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List Of Multivariable Calculus Topics
This is a list of multivariable calculus topics. See also multivariable calculus, vector calculus, list of real analysis topics, list of calculus topics. *Closed and exact differential forms *Contact (mathematics) *Contour integral *Contour line *Critical point (mathematics) *Curl (mathematics) *Current (mathematics) *Curvature *Curvilinear coordinates *Del *Differential form *Differential operator *Directional derivative *Divergence *Divergence theorem *Double integral *Equipotential surface *Euler's theorem on homogeneous functions *Exterior derivative *Flux *Frenet–Serret formulas *Gauss's law *Gradient *Green's theorem *Green's identities *Harmonic function *Helmholtz decomposition *Hessian matrix *Hodge star operator *Inverse function theorem *Irrotational vector field *Isoperimetry *Jacobian matrix *Lagrange multiplier *Lamellar vector field *Laplacian *Laplacian vector field *Level set *Line integral * Matrix calculus *Mixed derivatives *Monkey saddle *Multiple integral *Ne ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Multivariable Calculus
Multivariable calculus (also known as multivariate calculus) is the extension of calculus in one variable to calculus with functions of several variables: the differentiation and integration of functions involving multiple variables ('' multivariate''), rather than just one. Multivariable calculus may be thought of as an elementary part of calculus on Euclidean space. The special case of calculus in three dimensional space is often called ''vector calculus''. Introduction In single-variable calculus, operations like differentiation and integration are made to functions of a single variable. In multivariate calculus, it is required to generalize these to multiple variables, and the domain is therefore multi-dimensional. Care is therefore required in these generalizations, because of two key differences between 1D and higher dimensional spaces: # There are infinite ways to approach a single point in higher dimensions, as opposed to two (from the positive and negative direct ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Divergence Theorem
In vector calculus, the divergence theorem, also known as Gauss's theorem or Ostrogradsky's theorem, reprinted in is a theorem relating the '' flux'' of a vector field through a closed surface to the ''divergence'' of the field in the volume enclosed. More precisely, the divergence theorem states that the surface integral of a vector field over a closed surface, which is called the "flux" through the surface, is equal to the volume integral of the divergence over the region enclosed by the surface. Intuitively, it states that "the sum of all sources of the field in a region (with sinks regarded as negative sources) gives the net flux out of the region". The divergence theorem is an important result for the mathematics of physics and engineering, particularly in electrostatics and fluid dynamics. In these fields, it is usually applied in three dimensions. However, it generalizes to any number of dimensions. In one dimension, it is equivalent to the fundamental theorem of cal ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hessian Matrix
In mathematics, the Hessian matrix, Hessian or (less commonly) Hesse matrix is a square matrix of second-order partial derivatives of a scalar-valued Function (mathematics), function, or scalar field. It describes the local curvature of a function of many variables. The Hessian matrix was developed in the 19th century by the German mathematician Otto Hesse, Ludwig Otto Hesse and later named after him. Hesse originally used the term "functional determinants". The Hessian is sometimes denoted by H or \nabla\nabla or \nabla^2 or \nabla\otimes\nabla or D^2. Definitions and properties Suppose f : \R^n \to \R is a function taking as input a vector \mathbf \in \R^n and outputting a scalar f(\mathbf) \in \R. If all second-order partial derivatives of f exist, then the Hessian matrix \mathbf of f is a square n \times n matrix, usually defined and arranged as \mathbf H_f= \begin \dfrac & \dfrac & \cdots & \dfrac \\[2.2ex] \dfrac & \dfrac & \cdots & \dfrac \\[2.2ex] \vdots & \vdot ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Helmholtz Decomposition
In physics and mathematics, the Helmholtz decomposition theorem or the fundamental theorem of vector calculus states that certain differentiable vector fields can be resolved into the sum of an irrotational ( curl-free) vector field and a solenoidal (divergence-free) vector field. In physics, often only the decomposition of sufficiently smooth, rapidly decaying vector fields in three dimensions is discussed. It is named after Hermann von Helmholtz. Definition For a vector field \mathbf \in C^1(V, \mathbb^n) defined on a domain V \subseteq \mathbb^n, a Helmholtz decomposition is a pair of vector fields \mathbf \in C^1(V, \mathbb^n) and \mathbf \in C^1(V, \mathbb^n) such that: \begin \mathbf(\mathbf) &= \mathbf(\mathbf) + \mathbf(\mathbf), \\ \mathbf(\mathbf) &= - \nabla \Phi(\mathbf), \\ \nabla \cdot \mathbf(\mathbf) &= 0. \end Here, \Phi \in C^2(V, \mathbb) is a scalar potential, \nabla \Phi is its gradient, and \nabla \cdot \mathbf is the divergence of the vector fiel ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Harmonic Function
In mathematics, mathematical physics and the theory of stochastic processes, a harmonic function is a twice continuously differentiable function f\colon U \to \mathbb R, where is an open subset of that satisfies Laplace's equation, that is, \frac + \frac + \cdots + \frac = 0 everywhere on . This is usually written as \nabla^2 f = 0 or \Delta f = 0 Etymology of the term "harmonic" The descriptor "harmonic" in the name "harmonic function" originates from a point on a taut string which is undergoing harmonic motion. The solution to the differential equation for this type of motion can be written in terms of sines and cosines, functions which are thus referred to as "harmonics." Fourier analysis involves expanding functions on the unit circle in terms of a series of these harmonics. Considering higher dimensional analogues of the harmonics on the unit ''n''-sphere, one arrives at the spherical harmonics. These functions satisfy Laplace's equation and, over time, "harmon ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Green's Identities
