Advanced Functional Derivative Calculator

Calculator Inputs

This page uses a single stacked layout, while the calculator fields switch to 3 columns on large screens, 2 on smaller screens, and 1 on mobile.

Report title

Domain and sampling

x start

x end

Grid points

Recommended range: 31 to 121.

Profile y(x) parameters

Sine amplitude

Sine frequency

Sine phase

Cosine amplitude

Cosine frequency

Exponential amplitude

Exponential rate

Functional coefficients

c1 for y²

c2 for (y′)²

c3 for y·y′

c4 for x·y

c5 for x·y²

c6 for y⁴

c7 for sin(y)

c8 for cos(y)

c9 for exp(y)

c10 for s(x)·y

c11 for x·(y′)²

c12 for y·(y′)²

External source s(x)

Source frequency

Reset

Example data table

The table below shows sample profile values generated from the default settings bundled with this calculator.

x	y(x)	y′(x)	s(x)
0.0000	1.300000	1.175000	0.500000
0.5000	1.650635	0.098117	0.649499
1.0000	1.388485	-1.017813	0.428224
1.5000	0.867269	-0.783061	0.154494
2.0000	0.838206	0.791783	0.244117

Formula used

This calculator evaluates a configurable functional of the form shown below.


                    J[y] = ∫[ c1y² + c2(y′)² + c3yy′ + c4xy + c5xy² + c6y⁴ + c7sin(y) + c8cos(y) + c9e^y + c10s(x)y + c11x(y′)² + c12y(y′)² ] dx

The trial profile is built from polynomial, sine, cosine, and exponential parts.


                    y(x) = a0 + a1x + a2x² + a3x³ + Asin·sin(ws x + φ) + Acos·cos(wc x) + Aexp·e^rx

The Euler-Lagrange functional derivative used in the numerical report is:


                    δJ/δy = ∂L/∂y − d/dx(∂L/∂y′)


                    = 2c1y − 2c2y″ + c4x + 2c5xy + 4c6y³ + c7cos(y) − c8sin(y) + c9e^y + c10s(x)


                    − 2c11(y′ + xy″) − c12[(y′)² + 2yy″]

The constant-coefficient term c3yy′ is a total derivative, so its interior Euler-Lagrange contribution cancels. It still affects boundary momentum.

How to use this calculator

Choose the interval and number of sample points.
Set the profile coefficients that define your trial function y(x).
Enter the functional coefficients for each supported term.
Adjust the external source values if your model includes forcing.
Submit the form to compute J[y], stationarity metrics, and boundary momentum.
Inspect the plotted curves and the full data table.
Download CSV for raw values or PDF for a compact report.

FAQs

1. What does this calculator actually compute?

It evaluates a configurable variational functional, computes the Euler-Lagrange functional derivative numerically, and reports stationarity quality, boundary momentum, graphs, and tabulated sampled values.

2. Is this a fully general symbolic functional derivative engine?

No. It is an advanced numerical calculator for a rich predefined family of terms. That makes it practical, stable, and easy to extend inside one file.

3. Why is y·y′ included if its interior contribution cancels?

That term is useful educationally because it shows how a total derivative disappears from the interior Euler-Lagrange equation while still influencing the boundary momentum expression.

4. What does the RMS stationarity score mean?

It is the root-mean-square magnitude of δJ/δy across all sampled points. Smaller values mean the chosen profile is closer to satisfying the stationarity condition.

5. How should I choose the number of grid points?

Start with 41 or 61 points for smooth profiles. Increase the count when oscillations, sharp gradients, or strong exponential growth are present.

6. What is boundary momentum in this report?

Boundary momentum is ∂L/∂y′ evaluated at the interval endpoints. It helps analyze natural boundary behavior and terms that depend explicitly on derivatives.

7. Can I use negative coefficients and frequencies?

Yes. Negative values are accepted. They can reverse forcing, curvature, oscillation direction, or energetic contributions, so interpret large magnitudes with care.

8. What do the CSV and PDF downloads contain?

CSV exports the full sampled dataset. PDF exports the headline report values and a compact selection of computed rows for quick sharing or archiving.