Math101 · Independent Ontario learning library
Math101Vector Fields
A rigorous, example-driven guide to vector fields, including hypotheses, method choice, verification, and practice.
The central idea
A vector field assigns a vector to each point: $\mathbf F:D\subset\mathbb R^n\to\mathbb R^m$, often with $m=n$. In the plane, $\mathbf F=\langle P,Q\rangle$; in space, $\langle P,Q,R\rangle$. Domain, units, and input/output dimensions must be stated. A field may be radial, rotational, conservative, divergence-free, or curl-free under suitable definitions.
Definitions, hypotheses, and notation
A conservative field is a gradient $\nabla f$ and therefore has path-independent work on each connected potential domain. Curl-free is a local compatibility condition; simple connectivity is commonly added for the converse. Divergence-free fields have no local net source, but can still circulate strongly.
Field plots often normalize arrows for readability, hiding magnitude. A correct interpretation checks the plotting legend or computes norms. Streamlines solve $\mathbf r'(t)=\mathbf F(\mathbf r(t))$ and therefore depend on both the spatial arrow directions and the chosen time parametrization.
Conceptual meaning
A field is a distribution of arrows, not one moving arrow. It can represent fluid velocity, force, electric influence, or gradient direction. Integral curves follow the arrows; line and surface integrals measure interaction with paths and oriented surfaces.
A dependable method and decision rule
- Identify the field's domain and singularities.
- Evaluate components at representative points.
- Separate direction from magnitude and look for symmetry.
- Use divergence, curl, or potential tests only with required smoothness and topology.
- Interpret arrows with physical units and spatial scale.
