Calculator Inputs
Enter one scalar field and one vector field. The tool estimates partial derivatives numerically with central differences.
Field Visualization
The graph plots the scalar field surface over x and y while keeping z fixed at the chosen point.
Example Data Table
| Scalar Field | Vector Field | Point | Gradient | Divergence | Curl | Laplacian |
|---|---|---|---|---|---|---|
| x^2 + y^2 + z^2 | (xy, yz, zx) | (1,1,1) | (2,2,2) | 3 | (-1,-1,-1) | 6 |
| sin(x) + cos(y) + z^2 | (x, y, z) | (0,0,2) | (1,0,4) | 3 | (0,0,0) | 1 |
| x*y*z | (x^2, y^2, z^2) | (2,1,3) | (3,6,2) | 12 | (0,0,0) | 0 |
Formula Used
∇f = ( ∂f/∂x , ∂f/∂y , ∂f/∂z )
∇·F = ∂Fx/∂x + ∂Fy/∂y + ∂Fz/∂z
∇×F = ( ∂Fz/∂y - ∂Fy/∂z , ∂Fx/∂z - ∂Fz/∂x , ∂Fy/∂x - ∂Fx/∂y )
∇²f = ∂²f/∂x² + ∂²f/∂y² + ∂²f/∂z²
∂f/∂x ≈ [ f(x+h,y,z) - f(x-h,y,z) ] / 2h
∂²f/∂x² ≈ [ f(x+h,y,z) - 2f(x,y,z) + f(x-h,y,z) ] / h²
This calculator uses numerical differentiation. Smaller step sizes often improve accuracy, but extremely tiny values may introduce floating-point noise.
How to Use This Calculator
- Enter a scalar field using variables
x,y, andz. - Enter vector field components for
Fx,Fy, andFz. - Provide the evaluation point coordinates.
- Choose a small step size for numerical partial derivatives.
- Press the calculate button to view gradient, divergence, curl, and Laplacian.
- Review the generated Plotly surface for the scalar field.
- Export the results as CSV or PDF when needed.
FAQs
1. What does the del operator represent?
The del operator, written as ∇, is a vector differential operator. It helps measure change in scalar and vector fields through gradient, divergence, curl, and Laplacian calculations.
2. What is the difference between gradient and divergence?
Gradient applies to scalar fields and returns directional rates of change. Divergence applies to vector fields and measures net outward flow from a point.
3. What does curl tell me?
Curl measures local rotation in a vector field. It is useful in fluid flow, electromagnetism, and other systems involving circulation or spinning behavior.
4. Why does this calculator use a step size?
The step size supports numerical differentiation. The tool estimates partial derivatives using nearby values instead of symbolic algebra.
5. Can I enter trigonometric and exponential expressions?
Yes. You can use functions like sin, cos, tan, exp, log, sqrt, and abs. Use variables x, y, and z exactly as written.
6. What is the Laplacian used for?
Laplacian measures combined second-order curvature in a scalar field. It appears in heat flow, diffusion, wave behavior, and potential theory.
7. Why might my result look slightly approximate?
This tool uses finite differences and floating-point arithmetic. That method is practical, but results may show tiny rounding differences from exact symbolic answers.
8. Is this calculator useful for learning vector calculus?
Yes. It combines numeric outputs, surface plots, formulas, and examples. That makes it useful for checking homework, exploring patterns, and building intuition.