Covariant vectors vs reciprocal vectors

In summary, covariant vectors and reciprocal vectors are two types of vectors used in mathematics and physics. Covariant vectors are used to represent coordinate-independent physical quantities, while reciprocal vectors represent basis vectors of a coordinate system. They are related through the dot product and are important in understanding the transformation of vectors between different coordinate systems. Examples of covariant vectors include displacement, velocity, and force, while examples of reciprocal vectors include unit vectors in cartesian or polar coordinates. Covariant and reciprocal vectors differ from contravariant vectors and dual vectors in the way they transform under a change of coordinate systems. Overall, covariant and reciprocal vectors play a crucial role in understanding physical laws and equations, representing physical quantities and basis vectors in a coordinate-independent manner, and understanding
  • #1
ForMyThunder
149
0
If there is a contravariant vector

v=aa+bb+cc

with a reciprocal vector system where

[abc]v=xb×c+ya×c+za×b

would the vector expressed in the reciprocal vector system be a covariant vector?

Is there any connection between the reciprocal vector system of a covariant vector and a

covariant vector?
 
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  • #2
Nevermind, I think I understand it now.
 

What is the difference between covariant vectors and reciprocal vectors?

Covariant vectors and reciprocal vectors are two different types of vectors used in mathematics and physics. In short, covariant vectors are used to represent physical quantities that are independent of the choice of coordinate system, while reciprocal vectors are used to represent the basis vectors of a coordinate system.

How are covariant vectors and reciprocal vectors related?

Covariant vectors and reciprocal vectors are related through a mathematical operation called the dot product. The dot product of a covariant vector and a reciprocal vector gives a scalar value, which represents the magnitude of the original covariant vector. This relationship is important in understanding the transformation of vectors between different coordinate systems.

What are some examples of covariant vectors and reciprocal vectors?

Examples of covariant vectors include displacement, velocity, and force. These physical quantities do not change with the choice of coordinate system and can be represented by covariant vectors. Examples of reciprocal vectors include unit vectors in cartesian coordinates (i, j, k) or polar coordinates (r̂, θ̂). These basis vectors represent the direction of a coordinate system and can be used to construct other vectors.

How do covariant vectors and reciprocal vectors differ from contravariant vectors and dual vectors?

Covariant vectors and reciprocal vectors are related to each other, while contravariant vectors and dual vectors are related to each other. The main difference between these pairs is the way they transform under a change of coordinate systems. Covariant vectors and contravariant vectors transform differently, while reciprocal vectors and dual vectors also transform differently. In general, covariant and contravariant vectors are used to represent physical quantities, while reciprocal and dual vectors are used to represent basis vectors.

Why are covariant vectors and reciprocal vectors important in physics?

Covariant vectors and reciprocal vectors play a crucial role in understanding the physical laws and equations in physics. They allow us to represent physical quantities and basis vectors in a coordinate-independent manner, making it easier to solve problems and analyze physical systems. Additionally, the relationship between covariant and reciprocal vectors is essential in understanding the transformation of physical quantities and vectors between different coordinate systems, which is crucial in many areas of physics.

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