{ "cells": [ { "cell_type": "markdown", "id": "markdown-000", "metadata": {}, "source": [ "# Numerical differentiation and integration\n", "\n", "[Read the lesson](https://tirta.co/a-byte-of-physics/numerical-differentiation-integration). This notebook includes the lesson text and its Python examples in reading order. Run the setup cell, then run the remaining cells from top to bottom. In Colab, use File → Save a copy in Drive to keep your edits. You can run and edit this view without changing the course source. Examples use illustrative data unless stated otherwise." ] }, { "cell_type": "code", "id": "code-001", "metadata": {}, "source": [ "import importlib.metadata\n", "import subprocess\n", "import sys\n", "\n", "requirements = {\"numpy\": (2, 0), \"sympy\": (1, 13), \"matplotlib\": (3, 8)}\n", "to_install = []\n", "for package, minimum in requirements.items():\n", " try:\n", " installed = importlib.metadata.version(package)\n", " version = tuple(int(part) for part in installed.split(\".\")[:2])\n", " except importlib.metadata.PackageNotFoundError:\n", " version = (0, 0)\n", " if version < minimum:\n", " to_install.append(f\"{package}>={minimum[0]}.{minimum[1]}\")\n", "if to_install:\n", " subprocess.check_call([sys.executable, \"-m\", \"pip\", \"install\", *to_install])\n", "\n", "import numpy as np\n", "import sympy as sp\n", "import matplotlib\n", "import matplotlib.pyplot as plt\n", "\n", "print(\"Python\", sys.version.split()[0])\n", "print(\"NumPy\", np.__version__)\n", "print(\"SymPy\", sp.__version__)\n", "print(\"Matplotlib\", matplotlib.__version__)\n" ], "execution_count": null, "outputs": [] }, { "cell_type": "markdown", "id": "markdown-002", "metadata": {}, "source": [ "Measured motion often arrives as a table of positions and times rather than a formula.\n", "Numerical differentiation estimates velocity from nearby position samples. Numerical\n", "integration estimates displacement from velocity samples. Both operations approximate\n", "calculus with finite data, and their errors behave very differently.\n", "\n", "## From average slope to a derivative\n", "\n", "For a continuous position function, instantaneous velocity is\n", "\n", "\n", "$$\n", "v(t)=\\frac{\\mathrm{d}x}{\\mathrm{d}t}=\\lim_{h\\to0}\\frac{x(t+h)-x(t)}{h}\n", "$$\n", "\n", "\n", "Sampled data cannot take the limit because the interval is finite. A forward difference\n", "uses the current and next sample:\n", "\n", "\n", "$$\n", "v(t)\\approx\\frac{x(t+h)-x(t)}{h}\n", "$$\n", "\n", "\n", "A backward difference uses the current and previous sample, while a centered difference\n", "uses one on each side:\n", "\n", "\n", "$$\n", "v(t)\\approx\\frac{x(t+h)-x(t-h)}{2h}\n", "$$\n", "\n", "\n", "For a smooth function and exact arithmetic, forward and backward differences have\n", "leading truncation error proportional to $h$.\n", "The centered difference cancels that first-order term and has leading error proportional\n", "to $h^2$. This advantage does not\n", "make it usable at an endpoint where a sample on one side is missing.\n", "\n", "## A derivative example\n", "\n", "Let position be\n", "\n", "\n", "$$\n", "x(t)=t^2\\,\\mathrm{m}\n", "$$\n", "\n", "\n", "where the numerical time is expressed in seconds and the coefficient implicitly carries\n", "the units needed to make the result a length. The exact velocity at\n", "$t=2.0\\,\\mathrm{s}$ is\n", "$4.0\\,\\mathrm{m/s}$.\n", "With $h=0.5\\,\\mathrm{s}$,\n", "the forward difference gives\n", "$4.5\\,\\mathrm{m/s}$,\n", "the backward difference gives\n", "$3.5\\,\\mathrm{m/s}$,\n", "and the centered difference gives the exact\n", "$4.0\\,\\mathrm{m/s}$.\n", "\n", "The centered result is exact here because the symmetric formula differentiates this\n", "quadratic without error. It is not exact for every function." ] }, { "cell_type": "code", "id": "code-003", "metadata": {}, "source": [ "import numpy as np\n", "\n", "times_s = np.linspace(0, 4, 41)\n", "positions_m = 2 * times_s**2\n", "index = 20\n", "assert len(times_s) == len(positions_m)\n", "assert np.all(np.diff(times_s) > 0)\n", "velocity_m_per_s = (positions_m[index + 1] - positions_m[index - 1]) / (\n", " times_s[index + 1] - times_s[index - 1]\n", ")\n", "print(f\"At {times_s[index]:.1f} s: {velocity_m_per_s:.3f} m/s\")\n", "assert np.isclose(velocity_m_per_s, 4 * times_s[index])\n" ], "execution_count": null, "outputs": [] }, { "cell_type": "markdown", "id": "markdown-004", "metadata": {}, "source": [ "The denominator should come from the timestamps. Replacing it with a configured nominal\n", "interval gives a wrong slope when samples are irregular or one sample is missing.