Covector
High school
A covector is a rule that takes a vector and returns a number, in a way that respects adding vectors and scaling them. In coordinates, it is written as a row of numbers, which multiplies a column vector.
Undergraduate
Consider a vector space $V$ and its scalars. A covector on $V$ is a linear map from $V$ to its scalars; an element of the dual space $V^{\ast}$. In coordinates, covectors are often written as row vectors, which act on column vectors by matrix multiplication.
Graduate (differential geometry)
Consider a manifold $M$ and a point $x$ of $M$, with its cotangent space $T_x^{\ast} M$. A covector at $x$ is an element of $T_x^{\ast} M$. The differential $\mathrm{d}f$ of a function is a covector field (a one-form).
Examples
Examples
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The row vector $\begin{bmatrix} 1 & 2 & 3 \end{bmatrix}$
High school
The row vector $\begin{bmatrix} 1 & 2 & 3 \end{bmatrix}$ is a covector, because it maps a column vector $v$ to the number $v_1 + 2 v_2 + 3 v_3$.
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The temperature gradient, in degrees per meter
High school
The temperature gradient is a covector, because it takes a displacement and returns the change in temperature, a number.
Non-examples
Things that may be mistaken for examples.
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A velocity, in meters per second
High school
A velocity is a vector because it is an arrow in space, with units of length per time. It is not a covector, because it does not take a vector and return a number.
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The Euclidean norm $x \mapsto \sqrt{x_1^2 + x_2^2}$ on $\reals^2$
High school
It is not linear: the length of $-v$ is the length of $v$, not its negative.
Distinctions involving this term
- Vector vs. covector: A covector is a linear function that takes a vector and returns a number. In coordinates, both look like lists of numbers, but when the basis changes, their coordinates change in opposite ways.