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Visualizing vectors and fields

A vector diagram should make direction, magnitude, location, units, and scale easier to understand without changing their meaning. A single vector can be drawn as one arrow. A field requires sampled arrows, curves, colors, or a combination. Every visual choice is a mapping from physical data to screen geometry, and that mapping must be stated.

Drawing one vector

Suppose a vector begins at physical point \(\mathbf r_0\) and has components \(\mathbf v=(v_x,v_y)\). If one screen unit represents a physical scale \(s\), the physical endpoint before screen-coordinate conversion is

\[\mathbf r_1=\mathbf r_0+s\mathbf v\]

The scale has units that convert the vector quantity to plot distance. A velocity arrow cannot be added directly to a position without such a display convention. This arrow is not necessarily predicting where an object moves during a time interval; it is a glyph that represents a velocity at one point.

Many screen systems place the origin at the upper left and increase vertical pixel coordinates downward. A conventional Cartesian plot increases the physical vertical coordinate upward. The mapping therefore often includes a sign reversal:

\[p_x=p_{x0}+k(x-x_{\min}),\qquad p_y=p_{y0}-k(y-y_{\min})\]

The exact offset depends on the viewport. Mixing screen and physical coordinates too early causes reflected arrows, incorrect angles, and scale-dependent calculations.

import numpy as np
import matplotlib.pyplot as plt

origin = np.array([1.0, 1.0])
vector = np.array([3.0, 2.0])
fig, ax = plt.subplots()
ax.quiver(*origin, *vector, angles="xy", scale_units="xy", scale=1)
ax.set(xlim=(0, 5), ylim=(0, 4), xlabel="x (m)", ylabel="y (m)")
ax.set_aspect("equal")
plt.show()

The arrowhead should be sized in pixels for legibility, while the shaft length follows the declared physical-to-display scale. If the arrowhead itself scales without limit, large vectors can produce misleadingly bulky symbols.

Aspect ratio is part of correctness

If one metre occupies a different number of pixels horizontally and vertically, the display distorts angles and lengths. A physically perpendicular pair may no longer look perpendicular. Either use an equal spatial aspect ratio or explicitly warn that visual angles and magnitudes are distorted.

This matters for circles, radial fields, projectile paths, and any visual comparison of components. Labels and numeric readouts can remain correct while the geometry silently communicates something false.

Sampling a field with arrows

A quiver plot evaluates a vector field at selected positions and draws one arrow at each sample. For a two-dimensional field \(\mathbf F(x,y)\), a regular sampling lattice might use

\[x_i=x_{\min}+i\,\Delta x,\qquad y_j=y_{\min}+j\,\Delta y\]

Sampling density is a display choice. Too few arrows hide important spatial change. Too many overlap and create dark texture that is hard to read. Increasing the number of screen arrows does not increase the resolution of underlying measured data unless new samples actually exist.

coordinates = np.linspace(-2, 2, 9)
x, y = np.meshgrid(coordinates, coordinates)
u, v = -y, x  # A smooth rotational field in arbitrary field units.
fig, ax = plt.subplots()
ax.quiver(x, y, u, v)
ax.set(xlabel="x (m)", ylabel="y (m)")
ax.set_aspect("equal")
plt.show()

Undefined and masked regions should be visible. Drawing a zero arrow at a point-source singularity incorrectly implies a zero field. Omitting it without explanation may also mislead, so a legend or mask is useful.

Showing magnitude without losing direction

If arrow length is proportional to magnitude, strong regions can dominate the plot and weak arrows may disappear. If every arrow is normalized to equal length, direction is clear but magnitude information is lost. A responsible plot states which choice it uses.

One compromise uses a bounded nonlinear length such as

\[\ell(F)=\ell_{\max}\frac{F}{F+F_0}\]

Here \(F=|\mathbf F|\) and \(F_0\) sets the transition scale. This mapping preserves order for nonnegative magnitudes but is not linear. The legend must show that arrow length has been compressed.

