A covector is an element of the dual space, hence a linear functional on a vector space. In finite-dimensional coordinates, it acts as a row multiplied by a coordinate vector column. The row acts on to give five. The terminology emphasizes its role as a scalar-valued measurement and its behavior under a change of basis.
For example, the total derivative of at sends a small displacement to the linear change . This covector describes the first-order response. The gradient represents the same rule through the chosen Euclidean inner product. In differential geometry, the cotangent space collects these scalar-valued measurements of tangent vectors.
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