\n", "\n", "## Differentiation amplifies noise\n", "\n", "A difference subtracts nearby position measurements and divides by a small time. If\n", "each position includes noise of typical size\n", "$\\sigma_x$, the noise contribution\n", "to a simple derivative estimate grows roughly like\n", "$\\sigma_x/h$. Reducing\n", "the interval lowers truncation error but magnifies measurement noise and floating-point\n", "cancellation.\n", "\n", "There is therefore no universal instruction to use the smallest possible interval.\n", "Plot results across several intervals, compare with instrument resolution, and state any\n", "smoothing or fitted model used before differentiation. Smoothing changes the data and\n", "can erase real fast motion, so its window and assumptions belong in the report.\n", "\n", "At a data boundary, choose an explicit policy: use a one-sided difference, return no\n", "estimate, or fit a local model. Copying the nearest interior derivative silently invents\n", "information.\n", "\n", "## Integration as signed area\n", "\n", "Velocity is the rate of change of position, so displacement over an interval is\n", "\n", "\n", "$$\n", "\\Delta x=\\int_{t_0}^{t_1}v(t)\\,\\mathrm{d}t\n", "$$\n", "\n", "\n", "On a velocity-versus-time graph, this integral is signed area. Regions with negative\n", "velocity subtract from displacement. Integrating speed instead gives distance travelled,\n", "which is a different quantity.\n", "\n", "For samples separated by a constant interval, the left-rectangle rule is\n", "\n", "\n", "$$\n", "\\Delta x\\approx\\sum_{n=0}^{N-1}v_n\\Delta t\n", "$$\n", "\n", "\n", "The trapezoid rule averages the velocities at both ends of each interval:\n", "\n", "\n", "$$\n", "\\Delta x\\approx\\sum_{n=0}^{N-1}\\frac{v_n+v_{n+1}}{2}(t_{n+1}-t_n)\n", "$$\n", "\n", "\n", "Writing the actual time difference supports irregular samples and makes the units\n", "visible." ] }, { "cell_type": "code", "id": "code-005", "metadata": {}, "source": [ "from dataclasses import dataclass\n", "\n", "\n", "@dataclass(frozen=True)\n", "class MotionSample:\n", " time_seconds: float\n", " velocity_metres_per_second: float\n", "\n", "\n", "def trapezoid_displacement(\n", " samples: list[MotionSample],\n", ") -> float:\n", " displacement_metres = 0.0\n", "\n", " for current, next_sample in zip(samples, samples[1:]):\n", " interval_seconds = next_sample.time_seconds - current.time_seconds\n", "\n", " if interval_seconds <= 0:\n", " raise ValueError(\"sample times must increase\")\n", "\n", " average_velocity = (\n", " current.velocity_metres_per_second + next_sample.velocity_metres_per_second\n", " ) / 2\n", "\n", " displacement_metres += average_velocity * interval_seconds\n", "\n", " return displacement_metres\n" ], "execution_count": null, "outputs": [] }, { "cell_type": "markdown", "id": "markdown-006", "metadata": {}, "source": [ "## An integration example\n", "\n", "Suppose\n", "$v(t)=3t^2\\,\\mathrm{m/s}$\n", "from zero to $2.0\\,\\mathrm{s}$. The exact\n", "displacement is\n", "\n", "\n", "$$\n", "\\int_0^2 3t^2\\,\\mathrm{d}t=8.0\\,\\mathrm{m}\n", "$$\n", "\n", "\n", "Using trapezoids with\n", "$h=0.5\\,\\mathrm{s}$ gives\n", "$8.25\\,\\mathrm{m}$.\n", "The positive difference is consistent with trapezoids lying above this convex velocity\n", "curve. Refining the spacing should reduce the error quadratically while the function\n", "remains smooth and other error sources are negligible.\n", "\n", "## Bias accumulates under integration\n", "\n", "Integration is less sensitive to high-frequency random noise than differentiation, but\n", "it accumulates bias. If a velocity sensor has a constant offset\n", "$b$, the displacement error after\n", "duration $T$ contains\n", "\n", "\n", "$$\n", "\\Delta x_{\\mathrm{bias}}=bT\n", "$$\n", "\n", "\n", "A small offset can therefore create a large long-duration drift. More samples do not\n", "remove that systematic error. Calibration, independent position observations, or a\n", "state-estimation model may be needed.