Calculator Inputs
This page uses a single-column flow, with a responsive three-column form grid on large screens.
Example Data Table
These sample cases assume normalization is enabled for the displayed directional derivative.
| Case | Gradient ∇f | Direction v | Unit Direction û | Directional Derivative |
|---|---|---|---|---|
| 2D surface slope | (3, 4) | (4, 3) | (0.8000, 0.6000) | 4.8000 |
| 3D scalar field | (2, -1, 5) | (1, 2, 2) | (0.3333, 0.6667, 0.6667) | 3.3333 |
| 4D optimization step | (-6, 2, 1, 3) | (2, -1, 2, 1) | (0.6325, -0.3162, 0.6325, 0.3162) | -2.8460 |
Formula Used
The directional derivative measures how fast a scalar field changes along a selected direction at a point.
Gradient: ∇f = (fx, fy, fz, ...)
Direction magnitude: ||v|| = √(v12 + v22 + ...)
Unit direction: û = v / ||v||
Directional derivative: Dûf = ∇f · û = Σ giûi
Angle with gradient: cos θ = Dûf / ||∇f||
If normalization is disabled, the calculator also shows the raw dot product ∇f · v. That value includes the length of the entered direction vector, so it can be larger in magnitude than the unit-vector directional derivative.
How to Use This Calculator
- Select whether your problem uses 2, 3, or 4 variables.
- Optionally enter the evaluation point for documentation and exports.
- Enter gradient components already evaluated at that point.
- Enter the direction vector you want to test.
- Leave normalization enabled for the standard directional derivative.
- Set decimal places, then press the calculate button.
- Review the displayed derivative, angle, extrema rates, and interpretation.
- Use the CSV or PDF buttons to save your result.
FAQs
1. What does the directional derivative represent?
It shows the instantaneous rate of change of a scalar field when you move from a point in one chosen direction. Positive values indicate increase, negative values indicate decrease, and zero suggests local flatness in that direction.
2. Why should I normalize the direction vector?
Normalization removes the effect of vector length and keeps only direction. That produces the standard directional derivative. Without normalization, the dot product changes when you scale the same direction vector longer or shorter.
3. What inputs should I enter for the gradient?
Enter the partial derivative values already evaluated at your chosen point. For example, if ∇f(a, b, c) = (2, -1, 5), type those three numbers directly into the gradient fields.
4. What happens if the gradient is zero?
A zero gradient means the maximum local increase rate is zero. The directional derivative will also be zero in every direction, and the angle with the gradient is undefined because there is no nonzero gradient vector to compare.
5. Can I use this for 2D, 3D, and 4D problems?
Yes. The calculator supports two, three, or four variables. Extra fields remain available for flexible entry, but only the number of components matching the selected dimension are used in the final computation.
6. What is the maximum increase rate?
The maximum increase rate equals the gradient magnitude ||∇f||. It occurs when you move exactly in the gradient direction. The steepest decrease rate is the negative of that same magnitude.
7. Why does the raw dot product differ from the displayed result?
When normalization is enabled, the displayed result uses the unit direction vector. The raw dot product uses the original entered vector length, so both values match only when your entered direction vector already has magnitude one.
8. Do the point inputs affect the math directly?
No. They are included for documentation, reporting, and export clarity. The actual computation uses the gradient values you supply, which should already correspond to the point where you want the directional derivative.