Symbolic vector calculus

Divergence asks whether nearby arrows spread outward or inward. Curl asks whether their directions tend to circulate. Both can be estimated from nearby field samples.

Think like a programmer

These operators are structured local tests on a vector-valued function. Finite differences replace symbolic derivatives with small, explicit probes around a position.

Model checklist

Inputs: A vector field, a point, and a small finite-difference step.
State: Values sampled on either side of the point.
Rule: Subtract nearby components and divide by twice the step.
Output: Estimated divergence and z-curl.
Check: A linear field has constant known derivatives.

\[\\nabla\\cdot\\mathbf{F}=\\frac{\\partial F_x}{\\partial x}+\\frac{\\partial F_y}{\\partial y},\\qquad (\\nabla\\times\\mathbf{F})_z=\\frac{\\partial F_y}{\\partial x}-\\frac{\\partial F_x}{\\partial y}\]

Loading the interactive visual. The lesson text and model remain available while it starts.

const divergence = (field(x + h, y).x - field(x - h, y).x) / (2 * h)
  + (field(x, y + h).y - field(x, y - h).y) / (2 * h);

Try this experiment

Prediction: A radial outward field has positive divergence away from the source.

Inspect several arrows near a probe. Describe whether they spread or circulate, then state which operator represents that pattern.

Where this model breaks

Finite differences are approximate and can fail near singularities, noisy samples, or boundaries. Symbolic formulas also need domain assumptions; a single visual pattern is not a proof.

Summary

Treat divergence and curl as local calculations on a field function. Validate the code with a linear field whose derivative is known exactly.

Glossary

Divergence: local net outward flow of a vector field.
Curl: local rotational tendency of a vector field.
Partial derivative: change with respect to one input while holding others fixed.

Self-check

What samples does a central difference compare?
Which operator describes local spreading?
Why test a linear field first?

Sources

J. E. Marsden and A. Tromba, Vector Calculus.

Model contract

Treat the lesson as a small function before treating it as a fact to memorize. Give every value a unit, keep only the state needed for the next step, and make the output easy to inspect.

\[\text{observable output} = f(\text{inputs},\,\text{state})\]

Inputs: Quantities you set or measure, with units and useful bounds.
State: Values the program must retain to reproduce the next result.
Rule: The relationship or update that turns inputs and state into a result.
Check: A known limit, unit check, invariant, or measured result that can expose a bad model.

Implement the idea as a model

For Symbolic Vector Calculus, write down the quantities you can control, the values your program must retain, and the result a reader could inspect. In Vectors and Coordinate Systems, the useful program is not the drawing: it is the smallest explicit model that makes a prediction you can test.

Guided experiment

Prediction: changing one declared input while holding the others fixed should change only the outputs that the model connects to that input. Choose one input, predict the direction of change, then check a limiting case such as zero, a symmetric arrangement, or a familiar low-speed or small-change approximation.

Where this model breaks

This lesson is a teaching model, not a complete simulator. Before using it outside the stated question, check which interactions, scales, uncertainties, boundary conditions, and measurement limits it leaves out.

Summary

Treat Symbolic Vector Calculus as a contract: named inputs and units enter a rule, the rule produces an observable result, and a known limit or invariant checks whether the implementation deserves trust.

Glossary

Input: a measured value or chosen parameter supplied to a model.
State: the smallest set of values needed to continue or reproduce a model.
Validation: comparing an output with a known result, limit, invariant, or measurement.

Self-check

Which values are inputs, and which values must remain state?
What observable result would tell you the model is behaving as expected?
Which assumption would you test first before applying the model to a real system?

Model review: turn Symbolic Vector Calculus into a test

Connect divergence and curl notation to finite-difference code that samples a two-dimensional vector field.

Name the inputs and units that the vectors and coordinate systems model needs.
Separate the state you must keep from values you can calculate when needed.
Write one rule that maps the current state and inputs to an observable result.
Choose a limiting case, unit check, invariant, or known result before trusting an output.
State one assumption you would change before using this simplified model for a real decision.

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