\n", "\n", "Missing samples create gaps whose area is unknown. Interpolating across a gap is an\n", "assumption about the unobserved velocity. Record the gap and interpolation method rather\n", "than treating the reconstructed values as measurements.\n", "\n", "## Differentiate and integrate as cross-checks\n", "\n", "For a smooth analytic test trajectory, differentiate sampled positions and compare with\n", "the known velocity. Integrate sampled velocity and compare endpoint displacement. Then\n", "repeat with smaller intervals and observe the expected error trend.\n", "\n", "Differentiating and then integrating sampled data does not necessarily recover the\n", "original samples exactly. Boundary choices, noise, constants of integration, and both\n", "numerical approximations matter. Integration recovers position only up to an initial\n", "position:\n", "\n", "\n", "$$\n", "x(t)=x(t_0)+\\int_{t_0}^{t}v(\\tau)\\,\\mathrm{d}\\tau\n", "$$\n", "\n", "\n", "The starting value is not optional state.\n", "\n", "## Separate truncation, roundoff, and data error\n", "\n", "The difference between a computed result and the desired physical quantity can have\n", "several sources. Truncation error comes from replacing a derivative or integral with a\n", "finite formula. Roundoff comes from finite arithmetic. Data error includes measurement\n", "noise, timestamp uncertainty, bias, and quantization. Modeling error appears when the\n", "quantity being processed does not represent the intended physical system.\n", "\n", "A refinement study primarily diagnoses truncation error. Starting from one data set and\n", "interpolating it onto an increasingly fine grid does not create new measurement\n", "information. The numerical quadrature over the interpolant may converge, while the\n", "interpolant remains a poor account of what happened between observations.\n", "\n", "For differentiation, decreasing the interval initially reduces truncation error, but\n", "subtraction of nearby noisy or rounded values eventually dominates. Plotting estimated\n", "velocity against interval size can reveal a region where results are relatively stable.\n", "That stable region is evidence for a useful scale, not a universal optimum.\n", "\n", "For integration, compare multiple quadrature rules and refinements on a known smooth\n", "function before applying them to measurements. If results disagree more than the\n", "reported measurement uncertainty, investigate the sampling and method. If they agree\n", "far below a known sensor bias, do not claim the many matching digits as physical\n", "accuracy.\n", "\n", "Record each component that was estimated and each one that was omitted. A total error\n", "cannot be inferred solely from the numerical method's formal order.\n", "\n", "## What to report\n", "\n", "A numerical derivative or integral should include input units, timestamp policy, method,\n", "spacing, boundary treatment, missing-data policy, smoothing or interpolation, and an\n", "error comparison. Preserve unrounded samples and report uncertainty when the source data\n", "is measured.\n", "\n", "These operations translate geometric ideas - slope and signed area - into array\n", "algorithms. Their apparent simplicity should not hide the decisions made between the\n", "samples.\n", "\n", "## A worked Python notebook\n", "\n", "Differentiate a quadratic position curve and integrate its slope to recover displacement. The question for this calculation is precise: how does **position**\n", "change when we change **time**, while keeping the other choices fixed?\n", "The earlier sections explain the physical relationship. Here we turn one part of that\n", "relationship into a calculation that you can run, inspect, and change in Google Colab.\n", "\n", "This smooth reference is deliberately free of measurement noise. Keep that sentence beside the results. A formula can return a number outside\n", "its useful range, and a plotting library will usually draw that number without asking\n", "whether the physical assumptions still hold. The selected interval is from\n", "0 to 4 s. It gives us a specific domain to discuss\n", "instead of leaving the graph's range to a software default.