The vector area of an oriented surface is the integral of its unit normal vector times its area element. If its oriented boundary is a closed curve , Stokes theorem givesConsequently it depends only on the oriented boundary, not on the spanning surface. For a planar surface it is signed area times a unit normal vector. A current loop has magnetic dipole moment ; the scalar area of a nonplanar spanning surface is not interchangeable with this vector.
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Vector area is a concept in mathematics and physics that describes an area in two or three dimensions using a vector representation. It is particularly useful in fields like fluid dynamics, electromagnetism, and geometry. ### Definition: - **Vector Area**: The vector area of a surface is defined as a vector whose magnitude is equal to the area of the surface and whose direction is perpendicular to the surface in accordance with the right-hand rule.