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Shan K
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Will anyone help me to under stand the covariant and contravariant vector ? And can anyone show me the derivation of
dr=(dr/dx)dx + (dr/dy)dy + (dr/dz)dz
dr=(dr/dx)dx + (dr/dy)dy + (dr/dz)dz
Shan K said:Will anyone help me to under stand the covariant and contravariant vector ? And can anyone show me the derivation of
dr=(dr/dx)dx + (dr/dy)dy + (dr/dz)dz
Shan K said:Will anyone help me to under stand the covariant and contravariant vector ? And can anyone show me the derivation of
dr=(dr/dx)dx + (dr/dy)dy + (dr/dz)dz
cosmik debris said:Do a search, this comes up quite frequently. There are some excellent explanations in the Linear Algebra forum.
Covariant and contravariant vectors are two types of vectors used in multivariable calculus and differential geometry. The main difference between them is in how they transform under coordinate transformations. A covariant vector changes its components when the coordinate system is changed, while a contravariant vector keeps its components the same and changes its basis vectors.
Covariant and contravariant vectors are related through the metric tensor, which is a mathematical object that describes the relationship between the basis vectors of a coordinate system. The metric tensor is used to convert between covariant and contravariant vectors, allowing for calculations to be done in either coordinate system.
Yes, covariant and contravariant vectors can be represented graphically using a coordinate system. A covariant vector can be visualized as a set of arrows pointing in the direction of the basis vectors, while a contravariant vector can be represented as a set of arrows pointing in the opposite direction of the basis vectors.
Covariant and contravariant vectors are used in many fields, including physics, engineering, and computer graphics. They are particularly useful in the study of curved spaces, such as in general relativity, where the metric tensor is used to define the curvature of spacetime.
Covariant and contravariant vectors are only applicable in coordinate systems that have a metric tensor defined. In addition, they are only defined in vector spaces that have a notion of distance and angle, such as Euclidean spaces. They also have certain transformation properties that must be considered in order to use them accurately in calculations.