\n", "\n", "Use the notebook link above the lesson, run the setup cell, and then run the cells in\n", "order. The code below is also included there. NumPy handles the arrays of numerical\n", "values, SymPy keeps the equation symbolic, and Matplotlib draws the results. If these\n", "libraries or notebook cells are unfamiliar, start with\n", "[Your first physics notebook](https://tirta.co/a-byte-of-physics/your-first-physics-notebook).\n", "\n", "### State the relationship before choosing a graph\n", "\n", "In this small example, `q` stands for time, expressed in s. The output is\n", "position, expressed in m. The equation below uses the numerical values in those\n", "units. Any fixed coefficients therefore carry the units needed to make the relation\n", "consistent. A value supplied in another unit must be converted before it enters this\n", "expression.\n", "\n", "\n", "$$\n", "f(q) = 2 q^{2}\n", "$$\n", "\n", "\n", "This is a function of one changing input. It is not a statement that the physical\n", "system has only one variable. We deliberately hold the other inputs fixed so that one\n", "cause of variation is visible. Changing several inputs at once would make it harder\n", "to tell which change produced the output difference. Write down the held-fixed choices\n", "before trying a different experiment.\n", "\n", "At 2 s, the expression gives 8 m. This is our\n", "reference calculation. It is a prediction of the stated example, not a new measurement.\n", "Its displayed digits help us compare implementations. They should not be read as a\n", "claim that physical inputs are known to six significant figures." ] }, { "cell_type": "code", "id": "code-007", "metadata": {}, "source": [ "import numpy as np\n", "import sympy as sp\n", "import matplotlib.pyplot as plt\n", "\n", "q = sp.Symbol(\"q\", real=True)\n", "expression = sp.sympify(\"2*q**2\", locals={\"q\": q})\n", "reference_input = 2.0\n", "reference_output = float(expression.subs(q, reference_input))\n", "print(\"Input:\", reference_input, \"s\")\n", "print(\"Predicted position:\", reference_output, \"m\")\n" ], "execution_count": null, "outputs": [] }, { "cell_type": "markdown", "id": "markdown-008", "metadata": {}, "source": [ "`sp.Symbol` declares a mathematical symbol. It does not assign a numerical measurement\n", "to that symbol. `subs` then replaces the symbol with our chosen input, and `float`\n", "converts the resulting symbolic number into a standard Python numerical value.\n", "Keeping these steps separate lets us inspect the equation before numerical substitution.\n", "It also prevents a later plot from becoming the only surviving record of the model.\n", "\n", "### Compare a few cases you can explain\n", "\n", "Before drawing hundreds of points, calculate three. The lower endpoint gives\n", "0 m. The midpoint, at 2 s, gives\n", "8 m. The upper endpoint gives 32 m.\n", "Read these together with the assumptions: they describe different choices of time\n", "within the same example, with the other inputs held fixed.\n", "\n", "These three values are a useful first check, but they do not establish the shape between\n", "them. Two endpoints can be equal even when the curve bends or oscillates in between.\n", "Likewise, a large change between endpoints does not tell us whether the relationship is\n", "linear. We need the equation and intermediate samples to answer those questions." ] }, { "cell_type": "code", "id": "code-009", "metadata": {}, "source": [ "evaluate = sp.lambdify(q, expression, modules=\"numpy\")\n", "inputs = np.array([0.0, 2.0, 4.0])\n", "outputs = np.asarray(evaluate(inputs), dtype=float)\n", "outputs = np.broadcast_to(outputs, inputs.shape)\n", "for input_value, output_value in zip(inputs, outputs):\n", " print(f\"{input_value:.6g} s -> {output_value:.6g} m\")\n", "assert np.all(np.isfinite(outputs))\n" ], "execution_count": null, "outputs": [] }, { "cell_type": "markdown", "id": "markdown-010", "metadata": {}, "source": [ "`lambdify` turns the symbolic expression into a numerical function. We explicitly select\n", "NumPy as its numerical backend so it can accept an array. The same expression now has\n", "two useful forms: one for algebra and one for calculating many cases. The assertion\n", "checks that the chosen inputs produce finite outputs. It would catch an infinity or an\n", "undefined numerical result, but a finite value can still be physically inappropriate.