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 operators act. They are named after the mathematician George Green, who discovered Green's theorem. Green's first identity This identity is derived from the divergence theorem applied to the vector field while using an extension of the product rule that : Let and be scalar functions defined on some region , and suppose that is twice continuously differentiable, and is once continuously differentiable. Using the product rule above, but letting , integrate over . Then \int_U \left( \psi \, \Delta \varphi + \nabla \psi \cdot \nabla \varphi \right)\, dV = \oint_ \psi \left( \nabla \varphi \cdot \mathbf \right)\, dS=\oint_\psi\,\nabla\varphi\cdot d\mathbf where is the Laplace operator, is the boundary of region , is the outward pointing unit normal to the surface element and is the oriented surface element. This the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Green's Theorem
In vector calculus, Green's theorem relates a line integral around a simple closed curve to a double integral over the plane region (surface in \R^2) bounded by . It is the two-dimensional special case of Stokes' theorem (surface in \R^3). In one dimension, it is equivalent to the fundamental theorem of calculus. In three dimensions, it is equivalent to the divergence theorem. Theorem Let be a positively oriented, piecewise smooth, simple closed curve in a plane, and let be the region bounded by . If and are functions of defined on an open region containing and have continuous partial derivatives there, then \oint_C (L\, dx + M\, dy) = \iint_ \left(\frac - \frac\right) dA where the path of integration along is counterclockwise. Application In physics, Green's theorem finds many applications. One is solving two-dimensional flow integrals, stating that the sum of fluid outflowing from a volume is equal to the total outflow summed about an enclosing area. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Gradient
In vector calculus, the gradient of a scalar-valued differentiable function f of several variables is the vector field (or vector-valued function) \nabla f whose value at a point p gives the direction and the rate of fastest increase. The gradient transforms like a vector under change of basis of the space of variables of f. If the gradient of a function is non-zero at a point p, the direction of the gradient is the direction in which the function increases most quickly from p, and the magnitude of the gradient 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 as a stationary point. The gradient thus plays a fundamental role in optimization theory, where it is used to minimize a function by gradient descent. In coordinate-free terms, the gradient of a function f(\mathbf) may be defined by: df=\nabla f \cdot d\mathbf where df is the total infinitesimal change in f for a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Frenet–Serret Formulas
In differential geometry, the Frenet–Serret formulas describe the kinematic properties of a particle moving along a differentiable curve in three-dimensional Euclidean space \R^3, or the geometric properties of the curve itself irrespective of any motion. More specifically, the formulas describe the derivatives of the so-called tangent, normal, and binormal unit vectors in terms of each other. The formulas are named after the two French mathematicians who independently discovered them: Jean Frédéric Frenet, in his thesis of 1847, and Joseph Alfred Serret, in 1851. Vector notation and linear algebra currently used to write these formulas were not yet available at the time of their discovery. The tangent, normal, and binormal unit vectors, often called , , and , or collectively the Frenet–Serret basis (or TNB basis), together form an orthonormal basis that spans \R^3, and are defined as follows: * is the unit vector tangent to the curve, pointing in the direction of mot ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Flux
Flux describes any effect that appears to pass or travel (whether it actually moves or not) through a surface or substance. Flux is a concept in applied mathematics and vector calculus which has many applications in physics. For transport phenomena, flux is a vector quantity, describing the magnitude and direction of the flow of a substance or property. In vector calculus flux is a scalar quantity, defined as the surface integral of the perpendicular component of a vector field over a surface. Terminology The word ''flux'' comes from Latin: ''fluxus'' means "flow", and ''fluere'' is "to flow". As '' fluxion'', this term was introduced into differential calculus by Isaac Newton. The concept of heat flux was a key contribution of Joseph Fourier, in the analysis of heat transfer phenomena. His seminal treatise ''Théorie analytique de la chaleur'' (''The Analytical Theory of Heat''), defines ''fluxion'' as a central quantity and proceeds to derive the now well-known expre ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Exterior Derivative
On a differentiable manifold, the exterior derivative extends the concept of the differential of a function to differential forms of higher degree. The exterior derivative was first described in its current form by Élie Cartan in 1899. The resulting calculus, known as exterior calculus, allows for a natural, metric-independent generalization of Stokes' theorem, Gauss's theorem, and Green's theorem from vector calculus. If a differential -form is thought of as measuring the flux through an infinitesimal - parallelotope at each point of the manifold, then its exterior derivative can be thought of as measuring the net flux through the boundary of a -parallelotope at each point. Definition The exterior derivative of a differential form of degree (also differential -form, or just -form for brevity here) is a differential form of degree . If is a smooth function (a -form), then the exterior derivative of is the differential of . That is, is the unique -form such that ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