Color can encode magnitude while normalized arrows show direction. The color scale needs labeled numerical ticks and units. A sequential lightness scale is appropriate for nonnegative magnitude. A diverging scale is meaningful for signed scalar values, such as one component, but field magnitude itself is not signed. Color should not be the only encoding when the plot must be usable by readers with color-vision differences or in grayscale.

Logarithmic magnitude is helpful across many orders of magnitude, but \(\log 0\) is undefined and negative values cannot be passed to an ordinary logarithm. A floor, mask, or alternative mapping must be declared.

Field lines

A field line is a curve tangent to a nonzero vector field at every point. One convenient parameterization satisfies

\[\frac{d\mathbf r}{d\lambda}=\frac{\mathbf F(\mathbf r)}{|\mathbf F(\mathbf r)|}\]

Normalization makes the curve parameter track distance rather than field strength. It also fails at zeros of the field. Using the unnormalized field produces the same curve away from zeros but a different parameter speed.

Field lines can be approximated by numerical integration from seed points:

coordinates = np.linspace(-2, 2, 41)
x, y = np.meshgrid(coordinates, coordinates)
fig, ax = plt.subplots()
ax.streamplot(x, y, -y, x, density=1.0)
ax.set(xlabel="x (m)", ylabel="y (m)")
ax.set_aspect("equal")
plt.show()

Forward Euler is easy to understand but can curve poorly and drift when the field changes rapidly. A midpoint or Runge-Kutta step is usually more accurate for the same spacing. Whichever method is used, trace in both parameter directions if both halves of a line are desired, stop at boundaries and invalid regions, and refine the step until the visible curve stabilizes.

Field-line density is not automatically a measured quantity. Textbook conventions may draw more electric field lines where the field is stronger, but the number of drawn lines is chosen by the illustrator. Doubling the number of seed points does not double the physical field. Lines also should not cross at a regular point of a single-valued field: crossing would assign two directions to the same location. Apparent crossings in a two-dimensional projection may represent different depths in three dimensions.

Streamlines, pathlines, and trajectories

For a steady fluid velocity field, streamlines are tangent to the fluid velocity and fluid parcels follow the same curves. In a time-dependent flow, an instantaneous streamline and the pathline of a parcel can differ. A pathline solves

\[\frac{d\mathbf r}{dt}=\mathbf u(\mathbf r,t)\]

An electric field line is generally not a charged particle trajectory. A particle's state includes velocity and its acceleration follows \(m\mathbf a=q\mathbf E\). Inertia allows its velocity to point differently from the field. The plot should use the word “trajectory” only when the equations of motion have actually been solved.

Slices and projections of three-dimensional fields

A two-dimensional picture of a three-dimensional field may show a plane slice, a projection, or selected three-dimensional arrows. These are different reductions. A slice evaluates the field at a fixed coordinate. A projection discards or separately encodes the out-of-plane component. A perspective rendering can make arrow length depend on camera orientation and depth.

State the slice plane and what happens to the missing component. For example, showing only \((F_x,F_y)\) at \(z=0\) does not prove that \(F_z=0\). The hidden component may be encoded with color, symbols pointing into or out of the page, or a separate panel.

Avoiding visual singularity traps

Near an inverse-square point source, magnitude grows without bound in the ideal model. Auto-scaling from the largest sampled arrow can make the rest of the field appear zero. Clipping may improve readability but must be identified. A separate close-up, a logarithmic magnitude panel, or explicit excluded radius can be more honest.

Symmetry provides strong visual checks. A radial field should point directly along the source-to-sample displacement. A uniform field should have parallel arrows of equal length under a linear scale. A rotational field should display tangential directions. If these fail, inspect coordinate signs, basis order, screen-axis inversion, and source displacements before questioning the physical formula.

A reproducible visualization record

A useful figure caption or data record includes:

Test rendering with exact fields whose patterns are already known. Confirm arrow endpoints numerically, not only by eye. Resize the viewport and verify that physical directions and aspect ratio remain correct. Compare field-line traces after halving the integration step. Include zero fields, very weak fields, extreme magnitudes, domain edges, and singularities.