\n", "\n", "### Make a graph that answers the question\n", "\n", "The horizontal axis shows time; the vertical axis shows position. Every plotted\n", "point is a prediction of the same expression. We mark the reference case separately\n", "so that a reader can connect the earlier printed value to its place on the curve.\n", "The axis labels include units because a shape without a scale cannot tell us the size\n", "of a physical effect." ] }, { "cell_type": "code", "id": "code-011", "metadata": {}, "source": [ "input_grid = np.linspace(0.0, 4.0, 301)\n", "output_grid = np.asarray(evaluate(input_grid), dtype=float)\n", "output_grid = np.broadcast_to(output_grid, input_grid.shape)\n", "fig, ax = plt.subplots(figsize=(7, 4))\n", "ax.plot(input_grid, output_grid, label=\"Stated model\")\n", "ax.scatter([reference_input], [reference_output], label=\"Reference case\", zorder=3)\n", "ax.set_xlabel(\"Time (s)\")\n", "ax.set_ylabel(\"Position (m)\")\n", "ax.legend()\n", "ax.grid(alpha=0.25)\n", "fig.tight_layout()\n", "plt.show()\n" ], "execution_count": null, "outputs": [] }, { "cell_type": "markdown", "id": "markdown-012", "metadata": {}, "source": [ "Look for a sign change, a turning point, or a region where the curve becomes steep.\n", "Connect each feature to the expression rather than to the colour of the line. If the\n", "curve has a minimum or maximum, changing the input in opposite directions may have\n", "similar effects on the output near that point. If it is nearly straight over a short\n", "interval, a local linear approximation may be useful even when the complete expression\n", "is nonlinear.\n", "\n", "Increasing the number of samples makes the line smoother and may reveal features that\n", "a coarse grid missed. It does not improve the physical inputs or enlarge the model's\n", "validity range. For quantities that can take only discrete values, use the line as a\n", "guide to the relationship and interpret only the allowed input values. A plotting\n", "function does not enforce such restrictions for you.\n", "\n", "### Measure the local sensitivity\n", "\n", "A derivative asks a more specific question than the whole graph: how quickly does\n", "position change near the reference input? Its units are the output unit divided by\n", "the input unit. For this calculation, the derivative at 2 s is\n", "8 in those ratio units. Locally, the output increases as the\n", "input increases. This statement is local; the sign or size of the slope can be different\n", "elsewhere on the curve.\n", "\n", "\n", "$$\n", "f'(q) = 4 q\n", "$$\n", "\n", "\n", "For a small input change, multiply the derivative by that change to estimate the output\n", "change. Then compare that estimate with a fresh evaluation of the full expression.\n", "The distinction between these two calculations matters: the derivative gives a tangent\n", "approximation, while substitution evaluates the chosen expression at the new point." ] }, { "cell_type": "code", "id": "code-013", "metadata": {}, "source": [ "slope_expression = sp.diff(expression, q)\n", "slope_at_reference = float(slope_expression.subs(q, reference_input))\n", "changed_input = 2.02\n", "linear_prediction = reference_output + slope_at_reference * (\n", " changed_input - reference_input\n", ")\n", "full_prediction = float(evaluate(changed_input))\n", "print(\"Symbolic derivative:\", slope_expression)\n", "print(\"Tangent prediction:\", linear_prediction, \"m\")\n", "print(\"Full expression:\", full_prediction, \"m\")\n", "print(\"Approximation difference:\", full_prediction - linear_prediction, \"m\")\n" ], "execution_count": null, "outputs": [] }, { "cell_type": "markdown", "id": "markdown-014", "metadata": {}, "source": [ "Here the changed input is 2.02 s. The tangent estimate is\n", "8.16 m, while the full expression gives\n", "8.1608 m. If the two values are close, it means the chosen\n", "input change is small enough for this local approximation at this point. It does not\n", "mean the entire model is linear. If the slope is zero, a quadratic or higher-order term\n", "may supply the leading change instead.