In mathematics, a multivalued function, also called multifunction, many-valued function, set-valued function, is similar to a function, but may associate several values to each input. More precisely, a multivalued function from a domain X to a codomain Y associates each x in X to one or more values y in Y; it is thus a serial binary relation. Some authors allow a multivalued function to have no value for some inputs (in this case a multivalued function is simply a binary relation). However, in some contexts such as in complex analysis (X = Y = C), authors prefer to mimic function theory as they extend concepts of the ordinary (single-valued) functions. In this context, an ordinary function is often called a single-valued function to avoid confusion. The term multivalued function originated in complex analysis, from analytic continuation. It often occurs that one knows the value of a complex analytic function in some neighbourhood of a point . This is the case for functions defined by the implicit function theorem or by a Taylor series around . In such a situation, one may extend the domain of the single-valued function along curves in the complex plane starting at . In doing so, one finds that the value of the extended function at a point depends on the chosen curve from to ; since none of the new values is more natural than the others, all of them are incorporated into a multivalued function. For example, let be the usual square root function on positive real numbers. One may extend its domain to a neighbourhood of in the complex plane, and then further along curves starting at , so that the values along a given curve vary continuously from . Extending to negative real numbers, one gets two opposite values for the square root—for example ±i for –1—depending on whether the domain has been extended through the upper or the lower half of the complex plane. This phenomenon is very frequent, occurring for nth roots, logarithms, and inverse trigonometric functions. To define a single-valued function from a complex multivalued function, one may distinguish one of the multiple values as the principal value, producing a single-valued function on the whole plane which is discontinuous along certain boundary curves. Alternatively, dealing with the multivalued function allows having something that is everywhere continuous, at the cost of possible value changes when one follows a closed path (monodromy). These problems are resolved in the theory of Riemann surfaces: to consider a multivalued function as an ordinary function without discarding any values, one multiplies the domain into a many-layered covering space, a manifold which is the Riemann surface associated to . (Wikipedia).
11_2_1 The Geomtery of a Multivariable Function
Understanding the real-life 3D meaning of a multivariable function.
From playlist Advanced Calculus / Multivariable Calculus
Local linearity for a multivariable function
A visual representation of local linearity for a function with a 2d input and a 2d output, in preparation for learning about the Jacobian matrix.
From playlist Multivariable calculus
11_3_6 Continuity and Differentiablility
Prerequisites for continuity. What criteria need to be fulfilled to call a multivariable function continuous.
From playlist Advanced Calculus / Multivariable Calculus
Multivariable Calculus | Differentiability
We give the definition of differentiability for a multivariable function and provide a few examples. http://www.michael-penn.net https://www.researchgate.net/profile/Michael_Penn5 http://www.randolphcollege.edu/mathematics/
From playlist Multivariable Calculus
Continuity vs Partial Derivatives vs Differentiability | My Favorite Multivariable Function
In single variable calculus, a differentiable function is necessarily continuous (and thus conversely a discontinuous function is not differentiable). In multivariable calculus, you might expect a similar relationship with partial derivatives and continuity, but it turns out this is not th
Multivariable maxima and minima
A description of maxima and minima of multivariable functions, what they look like, and a little bit about how to find them.
From playlist Multivariable calculus
Worldwide Calculus: Multi-Component Functions of a Single Variable
Lecture on 'Multi-Component Functions of a Single Variable' from 'Worldwide Multivariable Calculus'. For more lecture videos and $10 digital textbooks, visit www.centerofmath.org.
From playlist Worldwide Multivariable Calculus
This is the simplest case of taking the derivative of a composition involving multivariable functions.
From playlist Multivariable calculus
Mateusz Skomora: Separation theorems in signed tropical convexities
The max-plus semifield can be equipped with a natural notion of convexity called the “tropical convexity”. This convexity has many similarities with the standard convexity over the nonnegative real numbers. In particular, it has been shown that tropical polyhedra are closely related to the
From playlist Workshop: Tropical geometry and the geometry of linear programming
Worldwide Calculus: Multivariable Functions
Lecture on 'Multivariable Functions' from 'Worldwide Multivariable Calculus'. For more lecture videos and $10 digital textbooks, visit www.centerofmath.org.
From playlist Worldwide Multivariable Calculus
Unit II: Lec 5 | MIT Calculus Revisited: Single Variable Calculus
Unit II: Lecture 5: Implicit Differentiation Instructor: Herb Gross View the complete course: http://ocw.mit.edu/RES18-006F10 License: Creative Commons BY-NC-SA More information at http://ocw.mit.edu/terms More courses at http://ocw.mit.edu
From playlist MIT Calculus Revisited: Single Variable Calculus
Toshitake Kohno: Quantum symmetry of conformal blocks and representations of braid [...]
Find this video and other talks given by worldwide mathematicians on CIRM's Audiovisual Mathematics Library: http://library.cirm-math.fr. And discover all its functionalities: - Chapter markers and keywords to watch the parts of your choice in the video - Videos enriched with abstracts, b
From playlist Geometry
UHCL 07a Graduate Database Course - ER Components Part 1
This video corresponds to the unit 2 notes for a graduate database (dbms) course taught by Dr. Gary D. Boetticher at the University of Houston - Clear Lake (UHCL). It discusses various components of an ER and EER diagram. Please note that this video is split into three parts. This is par
From playlist UHCL Graduate Database Course
Logarithms of Negative BASES and Negative Arguments! [ Log of Complex Numbers z ]
STEMweek Deal Chaos Double Pendulum! =D https://stemerch.com/products/chaos-double-pendulum German Version: https://youtu.be/FjfIeZN1CUU Today we are going to derive the equation for the (natural) log of negative arguments and negative base. We employ complex numbers for that and will tal
From playlist Random problems
A Continuous Transformation of a Double Cover of the Complex Plane into a Torus
To learn more about Wolfram Technology Conference, please visit: https://www.wolfram.com/events/technology-conference/ Speaker: Dominic Milioto Wolfram developers and colleagues discussed the latest in innovative technologies for cloud computing, interactive deployment, mobile devices, a
From playlist Wolfram Technology Conference 2017
Database Normalisation: First Normal Form
This video is part of a series about database normalisation. It explains how to transform a database into first normal form by working through an example. It covers the criteria for the first normal form including ensuring that a table does not contain composite or multi-valued attribute
From playlist Database Normalisation
How Large is the Shadow of a Symplectic Ball? - Alberto Abbondandolo
Alberto Abbondandolo University of Pisa, Italy February 8, 2012 I will discuss a middle-dimensional generalization of Gromov's Non-Squeezing Theorem. For more videos, visit http://video.ias.edu
From playlist Mathematics
ME565 Lecture 4: Cauchy Integral Formula
ME565 Lecture 4 Engineering Mathematics at the University of Washington Cauchy Integral Formula Notes: http://faculty.washington.edu/sbrunton/me565/pdf/L04.pdf Course Website: http://faculty.washington.edu/sbrunton/me565/ http://faculty.washington.edu/sbrunton/
From playlist Engineering Mathematics (UW ME564 and ME565)
11_3_1 The Gradient of a Multivariable Function
Using the partial derivatives of a multivariable function to construct its gradient vector.
From playlist Advanced Calculus / Multivariable Calculus
Anthony Nouy: Adaptive low-rank approximations for stochastic and parametric equations [...]
Find this video and other talks given by worldwide mathematicians on CIRM's Audiovisual Mathematics Library: http://library.cirm-math.fr. And discover all its functionalities: - Chapter markers and keywords to watch the parts of your choice in the video - Videos enriched with abstracts, b
From playlist Numerical Analysis and Scientific Computing