1. What is Spherical Gradient?

The spherical gradient represents the gradient of a scalar field in spherical coordinates (r, θ, φ). It shows how the field changes in each of the three spherical coordinate directions.

2. How Does the Calculator Work?

The calculator uses the spherical gradient formula:

Where:

• \( \frac{\partial f}{\partial r} \) — Partial derivative with respect to r
• \( \frac{\partial f}{\partial \theta} \) — Partial derivative with respect to θ
• \( \frac{\partial f}{\partial \phi} \) — Partial derivative with respect to φ
• \( r \) — Radial distance
• \( \theta \) — Polar angle (from z-axis)
• \( \hat{e}_r, \hat{e}_\theta, \hat{e}_\phi \) — Unit vectors in spherical coordinates

3. Applications of Spherical Gradient

Details: Spherical gradient is widely used in physics and engineering, particularly in electromagnetism, fluid dynamics, and quantum mechanics problems with spherical symmetry.

4. Using the Calculator

Tips: Enter all partial derivatives and spherical coordinates. The radial distance r must be positive. Angles should be in radians.

5. Frequently Asked Questions (FAQ)

Q1: What's the difference between spherical and Cartesian gradient?
A: Spherical gradient accounts for the non-uniform nature of spherical coordinates, with different scaling factors for each component.

Q2: What units should I use for the inputs?
A: Use consistent units for all quantities. The calculator doesn't perform unit conversions.

Q3: How do I convert from Cartesian to spherical gradient?
A: You need to transform both the coordinates and the derivatives using the appropriate transformation matrix.

Q4: What if my θ is 0 or π?
A: The φ component becomes undefined (division by zero) at these poles, indicating a coordinate singularity.

Q5: Can I use degrees instead of radians?
A: The calculator expects angles in radians. Convert degrees to radians first if needed.