\n", "\n", "This is also the beginning of uncertainty propagation. If the input has a small stated\n", "standard uncertainty, the magnitude of the slope times that uncertainty estimates one\n", "contribution to output uncertainty. Do not automatically use the same rule for a wide\n", "interval, a discontinuity, a turning point with zero first derivative, or several\n", "correlated uncertain inputs. Those situations need more of the model than one slope.\n", "\n", "### Check the derivative with numerical differences\n", "\n", "We can estimate the same slope from values on either side of the reference point.\n", "Subtract the lower output from the upper output and divide by twice the input step.\n", "Parentheses are essential: the entire output difference must be divided by the entire\n", "input interval. Without them, Python can execute a different mathematical expression\n", "while still producing a plausible number.\n", "\n", "The calculation below uses three decreasing step sizes. Each estimate is compared with\n", "the symbolic derivative, so the reference is independent of the numerical difference\n", "formula. Near an endpoint, it is acceptable here to evaluate a small extension only\n", "because the displayed mathematical expression is smooth there; such a check does not\n", "assert that extrapolated values describe additional physical cases." ] }, { "cell_type": "code", "id": "code-015", "metadata": {}, "source": [ "input_span = 4.0\n", "for fraction in [1e-2, 1e-3, 1e-4]:\n", " step = input_span * fraction\n", " upper = float(evaluate(reference_input + step))\n", " lower = float(evaluate(reference_input - step))\n", " numerical_slope = (upper - lower) / (2 * step)\n", " difference = numerical_slope - slope_at_reference\n", " print(f\"step={step:.6g}: slope={numerical_slope:.6g}, difference={difference:.6g}\")\n" ], "execution_count": null, "outputs": [] }, { "cell_type": "markdown", "id": "markdown-016", "metadata": {}, "source": [ "For a smooth function, reducing the step often reduces the approximation error at first.\n", "Very small steps can eventually make subtraction of nearly equal floating-point values\n", "less reliable. Therefore “smaller” is not an unlimited guarantee of “better.” Compare\n", "the observed behaviour over a useful range of steps instead of choosing the smallest\n", "number the computer can represent.\n", "\n", "An absolute slope difference is useful when the expected slope is near zero. A relative\n", "difference would divide by a very small reference and could look enormous even when the\n", "absolute discrepancy is tiny. Away from zero, a relative difference can help compare\n", "quantities with very different scales. The tolerance should reflect the question and\n", "the numerical method, rather than a universal decimal threshold copied into every test.\n", "\n", "### Change the example and preserve the reasoning\n", "\n", "Make one copy of the reference calculation before editing it. Choose a second value of\n", "time inside the stated interval, predict whether position will rise or fall,\n", "and then evaluate the expression. Place the prediction and the result in a text cell.\n", "If they disagree, identify whether the mistake was in the sign, the unit conversion,\n", "the algebra, or the assumption about how the curve behaves.\n", "\n", "Next choose a shorter plotting interval around the reference point. Compare the local\n", "curve with its tangent approximation. Then widen the interval again. The difference\n", "shows why a useful local estimate need not describe the whole range. Keep the axis\n", "labels and numerical scale visible when comparing the plots; automatic axis limits can\n", "make two different effects appear equally large.\n", "\n", "Finally, return to the physical limit stated at the start: This smooth reference is deliberately free of measurement noise. Describe one change\n", "to the setup that would violate that limit. State what new input, equation, or observation\n", "would be needed before trusting an answer in the changed situation. This last step\n", "connects the Python experiment to the physics instead of treating successful execution\n", "as the only criterion for a useful result.