Limit of a function of several variables
Summary
The limit of a function of several variables at a point is the value that approaches when approaches in Euclidean space. The approach can be from any direction and along any route. The distance in measures how close is to .
Reading path
To understand this page from the ground up, read these 4 pages in order. The foundations come first.
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- no evidence level
- Euclidean space object
- Function concept
- Function of several variables concept
- Limit of a function of several variables concept
Where this leads
234 pages build on this one.
- Exterior derivative
- Saddle point
- Change of variables in multiple integrals
- Arc length parametrisation
- Volume of a ball in R^n
- Additivity of line integrals under concatenation
- Differentiable function of several variables
- Strict differentiability
- Function with unequal mixed partial derivatives
- Symmetry of second derivatives
- Mean value inequality
- Divergence
- Higher partial derivatives
- Chain rule for Jacobian matrices
- Mean value theorem in several variables
- Jacobian determinant
- Lagrange multiplier with one constraint
- Parametrised surface
- Partial derivative
- Poincaré lemma for one-forms on convex sets
- Conservative vector field
- Laplacian as a sum of second partial derivatives
- Jacobian of spherical coordinates
- Tonelli's theorem
- Regular curve
- Simply connected domain
- Differentiable on an open set
- Area of a region as a line integral
- Derivative gives the directional derivatives
- Chain rule for a composition with a curve
- Stokes' theorem
- Differentiation under the integral sign
- Curl-free vector field on a convex set is conservative
- Path independence of line integrals
- Limit along different paths
- Tangent plane and normal vector of a surface
- Implicit function theorem
- Having a gradient
- Zero derivative on a connected open set implies constant
- Separately continuous but not continuous function
- Rectifiable curve
- Closed curve
- Tangent vector of a curve
- Surface integral
- Chain rule in several variables
- Orientable surface
- Iterated derivative
- Closed and exact forms
- Derivative vanishes at a local extremum
- Divergence of a curl is zero
- Scalar potential
- Reversing a path changes the sign of a line integral
- Cylindrical coordinates
- Derivative of the inverse function
- Curl
- Function for which the iterated integrals differ
- Curl-free field that is not conservative
- Triple integral
- Cavalieri's principle
- Moment of inertia
- Second derivative test in several variables
- Divergence theorem on a rectangle in the plane
- Changing the order of integration
- Generalised Stokes theorem
- Green's theorem
- Taylor's theorem in several variables
- Vector calculus identities
- Continuity in R^n
- Parametrised curve
- Divergence theorem
- Inverse function theorem
- Continuous partial derivatives imply differentiability
- Curl of a gradient is zero
- Line integral of a scalar field
- Centre of mass
- One-form
- Vector field
- Implicit function theorem for a surjective derivative
- Double integral
- Schwarz's theorem for C^2 functions
- Derivative of a map into R^n by components
- Gradient
- Second-order Taylor expansion
- Multi-index notation
- Differential form
- Implicit differentiation
- Fubini's theorem
- Polar coordinates
- Spherical coordinates
- Wedge product of forms
- Continuity in R^n
- Hessian matrix
- Gradient theorem
- Product rule in several variables
- C^1 functions are strictly differentiable
- Integration in polar coordinates
- C^k function characterised by continuous derivatives
- Bounded derivative on a convex set implies Lipschitz
- Partial derivatives do not imply differentiability
- Volume of a box
- Lagrange multipliers
- Gradient is orthogonal to level sets
- Line integral as an integral over the parameter interval
- Harmonic function
- Volume of a ball in R^3
- Surface area
- The exterior derivative squares to zero
- Laplacian
- Jacobian matrix
- Flux through a surface
- Line integral of a vector field
- Homotopy invariance of line integrals of closed forms
- Tangent plane to a graph
- Diffeomorphism
- Local homeomorphism from the inverse function theorem
- Gradient points in the direction of steepest ascent
- Morse–Smale system
- Geodesic curvature
- Shell theorem
- Holomorphic inverse function theorem
- Cauchy–Pompeiu formula
- Cheng–Yau gradient estimate
- Removable singularities for harmonic functions
- Markov partition
- Calculus of variations
- Jensen's formula
- Smoothing by convolution
- Affine differential geometry
- Monodromy theorem
- Wirtinger derivatives
- Harmonic majorant
- Spectral decomposition theorem
- Convolution power
- Weierstrass excess function
- Box–Muller transform
- Shape operator
- Computed tomography as an inverse problem
- Score
- Quasiregular map
- Cauchy–Kovalevskaya theorem
- Derivative of a convolution
- Brelot space
- Nash–Moser theorem
- Poisson–Jensen formula
- Multiplication formula for the Fourier transform
- Second fundamental form
- Rectangular function
- Smoothness and analyticity of harmonic functions
- Fáry–Milnor theorem
- Linear PDE
- Symplectic manifold
- Implicit function theorem in Banach spaces
- First fundamental form
- Sub-mean value property
- Lyapunov–Schmidt reduction
- Numerical continuation
- Smoothness classes
- Delta method
- Regular surface
- Quasilinear PDE
- Closing lemma
- Nontangential limit
- Harmonic measure
- Total curvature
- Anosov flow
- Fenchel's theorem
- Differential-algebraic system of equations
- Hermitian manifold
- Smale's problems on dynamics
- Schwartz space
- Integration in the transform variable
- Envelope
- Radon inversion formula
- Topological quantum field theory
- Associativity of convolution
- Smooth dynamical system
- Double layer potential
- Involute
- Conformal modulus
- Koebe's theorem on the mean value property
- Regular value theorem
- Area-minimizing current
- Inverse function theorem in Banach spaces
- Finsler manifold
- Curvature of a curve
- Compactness of families of harmonic functions
- Principal symbol
- Interchange of summation order
- Torsion of a curve
- Gradient estimates for harmonic functions
- Hyperbolic set
- Harnack's principle
- Poisson integral formula
- Partial differential equation
- Green's identities
- Least harmonic majorant
- First main theorem of Nevanlinna theory
- Liouville's theorem for harmonic functions
- Convolution theorem
- Whitney extension theorem
- Axiom A
- Liouville's theorem for planar harmonic functions
- Gauss's law for potentials
- Tautological one-form
- Extremal length
- Integral transforms of tempered distributions
- Fundamental theorem of space curves
- Graph embedding
- Slice theorem
- Besicovitch set of measure zero
- Range conditions for the Radon transform
- Envelope theorem
- Fubini's theorem for series
- Parabolic and hyperbolic Riemann surfaces
- Poisson integral for the half-space
- Lie symmetry method
- Anosov diffeomorphism
- Semilinear PDE
- Differentiable curve
- Adjoint state method
- Harmonic space
- Cartan–Kähler theorem
- Stability conjecture
- Prime end
- Bôcher's theorem
- Evolute
- Order of a PDE
- Osculating circle
- Four-vertex theorem
- Power series in several variables
- Hartogs lemma for subharmonic functions
- Frenet–Serret formulas
- Whitney–Graustein theorem
- Euler–Lagrange equation
Further reading
- Web https://en.wikipedia.org/wiki/Limit_of_a_function, Functions of more than one variable
About this page
| Tag | 0J8Y1NCY0K |
|---|---|
| URI | m2://0J8Y1NCY0K |
| Aliases | multivariable-limit |
| Kind | concept |
| Writing | W0 stub |
| MSC | 26B05 |
| Specializes | Limit of a function |
| Uses | Function of several variables, Euclidean space |
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Used by
- Differentiable function of several variables
- Partial derivative
- Limit along different paths
- Continuity in R^n
- Continuity in R^n
- Nontangential limit