Computed Results
The result block appears here after calculation.
Enter the coefficients, point, direction vector, and plot range. Then press the calculate button to display values here.
Calculator Inputs
The page uses one stacked layout. Only the input group becomes three columns on large screens, two on medium screens, and one on mobile.
Supported Function Model
This calculator evaluates a second-order scalar field in three variables.
f(x,y,z) = ax² + by² + cz² + dxy + exz + fyz + gx + hy + iz + j
It computes the gradient vector, its magnitude, a unit gradient, and the directional derivative along a custom direction vector.
Formula Used
∂f/∂x = 2ax + dy + ez + g∂f/∂y = 2by + dx + fz + h∂f/∂z = 2cz + ex + fy + i∇f(x,y,z) = (∂f/∂x, ∂f/∂y, ∂f/∂z)|∇f| = √[(∂f/∂x)² + (∂f/∂y)² + (∂f/∂z)²]Duf = ∇f · ûThe directional derivative uses the normalized direction vector.
The Plotly graph shows an x-y surface slice while holding z fixed.
How to Use This Calculator
- Enter coefficients for the quadratic, cross, linear, and constant terms.
- Enter the point where you want the gradient evaluated.
- Provide a direction vector for the directional derivative.
- Choose the graph range and the z slice value.
- Press Calculate Gradient to show the result block above the form.
- Review the vector, magnitude, normalized direction, and graph.
- Use the CSV or PDF buttons to export the current results.
Example Data Table
| Input or Output | Example Value |
|---|---|
| a | 1 |
| b | 2 |
| c | 1 |
| d | 1 |
| e | -1 |
| f | 2 |
| g | 3 |
| h | -2 |
| i | 1 |
| j | 5 |
| Point (x, y, z) | (1, 2, -1) |
| Direction Vector | (2, 1, -2) |
| Function Value | 12 |
| Gradient Vector | (8, 5, 2) |
| Gradient Magnitude | 9.643651 |
| Directional Derivative | 5.666667 |
Frequently Asked Questions
1. What does the gradient represent?
The gradient points toward the steepest increase of the scalar field. Its components are the partial derivatives with respect to x, y, and z.
2. Why does this calculator use a quadratic model?
A quadratic scalar field supports squared terms, cross terms, and linear terms. That gives a flexible structure while keeping derivatives exact and fast to compute.
3. What is the gradient magnitude?
The magnitude measures how strongly the function changes at the chosen point. Larger values mean steeper local change.
4. What is a directional derivative?
It measures the rate of change of the function along a chosen direction. The calculator normalizes the direction vector before taking the dot product.
5. Why is my directional derivative unavailable?
That happens when the direction vector is zero. A zero vector has no direction, so normalization is impossible.
6. What does the Plotly graph show?
The chart displays an x-y surface slice of the function while z stays fixed. It helps you inspect local shape around the selected point.
7. Can I use negative coefficients?
Yes. Negative coefficients are valid and often useful. They can invert curvature, rotate the surface behavior, or reduce gradient components.
8. When should I export CSV or PDF?
Export CSV for raw values and spreadsheets. Export PDF when you need a quick printable result snapshot for study notes or reports.