A vector field is a mathematical construct that assigns a vector to every point in a space. It can be thought of as a way to represent spatial variations in a quantity that has both magnitude and direction. Vector fields are widely used in physics and engineering to model phenomena such as fluid flow, electromagnetic fields, and gravitational fields, among others.
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On an open set , a vector field assigns a vector to each point . Its components are scalar functions. For a smooth manifold , the corresponding definition requires : the vector belongs to the tangent space at its base point. A smooth vector field is a smooth section of the tangent bundle.[ref]
Calculus Volume 3 by Gilbert Strang and others. section6.1, Vector Fields, planar and spatial definitions
For example, is tangent to circles centred at the origin. Its integral curves solve , , so is constant. This distinguishes the field, which assigns a direction and magnitude at every point, from any one trajectory following it. A fluid velocity is a physical vector field; its divergence measures local expansion, while its curl measures local rotation in three dimensions.
