. ...
Graph of example function, The mathematical concept of a function expresses the intuitive idea of deterministic dependence between two quantities, one of which is viewed as primary (the independent variable, argument of the function, or its input) and the other as secondary (the value of the function, or output). A...
- one dimension: the derivative is equal to zero or does not exist: it is points that are either stationary points or non-differentiable points.
- in general: there are two distinct concepts: either the derivative (Jacobian) vanishes, or it is not of full rank (or, in either case, the function in not differentiable); these agree in one dimension.
The value of a function at a critical point is called a critical value ("points" are inputs, "values" are outputs). Elements of the codomain of a function which are not critical values are called regular values. For a non-technical overview of the subject, see Calculus. ...
In mathematics, two mathematical objects are considered equal if they are precisely the same in every way. ...
For other uses, see zero or 0. ...
Stationary points (red pluses) and inflection points (green circles). ...
In vector calculus, the Jacobian is shorthand for either the Jacobian matrix or its determinant, the Jacobian determinant. ...
In differential topology, a critical value of a differentiable map between differentiable manifolds is the image of a critical point. ...
A codomain in mathematics is the set of output values associated with (or mapped to) the domain of inputs in a function. ...
Versus stationary point
- See also: Stationary point#Versus critical point
The term "critical point" is often confused with "stationary point". Critical point is more general: a critical point is either a stationary point or the derivative is not defined there. Stationary points (red pluses) and inflection points (green circles). ...
For a smooth function, these are interchangeable, hence the confusion; when a function is understood to be smooth, one can refer to stationary points as critical points, but when it may be non-differentiable, one should distinguish these notions. In mathematical analysis, a differentiability class is a classification of functions according to the properties of their derivatives. ...
In one dimension, critical point generally means "possibly non-differentiable", as in calculus. Calculus (from Latin, pebble or little stone) is a mathematical subject that includes the study of limits, derivatives, integrals, and infinite series, and constitutes a major part of modern university education. ...
In higher dimensions, critical point generally means "derivative is zero" (and the function is understood to be smooth), as in Morse theory. A Morse function is also an expression for an anharmonic oscillator In differential topology, the techniques of Morse theory give a very direct way of analyzing the topology of a manifold by studying differentiable functions on that manifold. ...
Optimization - See also: maxima and minima
By Fermat's theorem, maxima and minima of a function can occur either at its critical points or at points on its boundary. A graph illustrating local min/max and global min/max points In mathematics, maxima and minima, also known as extrema, are points in the domain of a function at which the function takes the largest (maximum), or smallest (minimum) value either within a given neighbourhood (local extrema), or on the...
Fermats theorem is a theorem in real analysis, named after Pierre de Fermat. ...
A graph illustrating local min/max and global min/max points In mathematics, maxima and minima, also known as extrema, are points in the domain of a function at which the function takes the largest (maximum), or smallest (minimum) value either within a given neighbourhood (local extrema), or on the...
A critical point is sometimes not a local maximum or minimum, in which case it is called a saddle point. Plot of y = x3 with a saddle-point at (0,0). ...
Several variables In this section, functions are assumed to be smooth.
For a smooth function of several real variables, the condition of being a critical point is equivalent to all of its partial derivatives being zero; for a function on a manifold, it is equivalent to the exterior derivative being zero. For a map between spaces of arbitrary finite or infinite dimension, it means that the derivative is zero as a linear map. In mathematics, the exterior derivative operator of differential topology, extends the concept of the differential of a function to differential forms of higher degree. ...
In mathematics, a linear transformation (also called linear operator or linear map) is a function between two vector spaces that respects the arithmetical operations addition and scalar multiplication defined on vector spaces, or, in other words, it preserves linear combinations. Definition and first consequences Formally, if V and W are...
If a critical point has a nonsingular Hessian matrix it is called nondegenerate, and the signs of the eigenvalues of the Hessian determine the function's local behavior. In the case of a real function of a real variable, the Hessian is simply the second derivative, and nonsingularity is equivalent to being nonzero. A nondegenerate critical point of a single-variable real function is a maximum if the second derivative is negative, and a minimum if it is positive. In general, the number of negative eigenvalues of a critical point is called its index, and a maximum occurs when all eigenvalues are negative (maximal index: the Hessian is negative definite) and a minimum occurs when all eigenvalues are positive (index zero: the Hessian is positive definite); otherwise it is a saddle point (the Hessian is indefinite (and nonsingular)). Morse theory studies both finite and infinite dimensional manifolds using these ideas. In mathematics, the Hessian matrix is the square matrix of second partial derivatives of a scalar-valued function. ...
In mathematics, a number is called an eigenvalue of a matrix if there exists a nonzero vector such that the matrix times the vector is equal to the same vector multiplied by the eigenvalue. ...
In mathematics, a definite bilinear form B is one for which B(v,v) has a fixed sign (positive or negative) when it is not 0. ...
In mathematics, a definite bilinear form B is one for which B(v,v) has a fixed sign (positive or negative) when it is not 0. ...
A Morse function is also an expression for an anharmonic oscillator In differential topology, the techniques of Morse theory give a very direct way of analyzing the topology of a manifold by studying differentiable functions on that manifold. ...
On a sphere, the sum of the angles of a triangle is not equal to 180°. A sphere is not a Euclidean space, but locally the laws of the Euclidean geometry are good approximations. ...
Gradient vector field In the presence of a Riemannian metric or a symplectic form, to every smooth function is associated a vector field (the gradient or Hamiltonian vector field). These vector fields vanish exactly at the critical points of the original function, and thus the critical points are stationary, i.e. constant trajectories of the flow associated to the vector field. In mathematics, in Riemannian geometry, the metric tensor is a tensor of rank 2 that is used to measure distance and angle in a space. ...
In mathematics, a symplectic manifold is a smooth manifold equipped with a closed, nondegenerate 2-form. ...
Vector field given by vectors of the form (-y, x) In mathematics a vector field is a construction in vector calculus which associates a vector to every point in a Euclidean space. ...
For other uses, see Gradient (disambiguation). ...
In mathematics and physics, the symplectic vector field, also known as the Hamiltonian vector field, is a vector field induced on a symplectic manifold by an energy function or Hamiltonian. ...
Alternative definition (not full rank) Critical points are also sometimes defined to be points where the derivative of a function is not of maximum rank. Sard's theorem states that the set of critical values, in this sense of critical point, of a differentiable function has measure zero. Sards lemma, also known as Sards theorem or the Morse-Sard theorem, is a result of mathematical analysis characterising the image of the critical points of a smooth function F from one Euclidean space to another as having Lebesgue measure 0 (and so small, in a definite sense). ...
Let μ be a measure on a sigma algebra Σ of subsets of a set X. An element A in Σ is said to have measure zero if μ(A)=0. ...
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