Tensors Explained Intuitively: Covariant, Contravariant, Rank
Why Mathematics Stack Exchange recommends it
Mathematics Stack Exchange users point people to it when they are confused about covariant versus contravariant components and want a visual picture before the formal definitions.
Recommended by 2 people across 2 different months, 2017 to 2018.
What it is
An 11-minute animated explainer by Eugene Khutoryansky that visualizes tensors of rank 1, 2 and 3. It introduces covariant and contravariant components geometrically, using dot-product projections and the parallelogram law.
What people said
Strong rec GRANZER · 2018-06 · Mathematics Stack Exchange · on “Vector components vs coordinates?”
This is a great explanation: [this video] But what they mean by "coordinates" is not given in the video. Coordinate means Coordinate functions (ie coordinate vector something like [1,0,0,0]) which would map vector v to its contravariant components.
Recommends Juan · 2018-10 · Mathematics Stack Exchange · on “Contravariance and Covariance of vectors”
I'm having trouble understanding the covariance and contravariance of a vector. So I watched this video [this video] and after the first 5 mins, this is how I understand it: contravariance and covariance are just two ways of identifying a vector. Contravariance means that the components change inversely to the change of the basis (i.e. if the basis vectors $e_1, e_2, e_3$ increase to $(2,0,0),(0,2,0),(0,0,2)$ then a vector $\vec{v}=(2,4,6)$ will now be decreasely displayed as $(1,2,3)$ ) Covariance means the covariant components change alike to the change of the basis. In the video, the covariant components is obtained by taking the dot product of $\vec{v}$ with the basis: $\vec{v}=(\vec{v} \cdot
Cites Antoni Parellada · 2017-07 · Mathematics Stack Exchange · on “What is the connection between a row vector and covariant vector (or column and contravariant)?”
This youtube video [this video] by Eugene Khutoryansky makes the distinction clear between the covariant coordinates of a vector (dot product of the vector with each of the basis vectors or vector projection [LINK] ), and the contravariant coordinates ( parallelogram law [LINK] ): [LINK] Is there a way you could explain to a lay person that this somehow underpins the following fact: Covariant vectors are representable as row vectors. Contravariant vectors are representable as column vectors. ? I would like to know, for example, if the idea carries beyond being able to calculate the length of a vector in non-Cartesian coordinates as the dot product of its covariant and contr$avariant expressions: