Vectors, metric and the connection 1 contravariant and. An informal discussion on tensor calculus chris tiee abstract. In multilinear algebra and tensor analysis, covariance and contravariance describe how the quantitative description of certain geometric or physical entities changes with a change of basis from one coordinate system to another. This blog post is about covariance, contravariance, and invariance of python types. Because of these restrictions, covariance is most commonly used in producers types that return something and immutable types. To get an intuitive idea of the difference between invariance, covariance, and contravariance, suppose we have an aquarium tank filled with water, and we define rectangular cartesian coordinates x,y,z to identify each point in the tank. Covariance and contravariance in generics microsoft docs. Vectors, metric and the connection 1 contravariant and covariant vectors 1. A change of scale on the reference axes corresponds to a change of units in the problem. Covariance and contravariance of functors, in category theory. Covariance and contravariance computer science, whether a type system preserves the ordering. It need to be expanded in some basis for practical calculation and its.
Now imagine that all of your rulers shrink by a factor of 10. If a and b are vectors of observations with equal length, cov a,b is the 2 by 2 covariance matrix. As per the above example classes, small is a base class for big and big is a base class for bigger. I hope many of you have come across the two buzz words covariance and contravariance while browsing through. Abstract vector is a physical quantity and it does not depend on any coordinate system. Given two types s and t such that s covariance and contravariance of vectors 3 for some invertible n. First of all, remember that the reason of having covariant or contravariant tensors is because you want to represent the same thing in a different coordinate system. Covariance and contravariance of vectors wikipedia. Covariance, in probability theory and statistics, the measure of how much two random. This post is intended to give a developerfocused explanation of the terms without diving into the category theory behind them too much.
All samples ive seen so far was just casting some object into system. These two topics will be then combined to write down the formalism for special. I have participated in interviews where the person interviewing couldnt ask questions about the subject clearly enough for the candidate to understand. Requirements 1 the derivative of a tensor must be a tensor. In multilinear algebra and tensor analysis, covariance and contravariance describe how the quantitative description of certain geometric or physical entities changes with a change of basis in physics, a basis is sometimes thought of as a set of reference axes. Covariance and contravariance are terms that many developers find puzzling. Intuitive way to understand covariance and contravariance. Here, each vector y j of the f basis is a linear combination of the vectors x i of the f basis, so that contravariant transformation a vector v in v is expressed uniquely as a linear combination of the elements of the f basis as 2. I aim to show that the knowledge of these concepts helps to write more reliable code and it is beneficial to all python programmers. At that point it becomes a form of hazing and esoteric blathering. The other set of base vectors is chosen to be the tangent vectors of the coordinate lines. In this video, i describe the meaning of contravariant and covariant vector components.
The basis set for dual vectors enables any dual vector p to be written. Generic type parameters support covariance and contravariance to provide greater flexibility in assigning and using generic types. Transformation laws and their relation to fancyschmancy tensors 17 2. Wikipedia covariance and contravariance of vectors tensor. The point to remember here is that a derived class will always. Change of basis, reciprocal basis vectors, covariant and. July 21, 2017 subtyping is a tricky topic in programming language theory. Wikipedia covariance and contravariance of vectors free download as pdf file. It is an idea that seems to originate in physics but in fact it is a more a mathematical concept. Chapter 10 vectors and tensors georgia institute of. Covariance and contravariance are collectively referred to as variance.
The correlation coefficient is a unitless version of the same thing. Understand the meaning of covariance and correlation. Be able to compute the covariance and correlation of two random variables. Covariant and contravariant vectors are so tied up with the formalism of tensors as used in general relativity that its quite hard to disentangle the notion and look at in a striaghtforward manner. This discussion has focused on scalars and vectors, but. Covariant derivative 22 mar 2012riemannchristoffel curvature tensor. This collection of notes attempts to demystify some of the vagaries of tensor analysis, to emphasize the connections between the old way of doing things and the new. For contravariant vectors or more correctly vectors with contravariant components, the matrix that transforms the vector components must be the inverse of the matrix that transforms the basis vectors. To understand contravariance, stop thinking of types in terms of is a more specialized type and switch the focus to the idea of acceptance. For other uses of covariant or contravariant, see covariance and contravariance. The mnemonic for producer extends, consumer super pecs, from the book effective java by joshua bloch gives an easy way to remember when to use covariance and contravariance. Net framework 4, variant type parameters are restricted to generic interface and generic. Imagine youve got some sort of coordinate system, so you can imagine measuring everything with n rulers. As shown on figure 1, the dual basis vectors are perpendicular to all basis vectors with a di erent index, and the scalar product of the dual basis vector with the basis vector of the same index is unity.
