A vector field assigns a vector to every point in a region of space. In a 2D plot, each arrow shows the direction and size of the vector at that point, such as velocity, force, or rate of change. Vector fields matter because they let us model motion and influence that vary from place to place, including wind, water flow, electric fields, and gravity.
Reading a vector field helps connect algebraic formulas to visual patterns in the plane.
A 2D vector field is often written as F(x, y) = <P(x, y), Q(x, y)>, where P gives the horizontal component and Q gives the vertical component. The arrow at each grid point starts at that point and points in the direction <P, Q>, with magnitude |F| = sqrt(P^2 + Q^2). Some vector fields come from a scalar potential function f(x, y), where F = grad f = <df/dx, df/dy>, and these are called gradient fields.
Gradient fields point in the direction of steepest increase of the potential function, which is useful in physics, optimization, and multivariable calculus.
Key Facts
- A 2D vector field has the form F(x, y) = <P(x, y), Q(x, y)>.
- The magnitude of a vector field at a point is |F(x, y)| = sqrt(P(x, y)^2 + Q(x, y)^2).
- To plot a vector field, evaluate F(x, y) at many grid points and draw an arrow with that direction and relative length.
- A gradient field is F = grad f = <df/dx, df/dy>, where f(x, y) is a scalar function.
- The divergence of F = <P, Q> is div F = dP/dx + dQ/dy, which measures local spreading or compression.
- The 2D curl of F = <P, Q> is curl F = dQ/dx - dP/dy, which measures local rotation.
Vocabulary
- Vector field
- A vector field is a rule that assigns a vector to each point in a region of space.
- Magnitude
- Magnitude is the length of a vector, representing the size or strength of the quantity at a point.
- Component
- A component is one coordinate part of a vector, such as the horizontal part P or vertical part Q in F(x, y) = <P, Q>.
- Gradient field
- A gradient field is a vector field made from the gradient of a scalar function, so its vectors point toward the direction of fastest increase.
- Divergence
- Divergence measures whether vectors near a point tend to flow outward from the point or inward toward it.
Common Mistakes to Avoid
- Confusing points with vectors: The point (x, y) is the location where the arrow is placed, while F(x, y) is the vector drawn at that location.
- Drawing every arrow with the same length: Arrow length should usually reflect magnitude, so larger values of |F| should appear longer or more intense than smaller values.
- Swapping the components P and Q: In F(x, y) = <P, Q>, P controls horizontal motion and Q controls vertical motion, so reversing them gives a different field.
- Assuming every vector field is a gradient field: A field must satisfy special conditions, such as matching cross partial behavior on a suitable domain, to come from a potential function.
Practice Questions
- 1 For F(x, y) = <2x, -y>, find the vector and its magnitude at the point (3, 4).
- 2 For F(x, y) = <y, x>, compute the vectors at (1, 2), (0, -3), and (-2, 1), then sketch those three arrows on a coordinate plane.
- 3 A vector field has arrows that point directly away from the origin and get longer as the distance from the origin increases. Explain what this suggests about the direction and magnitude of the field, and name one physical situation it could model.