Calculator Form
Choose a harmonic family, enter coefficients, set a point, and generate the conjugate pair with derivative checks.
Example Data Table
| Family | Sample coefficients | Point | u(x,y) | v(x,y) |
|---|---|---|---|---|
| Linear | a=2, b=1, c=0, k=0 | (1, 2) | 0 | 5 |
| Quadratic | a=1, b=0, c=0, d=0, e=0, k=0 | (2, 1) | 3 | 4 |
| Logarithmic | a=1, c=0, k=0 | (1, 1) | 0.346574 | 0.785398 |
| Reciprocal | a=3, c=0, k=0 | (2, 1) | 1.2 | -0.6 |
Formula Used
Harmonic conjugates are linked through the Cauchy Riemann equations. These are ux = vy and uy = -vx. When both relations hold in a region, u and v form the real and imaginary parts of an analytic function.
In physics, harmonic pairs appear in potential flow, electrostatics, and steady heat problems. One function may describe a potential. The conjugate function may describe a stream function or an orthogonal field pattern.
This calculator uses selected harmonic families instead of unrestricted symbolic algebra. That approach keeps the page fast and practical. It also allows reliable evaluation, graphing, and numerical derivative checks inside one file.
The linear family is useful for uniform fields. Quadratic and cubic families model richer local behavior. Logarithmic and reciprocal families model singular fields. The exponential family can represent oscillatory growth with orthogonal structure.
For logarithmic and angular families, the angle term uses atan2(y,x). That means the branch can change across the negative x axis. The pair remains useful, but interpretation should respect the selected branch.
How to Use This Calculator
- Select a harmonic family from the dropdown.
- Enter the coefficients used by that family.
- Set the x and y coordinates for evaluation.
- Set the graph range and the fixed y slice.
- Press the calculate button.
- Read the displayed pair, numeric values, and derivative checks.
- Review the Plotly graph for the selected slice.
- Use the CSV or PDF buttons to export the result.
Why Harmonic Conjugates Matter in Physics
Many two dimensional physical fields are easier to study through analytic functions. The real part and imaginary part form orthogonal structures. Equipotential lines intersect streamlines at right angles. That makes flow and field geometry easier to inspect.
In potential flow, velocity potential and stream function often behave as harmonic conjugates. In electrostatics, field potential in charge free regions is harmonic. In heat conduction, steady temperature fields also satisfy Laplace type behavior away from sources.
This page gives a compact workflow. You choose a family, evaluate the pair at a point, check the Cauchy Riemann relations, and inspect a graph slice. That is useful for classwork, demonstrations, and quick model exploration.
FAQs
1. What does this calculator return?
It returns a harmonic pair u and v for the selected family. It also evaluates both at a chosen point, checks numerical derivatives, and plots a graph slice.
2. Does it handle arbitrary symbolic input?
No. This version uses reliable harmonic families. That keeps the page stable in one file and still covers many common classroom and physics examples.
3. Why are Cauchy Riemann checks included?
They confirm that the selected functions behave like conjugate components locally. Small residuals suggest the pair is numerically consistent at the sampled point.
4. Why can logarithmic or angular results jump suddenly?
Those families depend on angle. Angle is branch dependent, so values can jump near the negative x axis even when the model remains mathematically useful.
5. What happens at the origin for singular families?
Logarithmic, angular, and reciprocal families are undefined at the origin. The calculator blocks direct evaluation there and marks invalid graph points as empty.
6. What does the graph show?
The Plotly graph shows a one dimensional slice along x while y stays fixed. That makes comparison between u and v simple and quick.
7. Can I export my results?
Yes. The page includes client side CSV and PDF export buttons. They capture the current result summary and the plotted sample table.
8. Is this useful for physics students?
Yes. It helps with potential flow, electrostatics, and steady field models. It is also useful for checking examples from complex variables courses.