How the covariant and contravariant base vectors are found.Orodruin said:Exactly what is it that confuses you?
not to sound slow but just to clarify.Orodruin said:It is a definition.
You choose those base vectors to be normal vectors to the coordinate functions. The normal vector of a function is given by its gradient.
The other set of base vectors is chosen to be the tangent vectors of the coordinate lines.
Covariance and contravariance are terms used in mathematics and computer science to describe the relationship between types and subtypes or supertypes. In simple terms, covariance refers to the ability of a subtype to be substituted for its supertype, while contravariance refers to the ability of a supertype to be substituted for its subtype.
In programming, covariance and contravariance are used to ensure type safety and to facilitate polymorphism. By allowing subtypes and supertypes to be substituted for each other, it allows for greater flexibility and extensibility in code without compromising type correctness.
One example of covariance is the use of a subclass object in place of its superclass in a method. For example, if a method expects an Animal object and we pass in a Dog object, which is a subclass of Animal, then the Dog object is being used covariantly. An example of contravariance is the use of a superclass object in place of its subclass. For instance, if a method expects a Dog object and we pass in an Animal object, which is a superclass of Dog, then the Animal object is being used contravariantly.
Using covariance and contravariance can lead to more concise and flexible code, as it allows for different types to be used interchangeably. It also helps to prevent errors and improve type safety by ensuring that the correct types are being used in different contexts.
While covariance and contravariance can improve code flexibility, it can also make code more complex and potentially lead to hard-to-debug errors if not used carefully. Additionally, some programming languages may not support covariance and contravariance, limiting their use in certain contexts.