A good vector-field visualization makes a mathematical function inspectable. It does not replace the function, add information absent from the data, or quietly turn display normalization into a physical claim.

A worked Python notebook

Follow the line y equals 1 through the rotational field with components minus y and x. The question for this calculation is precise: how does field magnitude change when we change horizontal coordinate, while keeping the other choices fixed? The earlier sections explain the physical relationship. Here we turn one part of that relationship into a calculation that you can run, inspect, and change in Google Colab.

This is a smooth test field for plotting; the units and physical interpretation must be supplied for a measured field. Keep that sentence beside the results. A formula can return a number outside its useful range, and a plotting library will usually draw that number without asking whether the physical assumptions still hold. The selected interval is from -3 to 3 m. It gives us a specific domain to discuss instead of leaving the graph's range to a software default.

Use the notebook link above the lesson, run the setup cell, and then run the cells in order. The code below is also included there. NumPy handles the arrays of numerical values, SymPy keeps the equation symbolic, and Matplotlib draws the results. If these libraries or notebook cells are unfamiliar, start with Your first physics notebook.

State the relationship before choosing a graph

In this small example, q stands for horizontal coordinate, expressed in m. The output is field magnitude, expressed in arbitrary field units. The equation below uses the numerical values in those units. Any fixed coefficients therefore carry the units needed to make the relation consistent. A value supplied in another unit must be converted before it enters this expression.

\[f(q) = \sqrt{q^{2} + 1}\]

This is a function of one changing input. It is not a statement that the physical system has only one variable. We deliberately hold the other inputs fixed so that one cause of variation is visible. Changing several inputs at once would make it harder to tell which change produced the output difference. Write down the held-fixed choices before trying a different experiment.

At 1 m, the expression gives 1.41421 arbitrary field units. This is our reference calculation. It is a prediction of the stated example, not a new measurement. Its displayed digits help us compare implementations. They should not be read as a claim that physical inputs are known to six significant figures.

import numpy as np
import sympy as sp
import matplotlib.pyplot as plt

q = sp.Symbol("q", real=True)
expression = sp.sympify("sqrt(q**2+1)", locals={"q": q})
reference_input = 1.0
reference_output = float(expression.subs(q, reference_input))
print("Input:", reference_input, "m")
print("Predicted field magnitude:", reference_output, "arbitrary field units")

sp.Symbol declares a mathematical symbol. It does not assign a numerical measurement to that symbol. subs then replaces the symbol with our chosen input, and float converts the resulting symbolic number into a standard Python numerical value. Keeping these steps separate lets us inspect the equation before numerical substitution. It also prevents a later plot from becoming the only surviving record of the model.

Compare a few cases you can explain

Before drawing hundreds of points, calculate three. The lower endpoint gives 3.16228 arbitrary field units. The midpoint, at 0 m, gives 1 arbitrary field units. The upper endpoint gives 3.16228 arbitrary field units. Read these together with the assumptions: they describe different choices of horizontal coordinate within the same example, with the other inputs held fixed.

These three values are a useful first check, but they do not establish the shape between them. Two endpoints can be equal even when the curve bends or oscillates in between. Likewise, a large change between endpoints does not tell us whether the relationship is linear. We need the equation and intermediate samples to answer those questions.

evaluate = sp.lambdify(q, expression, modules="numpy")
inputs = np.array([-3.0, 0.0, 3.0])
outputs = np.asarray(evaluate(inputs), dtype=float)
outputs = np.broadcast_to(outputs, inputs.shape)
for input_value, output_value in zip(inputs, outputs):
    print(f"{input_value:.6g} m -> {output_value:.6g} arbitrary field units")
assert np.all(np.isfinite(outputs))

lambdify turns the symbolic expression into a numerical function. We explicitly select NumPy as its numerical backend so it can accept an array. The same expression now has two useful forms: one for algebra and one for calculating many cases. The assertion checks that the chosen inputs produce finite outputs. It would catch an infinity or an undefined numerical result, but a finite value can still be physically inappropriate.

