Calculator Input
Formula Used
Supported scalar field
f(r,φ,z) = A·rn + B·φm + C·zp + D·r·z + E·r·φ + F·z·φ + G·sin(kφ) + H·cos(kφ) + J·eλr + C0
Partial derivatives
∂f/∂r = A·n·rn−1 + D·z + E·φ + J·λ·eλr
∂f/∂φ = B·m·φm−1 + E·r + F·z + G·k·cos(kφ) − H·k·sin(kφ)
∂f/∂z = C·p·zp−1 + D·r + F·φ
Cylindrical gradient: ∇f = er(∂f/∂r) + eφ(1/r)(∂f/∂φ) + ez(∂f/∂z)
- The angular coordinate φ should be entered in radians.
- The angular gradient component requires division by r.
- If r equals zero, the angular component can become undefined.
- The plotted line tracks gradient magnitude while φ and z stay fixed.
How to Use This Calculator
- Enter the evaluation point values for r, φ, and z.
- Fill in the coefficients and exponents for the scalar field model.
- Set graph radius bounds and the number of plotted points.
- Press Calculate Gradient to compute the field and gradient components.
- Review the result cards, table, and Plotly graph above the form.
- Use the CSV or PDF buttons to export the current calculation.
Example Data Table
This sample uses A=2, n=2, B=3, m=2, C=4, p=2, D=5, E=0, F=0, G=0, H=0, J=0, C₀=1 at r=2, φ=1, z=3.
| Example Item | Value | Explanation |
|---|---|---|
| Scalar field value | 78 | f(2,1,3) = 2(2²) + 3(1²) + 4(3²) + 5(2·3) + 1 |
| ∂f/∂r | 23 | 4r + 5z evaluated at r=2 and z=3 |
| ∂f/∂φ | 6 | 6φ evaluated at φ=1 |
| ∂f/∂z | 34 | 8z + 5r evaluated at z=3 and r=2 |
| Angular component | 3 | (1/r)(∂f/∂φ) = 6 / 2 |
| Gradient magnitude | 41.1582 | √(23² + 3² + 34²) |
FAQs
1) What does this calculator compute?
It computes the scalar field value, the three cylindrical gradient components, the partial derivatives, and the gradient magnitude at a chosen point.
2) Why is φ entered in radians?
The trigonometric terms sin(kφ) and cos(kφ) use radian measure in standard calculus formulas. Radians keep derivatives consistent and correct.
3) What happens when r equals zero?
The angular component uses (1/r)∂f/∂φ, so it may become undefined at the axis. The calculator flags this case clearly.
4) Can I model simple fields only?
No. You can combine polynomial, mixed-product, trigonometric, exponential, and constant terms, which makes the tool flexible for advanced practice.
5) What does the Plotly graph show?
It shows how the gradient magnitude changes with radius across your selected range while keeping φ and z fixed at the entered values.
6) Are the coefficients restricted to integers?
No. Coefficients can be decimal values. The exponents are entered as whole numbers here to keep real-valued evaluation stable and practical.
7) What is the difference between ∂f/∂φ and the angular gradient component?
In cylindrical coordinates, the angular gradient component is not just ∂f/∂φ. It must be scaled by 1/r to match the coordinate geometry.
8) Can I export the result?
Yes. After calculation, use the CSV button for spreadsheet-friendly data or the PDF button for a clean portable report.