A, in order to represent a linear transformation from x to y, or a linear transformation from y. I like picturing this in 2d, but it works in any number of dimensions. For other uses of covariant or contravariant, see covariance and contravariance disambiguation. In physics, a basis is sometimes thought of as a set of reference axes.
In this article, i will try to give an introduction to the concepts behind these buzz words. Covariance and contravariance of vectors semantic scholar. This blog post is also part of the fp complete haskell syllabus and part of our haskell training. On the other hand for vectors with covariant components, the components must be transformed by the same matrix as the change of basis matrix. Twenty years ago, in an article titled \covariance and contravariance. A basic explanation of the covariance and contravariance of vectors and of tensors too, since vectors are tensors of rank math1math is best done with the help of a geometric representation or illustration. Intuition behind covariant and contravariant vectors.
Covariance, contravariance, and invariance the ultimate. Covariance and contravariance allow us to be flexible when dealing with class hierarchy. I define these concepts and explain them in detail. Covariance and contravariance of vectors explained. Change of basis, reciprocal basis vectors, covariant and contravariant components of a vector and metric tensor. Chapter 10 vectors and tensors in this chapter we explain how a vector space v gives rise to a family of associated tensor spaces, and how mathematical objects such as linear maps or quadratic forms should be understood as being elements of these spaces. We wish to write the velocity vector at some time t 0. In multilinear algebra and tensor analysis, covariance and contravariance describe how the. Consider the following class hierarchy before we learn about covariance and contravariance. Or, in indices, a tangent vector is by definition a vector that is a linear combination of the coordinate partials. Linear functionals, volume forms, and rethinking integration 15 2. A brief summary of facts about variance in the common language runtime. Contours, covariance, contravariance, dual basis 5 note. Kolkers answer gives the gory detail, but heres a quick and dirty version.
I have been wanting for a long time to do a series of articles about covariance and contravariance which i will shorten to variance for the rest of this series. Covariant and contravariant vectors physics stack exchange. C cov a,b returns the covariance between two random variables a and b. Covariance and contravariance of vectors 5 then the frame f is related to the frame f by the inverse of the jacobian matrix of the coordinate transition. So i watched this video and after the first 5 mins, this is how i understand it. Data, covariance, and correlation matrix nathaniel e. Since you asked for an intuitive way to understand covariance and contravariance, i think this will do. What are the physical differences between contravariant. You choose those base vectors to be normal vectors to the coordinate functions.
In terms of the x coordinate system the contravariant components of p are x 1, x 2 and the covariant components are x 1, x 2. Covariance and contravariance in scala knoldus blogs. Covariance and contravariance of vectors wikimili, the. To understand in detail how the representations of vectors in different coordinate systems are related to each other, consider the displacement vector p in a flat 2dimensional space shown below. Every step is accompanied by a fairly straightforward code snippet. Covariance and contravariance of vectors, in mathematics and theoretical physics. Covariance and contravariance are terms that are used in different ways in the theory of objectoriented programming and they sound advanced and difficult but in fact the idea they they encapsulate is very, very simple. With the notion of contravariant and covariant components of a vector, we. The question is what is the difference between covariance and contravariance. Helwig u of minnesota data, covariance, and correlation matrix updated 16jan2017. For more information, please see the wikipedia page on covariance and contravariance. Covariant and contravariant vectors alok kumar1 iiser, bhopal iti campus gas rahat building govindpura, bhopal 23 india.
Covariance and contravariance fabulous adventures in coding. As mentioned in a previous video, tensors are invariant under coordinate transformations. In this article i will give an overview of covariance and contravariance. Covariance and contravariance are properties of a mapping function that associates one member of a set with another. As mentioned in a previous video, tensors are invariant. Cliquer ici pour le pdf en francais the foolish purpose of this post is to tackle the concept of tensor, while trying to keep it accessible to the widest audience possible. In multilinear algebra and tensor analysis, covariance and contravariance describe how the quantitative description of certain geometric or physical entities changes with a change of basis. Im having trouble understanding the covariance and contravariance of a vector. The normal vector of a function is given by its gradient. Covariance and contravariance computer science wikipedia. If a and b are matrices of observations, cov a,b treats a and b as vectors and is equivalent to cov a.
Discusses covariance and contravariance in generic and nongeneric delegates and provides a list of variant generic delegates in the. Lets work in the three dimensions of classical space forget time, relativity, four vectors etc. A generic type parameter that is not marked covariant or contravariant is referred to as invariant. This illustrates a subtle and slightly unfortunate design choice in the method type inference algorithm, which of course was designed long before covariance and contravariance were added to the language. Helwig assistant professor of psychology and statistics university of minnesota twin cities updated 16jan2017 nathaniel e. While covariance enables you to use a more derived type, contravariance allows you to take advantage of a more generic type that that was specified. Covariance and contravariance of vectors wikimili, the free encyclopedia.