Index
Definitions and theorems of Mathematical Analysis 2
The 84 definitions and 77 theorems of the theory of Mathematical Analysis 2, in the order of the topics. Each entry links to its box in the theory page.
Functions of several variables
- Definition The space ℝⁿ
- Definition Euclidean norm and distance
- Definition Circular neighborhood
- Definition Interior, exterior and boundary point
- Definition Accumulation point
- Definition Open set and closed set
- Definition Bounded set and compact set
- Definition Connected set
- Definition Function of two or more variables
- Definition Natural domain
- Definition Level curve
- Definition Level surface
- Definition Limit in two variables
- Theorem Restriction theorem
- Theorem Comparison theorem in two variables
- Proposition Useful upper bounds
- Theorem Bounded times infinitesimal
- Proposition Computation with polar coordinates
- Definition Continuous function
- Theorem Continuity of elementary functions
- Definition Continuous extension
- Theorem Weierstrass theorem in several variables
Differential calculus in several variables
- Definition Partial derivative
- Definition Gradient
- Definition Differentiable function
- Theorem Differentiability implies continuity
- Definition Tangent plane to the graph
- Theorem Total differential theorem
- Definition Directional derivative
- Theorem Gradient formula
- Theorem Direction of steepest ascent
- Theorem Chain rule
- Definition Second partial derivatives
- Theorem Schwarz theorem
- Definition Hessian matrix
- Theorem Second-order Taylor formula with Peano remainder
Implicit functions
- Definition Implicit equation and zero set
- Definition Implicitly defined function
- Theorem Dini's theorem (two variables)
- Proposition Second derivative of the implicit function
- Proposition Tangent line to an implicitly defined curve
- Theorem Dini's theorem (three variables)
- Proposition Tangent plane to a level surface
- Theorem Dini's theorem for systems
Free and constrained maxima and minima
- Definition Maximum and minimum, local and absolute
- Definition Critical point
- Theorem Fermat's theorem in several variables
- Definition Saddle point
- Definition Hessian matrix
- Definition Sign of a symmetric matrix
- Theorem Classification with the Hessian matrix
- Theorem Weierstrass's theorem
- Definition Constrained extremum
- Theorem Lagrange multiplier theorem
- Theorem Multipliers with two constraints
Curves and line integrals
- Definition Parametric curve and trace
- Definition Simple curves and closed curves
- Definition Regular curve
- Definition Tangent vector and tangent line
- Definition Piecewise regular curve
- Definition Equivalent curves
- Definition Orientation and opposite curve
- Definition Length of a curve
- Theorem Length formula
- Theorem Invariance of length under reparametrization
- Definition Arc length
- Definition Line integral of the first kind
- Definition Mass and center of mass of a wire
- Definition Vector field
- Definition Line integral of the second kind and work
- Definition Notation with differential forms
Vector fields and Green’s theorem
- Definition Vector field
- Definition Work and circulation
- Definition Conservative field and potential
- Theorem Work of a conservative field
- Theorem Characterization of conservative fields
- Definition Irrotational field (closed field)
- Theorem A conservative field of class C¹ is irrotational
- Definition Simply connected open set
- Theorem Irrotational on a simply connected set implies conservative
- Proposition Potential by successive integration
- Proposition Potential as work along a polygonal path
- Definition Regular domain and positively oriented boundary
- Theorem Green's theorem
- Proposition Area formulas
Double and triple integrals
- Definition Double integral over a rectangle
- Theorem Reduction theorem (Fubini) on rectangles
- Theorem Functions with separated variables
- Definition Normal domain
- Theorem Reduction formulas on normal domains
- Theorem Linearity, additivity, monotonicity
- Theorem Change of variables and the Jacobian
- Theorem Polar coordinates
- Theorem Odd functions on symmetric domains
- Definition Triple integral
- Theorem Integration by vertical lines
- Theorem Integration by slices
- Theorem Cylindrical coordinates
- Theorem Spherical coordinates
- Definition Area and volume as integrals
- Definition Mass and centroid
Surfaces, fluxes and the theorems of Gauss and Stokes
- Definition Parametric surface
- Definition Regular surface
- Definition Normal vector
- Definition Tangent plane to a surface
- Definition Cartesian surface
- Definition Area of a surface
- Theorem Area of a Cartesian surface
- Theorem Area of a surface of revolution
- Definition Surface integral
- Definition Oriented surface
- Definition Flux of a vector field
- Theorem Flux through a Cartesian surface
- Definition Divergence
- Definition Curl
- Theorem Properties of divergence and curl
- Definition Regular domain in space
- Theorem Divergence theorem (Gauss)
- Definition Compatible orientation of surface and boundary
- Theorem Stokes' theorem (curl theorem)
Sequences and series of functions
- Definition Sequence of functions
- Definition Pointwise convergence
- Definition Uniform convergence
- Theorem Sup criterion
- Theorem Continuity of the uniform limit
- Theorem Passing to the limit under the integral sign
- Theorem Differentiation theorem for sequences
- Definition Series of functions and pointwise convergence
- Definition Uniform convergence of a series
- Definition Total convergence
- Theorem Weierstrass test: total implies uniform
- Theorem Continuity of the sum
- Theorem Term-by-term integration
- Theorem Term-by-term differentiation
Power series and Taylor series
- Definition Power series
- Theorem Radius of convergence theorem
- Definition Set of convergence
- Theorem Ratio test for the radius
- Theorem Root test for the radius
- Theorem Total convergence on interior compact sets
- Theorem Continuity of the sum
- Theorem Abel's theorem
- Theorem Term-by-term differentiation
- Theorem Term-by-term integration
- Definition Taylor and Maclaurin series
- Theorem Uniqueness of the power series expansion
- Definition Analytic function
- Theorem Expandability criterion
- Theorem Standard expansions
Fourier series
- Definition Periodic function
- Definition Trigonometric polynomial
- Theorem Orthogonality of the trigonometric system
- Proposition Coefficients of a trigonometric polynomial
- Definition Fourier coefficients
- Definition Fourier series
- Proposition Cosine-only and sine-only series
- Definition Piecewise smooth function
- Theorem Pointwise convergence theorem
- Theorem Uniform convergence for continuous piecewise smooth functions
- Theorem Parseval's identity
- Definition Fourier series with period 2L