Enter Spherical Vector Field
Use variables r, theta, and phi.
Supported functions include sin, cos, tan,
sqrt, log, exp, and powers like r^2.
Curl Magnitude Graph
The chart varies the selected coordinate while other inputs stay fixed.
Formula Used
Vector field: A = Ar er + Aθ eθ + Aφ eφ
(∇ × A)r = [∂(Aφ sinθ)/∂θ − ∂Aθ/∂φ] / [r sinθ]
(∇ × A)θ = [(1/sinθ)∂Ar/∂φ − ∂(rAφ)/∂r] / r
(∇ × A)φ = [∂(rAθ)/∂r − ∂Ar/∂θ] / r
Magnitude = √[(curlr)² + (curlθ)² + (curlφ)²]
The calculator estimates partial derivatives with a central difference. Reduce h for smoother fields. Increase h if round-off noise appears.
How to Use This Calculator
- Enter each spherical component of the vector field.
- Use
thetafor the polar angle. - Use
phifor the azimuth angle. - Enter the evaluation point.
- Select radians or degrees.
- Choose the graph variable, range, and number of points.
- Press Calculate to view components and magnitude.
- Use CSV or PDF export for records and assignments.
Example Data Table
| Case | Ar | Aθ | Aφ | r | θ | φ | Expected idea |
|---|---|---|---|---|---|---|---|
| Swirl field | 0 | 0 | r*sin(theta) | 2 | 1.0472 | 0.5 | Nonzero radial and polar curl |
| Radial growth | r^2 | 0 | 0 | 3 | 1.2 | 0.7 | Zero curl for pure radial dependence |
| Mixed angle | sin(phi) | r*cos(theta) | theta*r | 1.5 | 0.9 | 0.4 | All components may appear |
Understanding Curl in Spherical Coordinates
Curl measures local rotation in a vector field. In spherical coordinates, it is useful for fields that fit round shapes. Many mathematics and physics problems use this system. Examples include magnetic fields, fluid flow, angular motion, and radial force models. The coordinates are radius r, polar angle theta, and azimuth angle phi. Each coordinate follows a curved direction. That is why the curl formula needs extra scale factors.
Why Spherical Curl Is Different
Cartesian curl uses direct partial derivatives. Spherical curl is more careful. The basis vectors change as position changes. The distance covered by an angle also depends on radius. The formula includes r and sin theta. These factors adjust rotation for curved coordinate lines. Without them, the answer would describe the wrong geometry. This matters near poles and spherical shells.
What the Components Mean
The radial component describes rotation around a direction normal to a sphere. The theta component describes rotation linked with polar movement. The phi component describes rotation linked with azimuth motion. A large magnitude means stronger local circulation. A zero result means the field is locally irrotational at that point. Direction also matters. Positive and negative signs show the orientation of rotation.
Numerical Evaluation
This page evaluates the field at a chosen point. It then estimates the needed partial derivatives. Central difference is used because it is accurate for smooth expressions. The step h controls the small change used for each derivative. Very small values can cause round-off error. Large values can miss sharp changes. A good starting value is usually near 0.00001.
Practical Use
Use this calculator to check homework, model vector fields, or test symbolic work. Enter expressions with standard functions. Keep theta away from zero and pi. At those poles, sin theta becomes zero. The spherical formula becomes singular there. The graph helps compare curl strength across a coordinate range. Exports help save results for notes and reports.
FAQs
1. What does curl in spherical coordinates measure?
It measures local rotation of a vector field written with radial, polar, and azimuth components. The result is another vector with r, theta, and phi components.
2. Which variables can I use?
Use r, theta, and phi. You can also use constants pi and e. Common functions such as sin, cos, tan, sqrt, log, exp, and powers are supported.
3. Should angles be in radians or degrees?
You can choose either option. Internally, trigonometric functions use radians. When degrees are selected, entered theta and phi values are converted before calculation.
4. Why can theta not be zero?
The formula divides by sin theta. At theta equal to zero or pi, sin theta is zero. That creates a pole singularity in spherical coordinates.
5. What is derivative step h?
It is the small coordinate change used for numerical partial derivatives. Smaller h can improve accuracy for smooth fields, but too small may increase round-off error.
6. Can this calculator solve symbolic derivatives?
It focuses on numerical evaluation at a point. It displays the formula and derivative terms, but it does not simplify a full symbolic curl expression.
7. What does the graph show?
The graph shows curl magnitude while one selected coordinate changes. The other coordinates and all vector field expressions remain fixed.
8. Why is my result undefined?
An undefined result can come from division by zero, invalid functions, pole values, or expressions outside their domain. Check r, theta, and function inputs.