A vector field assigns a vector, not just a number, to every point. Wind maps, water currents and force fields are all vector fields.
Step 1: Let's Learn
Read it, or press Listen and follow the words.
Reading one
Follow the arrows as though you were a leaf dropped in. The path you would take is called a flow line.
Sources and sinks
Arrows pointing away from a point mark a source. Arrows converging on one mark a sink.
Circulation
Arrows curling around a point mean circulation. A paddle wheel dropped there would spin.
Gradient fields
Some vector fields are the gradient of a scalar function. Those are the ones that flow uphill on some hidden surface.
Not every field is one
A purely rotating field cannot be a gradient field. There is no surface whose uphill direction goes in circles forever.
An arrow at every point
A vector field assigns a vector to each point of a region. Wind velocity, fluid flow, gravitational and electric fields are all vector fields, and drawing a sample of arrows is how they are visualised.
Flow lines
A curve everywhere tangent to the field traces the path a particle would follow. Flow lines are to vector fields what solution curves are to slope fields in differential equations.
Divergence measures outflow
Divergence at a point measures the net flux out of a tiny region around it — whether the point acts as a source or a sink. It converts a vector field into a scalar field.
Curl measures rotation
Curl measures the tendency of the field to rotate around a point. A paddlewheel placed in the flow would spin where the curl is nonzero, which is the standard physical picture.
Step 2: Try It Yourself
Tap and try it out.
- Released from(3, 2)
Every arrow points toward the origin, so trajectories settle there. The origin is stable.
- Released from(2, 0)
Every arrow is perpendicular to the line from the origin, so trajectories circle rather than approach.
Step 3: Watch an Example
One step at a time.
Watch Farid Classify a Field
Farid examines the field F = ⟨−x, −y⟩.
- Step 1
He evaluates it at (1, 0), getting ⟨−1, 0⟩, which points back toward the origin.
Step 4: Your Turn
Practice makes it stick.
The Field Value
Problem 1 of 2
F = ⟨2x, y⟩. What is the x-component of F at the point (3, 5)?
The Classification
Problem 2 of 2
Arrows all point away from the origin. Enter 1 for source, 2 for sink.
Read the Arrows
1 of 8
F = ⟨x, y⟩ at (4, 1). What is the x-component?
2 of 8
F = ⟨−y, x⟩ at (1, 0). What is the y-component?
3 of 8
Arrows converging on a point. Enter 1 for source, 2 for sink.
4 of 8
F = ⟨−y, x⟩. Does a paddle wheel at the origin spin? 1 yes, 0 no.
5 of 8
F = ⟨3, 0⟩ everywhere. What is the y-component at any point?
6 of 8
F = ⟨2x, 2y⟩ is the gradient of x² + y². Is it a gradient field? 1 yes, 0 no.
7 of 8
Match each field to what it does.
Tap a card on the left to start.
8 of 8
F = ⟨0, 0⟩ at every point. Does anything move? 1 yes, 0 no.
Step 5: Quick Check
Show what you know.
Question 1 of 2
F = ⟨y, x⟩ at (2, 7). What is the x-component?
Question 2 of 2
What does a vector field assign to each point?
What You Learned
- A vector field assigns a vector to every point, so it is drawn as a field of arrows.
- Arrows flowing outward mark a source; arrows flowing inward mark a sink.
- Some fields are gradients of a surface, and a purely rotating field never is.