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extrads
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If the notions of covariant and contravariant tensors were not introduced,what would happen?E.g. what form will the Einstein E.q. Guv=8πTuv be changed into ?
extrads said:If the notions of covariant and contravariant tensors were not introduced,what would happen?
Covariant and contravariant refer to the way in which a mathematical quantity changes when the coordinate systems used to measure it are transformed. In general, covariant quantities change in the same direction as the coordinate system changes, while contravariant quantities change in the opposite direction.
Covariant and contravariant vectors are related through a mathematical operation called the metric tensor. The metric tensor relates the components of a covariant vector to the components of a contravariant vector, allowing for the transformation between the two types of vectors.
Covariant and contravariant are essential concepts in tensor calculus, as they allow for the proper transformation of tensors between different coordinate systems. Without taking into account the covariant and contravariant nature of tensors, calculations and equations in tensor calculus may result in incorrect results.
In tensor notation, covariant quantities are represented with lower indices, while contravariant quantities are represented with upper indices. This notation helps to distinguish between the two types of quantities and is used in various mathematical operations involving tensors.
A common example of a covariant quantity is length, as it changes in the same direction as the coordinate system changes. An example of a contravariant quantity is velocity, as it changes in the opposite direction to the coordinate system changes. This can be seen when comparing the velocity of an object from an observer on the ground and an observer on a moving train.