Math101 · Independent Ontario learning library
Math101learn.math101.caVector 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.
Fully worked example
Graphical or geometric meaning
Along each circle the arrows are tangent in the counterclockwise direction. The origin has zero vector, and arrow lengths increase linearly outward. This pattern suggests rotation and is confirmed by scalar curl two.
Common mistakes and why they fail
Verification and reasonableness checks
- Evaluate at symmetric points and compare expected transformations.
- Compute dot products with radial or tangent directions.
- Verify output dimension and units match the model.
Domain and geometry come before computation
A vector field assigns a vector to each point of its domain. Plotting representative arrows reveals direction, relative magnitude, sources, sinks, or circulation, but the scale should be stated because drawings often normalize arrows. Determine the domain before differentiating: denominators and radial factors can create singularities that affect conservativeness and integral theorems. Streamlines are curves tangent to the field, whereas a time-dependent particle path may differ when the field itself changes with time. Check units componentwise and evaluate all components at the same point. Symmetry can predict zero or sign patterns in flux and circulation, but it does not repair a field undefined on part of the region.
Practice
- Evaluate $\langle-y,x\rangle$ at $(3,4)$.
- Is that field radial or tangential to origin-centered circles?
- What is the domain of $\langle x,y\rangle/(x^2+y^2)$?
Answers and brief solutions
- $\langle-4,3\rangle$.
- Tangential.
- $\mathbb R^2\setminus\{(0,0)\}$.
Connections and next steps
Try it yourself
Hints are part of learning. Open one whenever it makes the next step feel possible.
For F(x,y)=⟨−y,x⟩, what is F(2,−1)?
- The first component is −y=1.
- The second component is x=2.
- Thus F(2,−1)=⟨1,2⟩.
End of lesson
Nice work making it this far.
Understanding grows through return visits. Save this lesson, try the practice, or continue when you are ready.
