Measure directional change using gradients and unit vectors. Analyze local behavior at any chosen point. Graph paths, export findings, and compare worked examples easily.
Scalar field: f(x,y,z)=ax²+by²+cz²+dxy+eyz+fzx+gx+hy+iz+j
Gradient: ∇f(x,y,z)=⟨2ax+dy+fz+g, 2by+dx+ez+h, 2cz+ey+fx+i⟩
Unit direction: û = v / ||v||
Directional derivative: Dᵤf = ∇f · û
Steepest ascent rate: ||∇f||, and the steepest descent rate is −||∇f||
The gradient points toward the fastest local increase of the scalar field. When you project the gradient onto a chosen unit direction, you obtain the directional derivative. A positive value means the field rises along that direction. A negative value means it falls.
| Example | Scalar Field Coefficients | Point P | Direction v | Gradient at P | Directional Derivative | Field Value f(P) |
|---|---|---|---|---|---|---|
| 1 | a=2, b=1, c=3, d=1, e=-2, f=0.5, g=4, h=-1, i=2, j=5 | (1, 2, -1) | (3, -2, 6) | ⟨9.5, 6, -7.5⟩ | -4.071429 | 19.5 |
| 2 | a=1, b=2, c=1, d=0, e=1, f=-1, g=0, h=3, i=-2, j=0 | (2, -1, 1) | (1, 4, 2) | ⟨3, 0, -3⟩ | -0.654654 | -1 |
It measures how quickly the scalar field changes at a point when you move in a chosen direction. Positive values indicate increase, negative values indicate decrease, and zero suggests no first-order change in that direction.
Normalization removes the effect of vector length. That lets the result represent change per unit distance, which is the standard meaning of a directional derivative.
A zero vector has no direction, so it cannot be normalized. The calculator blocks this case and asks for a valid non-zero vector.
The gradient contains all first-order local change information. It points toward the direction of greatest increase, and its magnitude gives the steepest ascent rate.
It shows how aligned your chosen direction is with the steepest ascent direction. Smaller angles mean stronger increase, angles above ninety degrees mean decrease, and ninety degrees implies local orthogonality.
The graph plots the exact field value along the selected line through the point. It also plots a linear approximation so you can compare local behavior with the actual curve.
This version is built for a general quadratic field in three variables. It is flexible enough for many classroom, engineering, and optimization examples, but it is not a symbolic parser.
It is the negative of the gradient magnitude. Moving opposite the gradient gives the fastest local decrease of the scalar field.
Important Note: All the Calculators listed in this site are for educational purpose only and we do not guarentee the accuracy of results. Please do consult with other sources as well.