Introduction to Pandas for Physics: Reshaping Data & Pivot Tables in Grid Measurements - Computational Physics Formulation & Simulation

Program 1

Reshaping Data & Pivot Tables in Grid Measurements

Python 3.11 Ready

import pandas as pd
import numpy as np

# Spatial Magnetic Field Mapping: B_z(x, y) in milliTesla (mT)
# Measured across a 5x5 transverse grid (-2 cm to +2 cm) inside a Helmholtz coil
x_coords = [-2, -1, 0, 1, 2]  # x positions (cm)
y_coords = [-2, -1, 0, 1, 2]  # y positions (cm)

# Center field B0 = 10.00 mT, coil radius R = 10 cm
B0 = 10.00
scan_log = []

for x in x_coords:
    for y in y_coords:
        r_sq = x**2 + y**2
        # Off-axis expansion drop-off B(r) = B0 * (1 - 0.0075 * r^2) + noise
        b_measured = B0 * (1.0 - 0.0075 * r_sq) + np.random.normal(0, 0.012)
        scan_log.append({
            'x_cm': x,
            'y_cm': y,
            'B_z_mT': round(b_measured, 3),
            'probe_id': 'HALL-PROBE-01'
        })

df_field = pd.DataFrame(scan_log)

print("=== Sequential Scanning Log (Long Format, First 5 Records) ===")
print(df_field.head(5))

# 1. Reshape Long-Format into 2D Spatial Grid Matrix using pd.pivot_table
grid_matrix = df_field.pivot_table(index='y_cm', columns='x_cm', values='B_z_mT')

# Invert y index so top row corresponds to positive y (Cartesian orientation)
grid_matrix = grid_matrix.sort_index(ascending=False)

print("\n=== 2D Magnetic Field Matrix B_z(x, y) [mT] (Wide Format) ===")
print(grid_matrix.to_string())

# 2. Central Field Homogeneity Assessment: Delta_B / B_0 (%)
B_center = grid_matrix.loc[0, 0]
deviation_matrix = 100.0 * (grid_matrix - B_center) / B_center

print("\n=== Percentage Field Deviation Relative to Center Delta_B / B_0 (%) ===")
print(deviation_matrix.round(2).to_string())

# 3. MultiIndex Stacking: Convert 2D Grid back into Hierarchical Series
stacked_data = grid_matrix.stack()
print("\n=== Stacked Coordinate MultiIndex Series (Sample) ===")
print(stacked_data.head(6))

# Max spatial field deviation across 4 cm x 4 cm working region
max_drop = np.abs(grid_matrix.values - B_center).max()
print(f"\nCoil Center Field B(0,0): {B_center:.3f} mT")
print(f"Max Field Inhomogeneity within r = 2 cm: {max_drop:.3f} mT ({(max_drop/B_center)*100:.2f}%)")

Click "Run Code" to execute script and render figures.

Mathematical Problem Formulation

Physical Law: Spatial Magnetic Field Uniformity in a Helmholtz Coil:

A Helmholtz coil pair consists of two identical circular coils of radius $R$, separated by distance equal to their radius ($d = R$). On the central transverse plane ($z = 0$), the axial magnetic field $B_z(x, y)$ exhibits maximal uniformity due to cancellation of second-order spatial derivatives:

$$\left. \frac{\partial B_z}{\partial z} \right|_{0} = 0, \quad \left. \frac{\partial^2 B_z}{\partial z^2} \right|_{0} = 0$$

Off-axis in the transverse plane, the field diminishes quadratically with radial distance $r = \sqrt{x^2 + y^2}$ according to the multipole expansion:

$$B_z(r) \approx B_0 \left( 1 - \alpha \frac{r^2}{R^2} \right)$$

Spatial Matrix Transformation (Pivot Table):

Laboratory data collected as sequential serial tuples $(x_i, y_i, B_i)$ in "long format" is mapped via a pivot transformation into a 2D Cartesian spatial grid matrix $\mathbf{B} \in \mathbb{R}^{M \times N}$:

$$\mathbf{B}_{j, k} = B_z(x_k, y_j)$$

Theoretical Background & Computational Mechanics

1. Long vs Wide Formats in Experimental Physics:

Laboratory automated scanning rigs (such as 2D Hall probe positioners or automated spectrometer goniometers) output continuous streaming logs where each line records a point coordinate and a sensor reading ("long format"). However, physical analysis and 2D contour visualization require a 2D matrix structure ("wide format").

2. pd.pivot_table() for Spatial Grid Construction:

The pivot_table() function maps coordinates to index rows and columns, reorganizing 1D lists of records into coordinate matrices. If multiple measurements exist at the same coordinate $(x, y)$, it automatically aggregates them via mean or median.

3. Stacking & Unstacking:

.unstack(): Pivots an index level into column headers.
.stack(): Condenses column headers back into a compact multi-index Series.

4. Field Homogeneity Evaluation:

By subtracting the central field $B(0, 0)$ across the matrix, researchers quantify spatial field deviations $\frac{\Delta B}{B_0}\%$, establishing the effective uniform working volume of the magnet apparatus.

Theoretical Principles & Computational Physics Guide

Reshaping Data & Pivot Tables in Grid Measurements: Computational Physics Modeling & Analysis

Dr. Alorika Chatterjee & Dr. Anirban Shaw Science & Physics Learners Python 3 / NumPy / SciPy

1. Core Principles & Theoretical Framework

Core Concept: Computational formulation and simulation of Reshaping Data & Pivot Tables in Grid Measurements within the study of Introduction to Pandas for Physics.

Algorithmic Translation: Translates theoretical governing equations into clean, discrete Python procedures and verified numerical solvers.

Numerical Verification: Evaluates numerical accuracy, parameter sensitivity, and error bounds for reliable computational results.

2. Mathematical Formulation & Algorithmic Pipeline

Governing computational formulation for Reshaping Data & Pivot Tables in Grid Measurements:

$$ \mathcal{F}_{\text{model}}(\mathbf{x}, t; \mathbf{p}) = 0 \implies \mathbf{y}_{n+1} = \mathcal{S}(\mathbf{y}_n, \Delta t, \mathbf{p}) $$

1. Define Physical Parameters & Boundaries → 2. Execute Numerical Simulation Steps → 3. Evaluate Observables & Plot Results

3. Numerical Mechanics & Implementation Logic

Parameter Initialization: Set physical constants, initial states, and coordinate boundary bounds for Reshaping Data & Pivot Tables in Grid Measurements.
Step Execution: Advance the computational state using optimized vector operations and scientific libraries.
Convergence Control: Monitor stability and ensure simulation metrics remain physically meaningful.

4. Relevant Computational Practice Exercises

Exercise 1

Implement Reshaping Data & Pivot Tables in Grid Measurements and observe how the output responds to variations in primary physical parameters.

Exercise 2

Refine the calculation grid or step size to observe convergence and reduction in numerical approximation error.

Exercise 3

Compare the simulated computational results against an analytical benchmark or limiting physical case.

Standard Literature & References:

Mark Newman, "Computational Physics", University of Michigan (2013).
Hans Petter Langtangen, "A Primer on Scientific Programming with Python", Springer (2016).

Ch. 8.8 Reshaping Data & Pivot Tables in Grid Measurements