Curl in Spherical Calculator

Calculator Inputs

Use the responsive grid below. Large screens show three columns, smaller screens show two, and mobile shows one.

Angular derivatives such as ∂A/∂θ and ∂A/∂φ should be entered per radian. Angle mode only changes the point angle conversion.

Radius r

Polar angle θ

Azimuth angle φ

Angle unit

A_r

A_θ

A_φ

∂A_r/∂θ

∂A_r/∂φ

∂A_θ/∂r

∂A_θ/∂φ

∂A_φ/∂r

∂A_φ/∂θ

Decimal places

Formula Used

For a vector field A = A_re_r + A_θe_θ + A_φe_φ, the curl in spherical coordinates is:

(∇ × A)_r = 1 / (r sinθ) × [∂(A_φ sinθ)/∂θ − ∂A_θ/∂φ]

(∇ × A)_θ = 1 / r × [(1 / sinθ)∂A_r/∂φ − ∂(rA_φ)/∂r]

(∇ × A)_φ = 1 / r × [∂(rA_θ)/∂r − ∂A_r/∂θ]

This page expands the product terms automatically: ∂(A_φsinθ)/∂θ = sinθ·∂A_φ/∂θ + A_φcosθ, ∂(rA_φ)/∂r = A_φ + r·∂A_φ/∂r, and ∂(rA_θ)/∂r = A_θ + r·∂A_θ/∂r.

How to Use This Calculator

Enter the spherical point using r, θ, and φ.
Select whether your entered angles are in degrees or radians.
Provide the vector field components A_r, A_θ, and A_φ at that point.
Enter the required partial derivatives.
Choose the decimal precision.
Press Calculate Curl to display the result above the form.
Use the CSV and PDF buttons to export the computed summary.
Review the chart and substitution table to validate each component.

Example Data Table

r	θ	φ	A_r	A_θ	A_φ	∂A_r/∂θ	∂A_r/∂φ	∂A_θ/∂r	∂A_θ/∂φ	∂A_φ/∂r	∂A_φ/∂θ	(∇ × A)_r	(∇ × A)_θ	(∇ × A)_φ	\|∇ × A\|
3.000000	45.000000°	60.000000°	2.100000	1.400000	0.900000	0.500000	-0.200000	0.300000	0.400000	-0.100000	0.600000	0.311438	-0.294281	0.600000	0.737289

FAQs

1. What does this calculator compute?

It computes the curl vector of a field written in spherical coordinates. You get the r, θ, and φ curl components, magnitude, intermediate terms, and a comparison chart.

2. Why is θ = 0 or θ = π a problem?

The spherical curl equations contain 1/sinθ. At the poles, sinθ becomes zero, so the standard component form becomes singular and direct evaluation is not reliable.

3. Do I enter derivatives in degrees or radians?

Enter angular derivatives per radian. The angle unit selector only converts the point angle you enter for θ and φ, not the derivative scaling.

4. Is φ always needed in the formulas?

The explicit formulas shown do not use the numeric φ value directly, but the point definition is still important for documenting where the vector field and derivatives were evaluated.

5. Can I use negative component values?

Yes. Field components and derivatives may be positive, negative, or zero. The calculator handles signed values and shows the final sign for each curl component.

6. What is the curl magnitude useful for?

The magnitude summarizes total local rotational strength at the chosen point. It is useful when you want one scalar measure instead of tracking three separate directional components.

7. Why are product terms expanded automatically?

Terms like ∂(Aφsinθ)/∂θ and ∂(rAθ)/∂r require product rules. Expanding them avoids manual algebra mistakes and makes the result easier to audit.

8. Can this be used for homework checking?

Yes. It is helpful for checking hand calculations, validating intermediate substitutions, and comparing the final vector curl with your own derivation at the same point.