Make a graph that answers the question

The horizontal axis shows horizontal coordinate; the vertical axis shows field magnitude. Every plotted point is a prediction of the same expression. We mark the reference case separately so that a reader can connect the earlier printed value to its place on the curve. The axis labels include units because a shape without a scale cannot tell us the size of a physical effect.

input_grid = np.linspace(-3.0, 3.0, 301)
output_grid = np.asarray(evaluate(input_grid), dtype=float)
output_grid = np.broadcast_to(output_grid, input_grid.shape)
fig, ax = plt.subplots(figsize=(7, 4))
ax.plot(input_grid, output_grid, label="Stated model")
ax.scatter([reference_input], [reference_output], label="Reference case", zorder=3)
ax.set_xlabel("Horizontal coordinate (m)")
ax.set_ylabel("Field magnitude (arbitrary field units)")
ax.legend()
ax.grid(alpha=0.25)
fig.tight_layout()
plt.show()

Look for a sign change, a turning point, or a region where the curve becomes steep. Connect each feature to the expression rather than to the colour of the line. If the curve has a minimum or maximum, changing the input in opposite directions may have similar effects on the output near that point. If it is nearly straight over a short interval, a local linear approximation may be useful even when the complete expression is nonlinear.

Increasing the number of samples makes the line smoother and may reveal features that a coarse grid missed. It does not improve the physical inputs or enlarge the model's validity range. For quantities that can take only discrete values, use the line as a guide to the relationship and interpret only the allowed input values. A plotting function does not enforce such restrictions for you.

Measure the local sensitivity

A derivative asks a more specific question than the whole graph: how quickly does field magnitude change near the reference input? Its units are the output unit divided by the input unit. For this calculation, the derivative at 1 m is 0.707107 in those ratio units. Locally, the output increases as the input increases. This statement is local; the sign or size of the slope can be different elsewhere on the curve.

\[f'(q) = \frac{q}{\sqrt{q^{2} + 1}}\]

For a small input change, multiply the derivative by that change to estimate the output change. Then compare that estimate with a fresh evaluation of the full expression. The distinction between these two calculations matters: the derivative gives a tangent approximation, while substitution evaluates the chosen expression at the new point.

slope_expression = sp.diff(expression, q)
slope_at_reference = float(slope_expression.subs(q, reference_input))
changed_input = 1.03
linear_prediction = reference_output + slope_at_reference * (
    changed_input - reference_input
)
full_prediction = float(evaluate(changed_input))
print("Symbolic derivative:", slope_expression)
print("Tangent prediction:", linear_prediction, "arbitrary field units")
print("Full expression:", full_prediction, "arbitrary field units")
print(
    "Approximation difference:",
    full_prediction - linear_prediction,
    "arbitrary field units",
)

Here the changed input is 1.03 m. The tangent estimate is 1.43543 arbitrary field units, while the full expression gives 1.43558 arbitrary field units. If the two values are close, it means the chosen input change is small enough for this local approximation at this point. It does not mean the entire model is linear. If the slope is zero, a quadratic or higher-order term may supply the leading change instead.

This is also the beginning of uncertainty propagation. If the input has a small stated standard uncertainty, the magnitude of the slope times that uncertainty estimates one contribution to output uncertainty. Do not automatically use the same rule for a wide interval, a discontinuity, a turning point with zero first derivative, or several correlated uncertain inputs. Those situations need more of the model than one slope.

Check the derivative with numerical differences

We can estimate the same slope from values on either side of the reference point. Subtract the lower output from the upper output and divide by twice the input step. Parentheses are essential: the entire output difference must be divided by the entire input interval. Without them, Python can execute a different mathematical expression while still producing a plausible number.