\n", "\n", "Before saving, restart the runtime and run all cells in order. A result that depends on\n", "an earlier, now-deleted input cell is not reproducible from the saved notebook. Keep\n", "unrounded values for later calculations, round only the reported output, and label any\n", "new data as measured, assumed, or generated. The graph and printed values should remain\n", "traceable to the same expression and the same choices of inputs.\n", "\n", "### Recover the output change by integrating the slope\n", "\n", "There is another way to check that the expression and its derivative describe the same\n", "relationship. Add up the small output changes predicted by the slope across the chosen\n", "input interval. In the continuous limit, this integral equals the output at the upper\n", "endpoint minus the output at the lower endpoint. The result has units of m:\n", "the input unit in the integration cancels the input unit in the denominator of the slope.\n", "\n", "This statement is about the mathematics of this particular relationship. It does not\n", "assign a new physical meaning to an arbitrary area under a graph. For example, an area\n", "under force versus distance represents work, while an area under a different pair of\n", "quantities may represent something else or have no standard name. Always carry the\n", "axis units through the calculation before naming an integral." ] }, { "cell_type": "code", "id": "code-017", "metadata": {}, "source": [ "slope_function = sp.lambdify(q, slope_expression, modules=\"numpy\")\n", "endpoint_change = float(evaluate(4.0) - evaluate(0.0))\n", "for points in [101, 201, 401]:\n", " grid = np.linspace(0.0, 4.0, points)\n", " slopes = np.asarray(slope_function(grid), dtype=float)\n", " slopes = np.broadcast_to(slopes, grid.shape)\n", " integrated_change = np.trapezoid(slopes, grid)\n", " print(\n", " points,\n", " \"integrated change:\",\n", " integrated_change,\n", " \"endpoint change:\",\n", " endpoint_change,\n", " )\n" ], "execution_count": null, "outputs": [] }, { "cell_type": "markdown", "id": "markdown-018", "metadata": {}, "source": [ "The trapezoidal rule joins neighbouring slope samples with straight segments and adds\n", "their signed areas. A negative contribution remains negative; replacing every slope\n", "by its magnitude would calculate a different quantity. Increasing the sample count\n", "provides a practical refinement check. If the values do not settle toward the endpoint\n", "difference, first inspect the interval, singularities, and array ordering.\n", "\n", "Do not conclude that two agreeing calculations prove the model is physically complete.\n", "Both use the same selected relationship, so an omitted effect would be omitted from\n", "both. Their agreement is evidence that the numerical tools implement that relationship\n", "consistently. Comparing with an independent experiment or a more appropriate physical\n", "model is a separate test.\n", "\n", "Keep this comparison in the notebook when you change the expression. It is particularly\n", "useful after adding a coefficient, changing a sign convention, or converting an input\n", "unit. A code cell that produces a smooth graph can conceal these mistakes; a second\n", "calculation of the same output change gives you another place to notice them.\n", "\n", "## Sources\n", "- [OpenStax, *University Physics Volume 1*, Sections 3.1-3.3, position, velocity, and acceleration](https://openstax.org/books/university-physics-volume-1/pages/3-introduction).\n", "- [MIT OpenCourseWare, *8.01SC Classical Mechanics*, Chapter 4, \"One Dimensional Kinematics\"](https://ocw.mit.edu/courses/8-01sc-classical-mechanics-fall-2016/resources/mit8_01scs22_chapter4/).\n", "- [Benjamin Crowell, *Light and Matter*, Sections 3.4-3.7, varying acceleration, graphical integration, and calculus](https://open.umn.edu/opentextbooks/textbooks/light-and-matter).\n", "- [The Physics Hypertextbook, \"Calculus\"](https://physics.info/calculus/)." ] } ], "metadata": { "kernelspec": { "display_name": "Python 3", "language": "python", "name": "python3" }, "language_info": { "name": "python", "version": "3.12" }, "colab": { "name": "numerical-differentiation-integration.ipynb", "provenance": [] }, "tirta": { "lesson": "https://tirta.co/a-byte-of-physics/numerical-differentiation-integration", "source_sha256": "4bc0c3680bc848ba35b04f470a0f38a7da178d6481086592528fc7bb177a5d7d" } }, "nbformat": 4, "nbformat_minor": 5 }