Vector Fields: Step-by-Step Guide
A vector field assigns a vector to every point in a region of space. Common examples include velocity fields, force fields, and gradient fields. Key questions about a vector field include whether it is conservative (the curl of F equals zero), whether it is incompressible (the divergence of F equals zero), and what its flow lines look like.
Math Racer skill guide · Calculus 3 · College
How to solve it, step by step
- Write F in component form. Express F(x, y, z) = (P, Q, R), identifying each scalar component as a function of position.\[\mathbf{F}(x,y,z) = \langle P(x,y,z),\, Q(x,y,z),\, R(x,y,z) \rangle\]
- Sketch or evaluate at sample points. Plug in several (x, y) or (x, y, z) values to get vectors, then draw the arrows at those points to visualize the field.\[\mathbf{F}(P_i) \text{ at sample points } P_i\]
- Test if F is conservative. Compute curl F. If curl F = 0 on a simply connected domain, F is conservative; find a scalar potential f with grad f = F.\[\nabla \times \mathbf{F} = \mathbf{0} \;\Rightarrow\; \mathbf{F} = \nabla f\]
- Find the potential function (if conservative). Integrate P with respect to x to get f up to a function g(y, z); differentiate with respect to y to match Q and solve for g; repeat for z.\[f(x,y,z) = \int P\,dx + g(y,z)\]
- Compute divergence and curl as needed. div F gives source/sink behavior; curl F gives rotation. These describe the field's local structure.\[\nabla \cdot \mathbf{F},\;\; \nabla \times \mathbf{F}\]
Worked example
Determine whether F(x, y) = (2xy, x^2) is conservative, and if so, find a potential function.
Step 1: Identify P = 2xy and Q = x^2.
Step 2: Compute the 2D curl test: P_y vs Q_x. P_y = 2x and Q_x = 2x. They match.
Step 3: Since P_y = Q_x on R^2 (simply connected), F is conservative.
Step 4: Find potential. Integrate P = 2xy with respect to x: f = x^2 y + g(y).
Step 5: Differentiate with respect to y: f_y = x^2 + g'(y). Set equal to Q = x^2, so g'(y) = 0, hence g(y) = C.
Conclusion: A potential is f(x, y) = x^2 y. Verification: grad f = (2xy, x^2) = F. Matches.
Tips and common mistakes
- For a 2D field F = (P, Q), the conservative test is simply P_y = Q_x.
- For a 3D field, all three components of curl F must vanish.
- Conservative fields require a simply connected domain; on annular regions the curl test can fail to be sufficient.
- If F is conservative, line integrals depend only on endpoints, and closed-loop integrals vanish.
- Gradient fields F = grad f are always conservative.
- Use sample plots to build intuition: F(x,y) = (-y, x) rotates counterclockwise; F = (x, y) points outward radially.
- div F > 0 indicates a source; div F < 0 indicates a sink.
- When finding a potential, double-check by computing grad f and confirming you recover F.