The calculation below uses three decreasing step sizes. Each estimate is compared with the symbolic derivative, so the reference is independent of the numerical difference formula. Near an endpoint, it is acceptable here to evaluate a small extension only because the displayed mathematical expression is smooth there; such a check does not assert that extrapolated values describe additional physical cases.

input_span = 6.0
for fraction in [1e-2, 1e-3, 1e-4]:
    step = input_span * fraction
    upper = float(evaluate(reference_input + step))
    lower = float(evaluate(reference_input - step))
    numerical_slope = (upper - lower) / (2 * step)
    difference = numerical_slope - slope_at_reference
    print(f"step={step:.6g}: slope={numerical_slope:.6g}, difference={difference:.6g}")

For a smooth function, reducing the step often reduces the approximation error at first. Very small steps can eventually make subtraction of nearly equal floating-point values less reliable. Therefore “smaller” is not an unlimited guarantee of “better.” Compare the observed behaviour over a useful range of steps instead of choosing the smallest number the computer can represent.

An absolute slope difference is useful when the expected slope is near zero. A relative difference would divide by a very small reference and could look enormous even when the absolute discrepancy is tiny. Away from zero, a relative difference can help compare quantities with very different scales. The tolerance should reflect the question and the numerical method, rather than a universal decimal threshold copied into every test.

Change the example and preserve the reasoning

Make one copy of the reference calculation before editing it. Choose a second value of horizontal coordinate inside the stated interval, predict whether field magnitude will rise or fall, and then evaluate the expression. Place the prediction and the result in a text cell. If they disagree, identify whether the mistake was in the sign, the unit conversion, the algebra, or the assumption about how the curve behaves.

Next choose a shorter plotting interval around the reference point. Compare the local curve with its tangent approximation. Then widen the interval again. The difference shows why a useful local estimate need not describe the whole range. Keep the axis labels and numerical scale visible when comparing the plots; automatic axis limits can make two different effects appear equally large.

Finally, return to the physical limit stated at the start: This is a smooth test field for plotting; the units and physical interpretation must be supplied for a measured field. Describe one change to the setup that would violate that limit. State what new input, equation, or observation would be needed before trusting an answer in the changed situation. This last step connects the Python experiment to the physics instead of treating successful execution as the only criterion for a useful result.

Before saving, restart the runtime and run all cells in order. A result that depends on an earlier, now-deleted input cell is not reproducible from the saved notebook. Keep unrounded values for later calculations, round only the reported output, and label any new data as measured, assumed, or generated. The graph and printed values should remain traceable to the same expression and the same choices of inputs.

Recover the output change by integrating the slope

There is another way to check that the expression and its derivative describe the same relationship. Add up the small output changes predicted by the slope across the chosen input interval. In the continuous limit, this integral equals the output at the upper endpoint minus the output at the lower endpoint. The result has units of arbitrary field units: the input unit in the integration cancels the input unit in the denominator of the slope.

This statement is about the mathematics of this particular relationship. It does not assign a new physical meaning to an arbitrary area under a graph. For example, an area under force versus distance represents work, while an area under a different pair of quantities may represent something else or have no standard name. Always carry the axis units through the calculation before naming an integral.

slope_function = sp.lambdify(q, slope_expression, modules="numpy")
endpoint_change = float(evaluate(3.0) - evaluate(-3.0))
for points in [101, 201, 401]:
    grid = np.linspace(-3.0, 3.0, points)
    slopes = np.asarray(slope_function(grid), dtype=float)
    slopes = np.broadcast_to(slopes, grid.shape)
    integrated_change = np.trapezoid(slopes, grid)
    print(
        points,
        "integrated change:",
        integrated_change,
        "endpoint change:",
        endpoint_change,
    )

The trapezoidal rule joins neighbouring slope samples with straight segments and adds their signed areas. A negative contribution remains negative; replacing every slope by its magnitude would calculate a different quantity. Increasing the sample count provides a practical refinement check. If the values do not settle toward the endpoint difference, first inspect the interval, singularities, and array ordering.

Do not conclude that two agreeing calculations prove the model is physically complete. Both use the same selected relationship, so an omitted effect would be omitted from both. Their agreement is evidence that the numerical tools implement that relationship consistently. Comparing with an independent experiment or a more appropriate physical model is a separate test.

Keep this comparison in the notebook when you change the expression. It is particularly useful after adding a coefficient, changing a sign convention, or converting an input unit. A code cell that produces a smooth graph can conceal these mistakes; a second calculation of the same output change gives you another place to notice them.

Sources

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