Calculator Inputs
The page stays single-column, while the form uses three columns on large screens, two on medium screens, and one on mobile.
Formula Used
General spherical gradient formula
- ∇f = er(∂f/∂r) + eθ(1/r)(∂f/∂θ) + eφ(1/(r sinθ))(∂f/∂φ)
- gr = ∂f/∂r
- gθ = (1/r)(∂f/∂θ)
- gφ = (1/(r sinθ))(∂f/∂φ)
- |∇f| = √(gr2 + gθ2 + gφ2)
Preset 1: Power-Trig Field
- f(r,θ,φ) = A rn sinm(θ) cos(kφ)
- ∂f/∂r = A n rn-1 sinm(θ) cos(kφ)
- ∂f/∂θ = A m rn sinm-1(θ) cos(θ) cos(kφ)
- ∂f/∂φ = -A k rn sinm(θ) sin(kφ)
How to Use This Calculator
- Choose a scalar field preset that matches your problem.
- Enter model parameters A, n, m, k, and a where needed.
- Provide the evaluation point using r, θ, and φ.
- Set the radial sweep range for the Plotly graph.
- Click the calculate button to display the result section.
- Review scalar value, partial derivatives, gradient components, and magnitude.
- Use the CSV button for tabular export.
- Use the PDF button for a compact report download.
Example Data Table
| Preset | A | n | m | k | a | r | θ | φ | gr | gθ | gφ | |∇f| |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Power-Trig | 2.0 | 3 | 1 | 2 | 0.35 | 2.0 | 45° | 30° | 8.485281 | 2.828427 | -13.856406 | 16.492423 |
| Exponential-Angular | 1.8 | 2 | 2 | 1 | 0.35 | 1.5 | 60° | 45° | 0.188265 | -1.242228 | 0.414076 | 1.322889 |
| Log-Trig | 3.2 | 2 | 2 | 3 | 0.00 | 2.4 | 50° | 20° | 0.391216 | 0.574779 | -2.323193 | 2.425004 |
| Inverse Harmonic | 4.0 | 2 | 1 | 2 | 0.00 | 3.0 | 40° | 35° | -0.065140 | 0.038815 | -0.278427 | 0.288568 |
FAQs
1) What does this calculator compute?
It computes the gradient of a scalar field in spherical coordinates. You get partial derivatives, spherical gradient components, magnitude, direction, a radial sweep graph, and export-ready output.
2) Which angle definition does the calculator use?
Theta θ is measured downward from the positive z-axis. Phi φ is the azimuth angle in the xy-plane. This is the common physics and vector-calculus convention.
3) Why can the φ-component become undefined?
The azimuthal term contains 1/(r sinθ). At θ = 0° or 180°, sinθ becomes zero, so the spherical basis is singular there. The calculator warns you when this occurs.
4) What is the meaning of gr, gθ, and gφ?
These are the orthonormal spherical gradient components. They describe how strongly the scalar field changes in the radial, polar, and azimuthal directions at the chosen point.
5) Why include several scalar field presets?
Different presets help test power-law, exponential, logarithmic, and inverse behaviors. This makes the page more useful for practice, verification, and quick sensitivity studies across multiple field types.
6) How should I interpret the Plotly graph?
The graph sweeps radius while holding θ and φ fixed. It shows how each gradient component and the full magnitude change as you move outward or inward along the same direction.
7) What is included in the CSV and PDF exports?
The exports include model settings, evaluation coordinates, main gradient results, and the radial sweep table. They are useful for reports, homework checks, or engineering-style documentation.
8) Can this calculator replace symbolic algebra software?
It is best for structured presets and fast numerical evaluation. For arbitrary custom symbolic expressions, a dedicated symbolic algebra tool remains better suited than a preset-based calculator.