2020年4月6日 · Technically contravariant vectors are in one vector space, and covariant vectors are in a different space, the dual space. But there is a clear 1-1 correspondence between the space and its dual, and we tend to think of the contravariant and covariant vectors as different descriptions of the same vector.

2015年12月31日 · For covariant vector it is the opposit: covariant vector has components that change oppositely to the coordinates or, equivalently, transform like the reference axes. Wikipedia . with the classical example being the gradient.

2017年12月7日 · The visual picture of a "covariant" vector (dual vector) is a series of surfaces. See Schutz (2009, $\S3.3$, "Picture of a one-form") or Misner, Thorne & Wheeler ($\S2.5$). A [contravariant] vector acts on a dual vector to produce a scalar, and the visual picture of this is the number of surfaces the arrow crosses.

2016年7月14日 · You can verify that the normal vector has covariant components by recalling that the normal can be defined through a cross product of tangent vectors (which have contravariant components; the cross product of true vectors is a pseudovector, which has covariant components), for instance.

As a very basic example, the covariant vector of a column vector is a row vector. For a complex number, a similar definition yields its conjugate. When it comes to inner products, it usually depends upon what space you are working with.

2016年10月22日 · $\begingroup$ I say "clunky" because saying a vector has "covariant/contravariant components" totally wipes out the geometric interpretations of vectors and covectors, shown in Alfred's answer; it makes it harder to see why a covector is a linear function from vectors to $\mathbb{R}$, for instance. $\endgroup$

The first one is to say there is only a single entity - the vector - which has covariant and contravariant components. This is inspired by classical tensor calculus: when doing calculations, we often do not care about the placement of the indices of a particular tensor - after all, we can always lower or raise them (ie go from column vectors to ...

2015年10月20日 · I am trying to do exercise 3.2 of Sean Carroll's Spacetime and geometry. I have to calculate the formulas for the gradient, the divergence and the curl of a vector field using covariant derivatives...

2019年12月25日 · I'm reading a Quora answer on an intuitive explanation of covariant/contravariant components of vectors.If we have a coordinate system with straight coordinate axes, the geometric explanation given is that a vector's covariant components in such a system will be perpendicular projections on the axes, whereas its contravariant components …

2024年3月16日 · $\begingroup$ @SSsaha Just want to point out that the contravariant/covariant system of terminology is older, but with well established meaning in physics. I am not yet forty, however, my professors were old guys nearly retired, and that is the manner in which they spoke.