Gradient Calculator Inputs
Provide the point and partial derivatives of the scalar field.
Formula Used
The cylindrical system is an orthogonal curvilinear coordinate system. Its scale factors are hr = 1, hθ = r, and hz = 1.
General orthogonal formula: ∇f = Σ (1/hi)(∂f/∂ui)ei Cylindrical substitution: ∇f = (∂f/∂r)er + (1/r)(∂f/∂θ)eθ + (∂f/∂z)ez Basis conversion: er = (cosθ, sinθ, 0), eθ = (−sinθ, cosθ, 0), ez = (0, 0, 1) Cartesian components: gx = grcosθ − gθsinθ, gy = grsinθ + gθcosθ, gz = ∂f/∂zThis page assumes you already know the scalar field partial derivatives at the chosen cylindrical point.
Derivation Notes
- Start from the orthogonal curvilinear gradient operator. Every coordinate contributes one derivative term divided by its scale factor.
- For cylindrical coordinates, the differential displacement is dr = erdr + eθr dθ + ezdz.
- From that displacement, the scale factors become 1, r, and 1. Only the angular direction stretches with radius.
- Insert the scale factors into the general operator. That immediately creates the factor (1/r) in front of ∂f/∂θ.
- The result is the standard cylindrical gradient formula, which can then be converted into Cartesian components using the basis-vector identities.
How to Use This Calculator
- Enter the cylindrical point using r, θ, and z.
- Choose degrees or radians for the angle input.
- Supply the evaluated partial derivatives ∂f/∂r, ∂f/∂θ, and ∂f/∂z.
- Press Calculate Gradient to generate the cylindrical vector, Cartesian vector, magnitude, derivation snapshot, and Plotly graph.
- Use the CSV and PDF buttons to export the generated results table.
Example Data Table
| r | θ | z | ∂f/∂r | ∂f/∂θ | ∂f/∂z | gr | gθ | gz | |∇f| |
|---|---|---|---|---|---|---|---|---|---|
| 2 | 30° | 1 | 4 | 6 | -1 | 4 | 3 | -1 | 5.099 |
| 5 | 90° | 2 | 2.5 | 10 | 4 | 2.5 | 2 | 4 | 5.123 |
| 3 | 45° | -2 | 1.2 | 9 | 0.5 | 1.2 | 3 | 0.5 | 3.269 |
Frequently Asked Questions
1. Why does the angular term include 1/r?
The angular arc length is r dθ, not just dθ. That stretch changes the scale factor, so the gradient divides the angular derivative by r.
2. Why is r = 0 blocked?
The cylindrical basis becomes singular on the axis. The term (1/r)(∂f/∂θ) is not generally defined there, so this calculator requires a positive radius.
3. Does the calculator perform symbolic differentiation?
No. It evaluates the gradient from partial derivative values you already know. That makes the tool practical for solved homework, checkpoints, and verification.
4. What does gθ represent here?
It is the actual cylindrical gradient component along eθ. The tool computes it as (1/r)(∂f/∂θ), after applying the cylindrical scale factor.
5. Why convert the result into Cartesian components?
Cartesian components help compare the vector with standard x, y, z geometry. They also make plotting, verification, and cross-checking with other methods easier.
6. Is θ entered in degrees or radians?
Either format works. Select the matching angle unit before calculation, and the page converts the value internally for the trigonometric basis conversion.
7. What is the magnitude used for?
The magnitude measures the maximum local rate of increase of the scalar field. It summarizes how strongly the field changes at the chosen point.
8. When is this calculator most useful?
It is helpful during vector calculus study, coordinate-system comparisons, worked examples, engineering field analysis, and quick validation of cylindrical gradient computations.