Map gradients from custom scalar fields quickly. Inspect direction, strength, and turning patterns. See steepest change clearly across every plotted point.
Choose a scalar field, enter coefficients, define a grid, and generate a gradient field table with a vector plot.
This sample shows how the generated data table will look.
| # | x | y | f(x,y) | ∂f/∂x | ∂f/∂y | |∇f| | Direction (°) |
|---|---|---|---|---|---|---|---|
| 1 | -2 | -2 | 2.4 | -2.6 | -2 | 3.280244 | -142.431408 |
| 2 | -1 | 0 | 0.2 | -1.4 | 0.1 | 1.403567 | 175.914384 |
| 3 | 0 | 1 | 0.3 | 0.4 | 1.1 | 1.17047 | 70.016893 |
| 4 | 2 | 2 | 5.2 | 4.6 | 2.9 | 5.43783 | 32.228758 |
Quadratic example: f(x,y)=ax²+by²+cxy+dx+ey+g
Gradient: ∇f(x,y)=(∂f/∂x, ∂f/∂y)=(2ax+cy+d, 2by+cx+e)
A gradient field assigns a vector to every point in the plane. That vector points toward the fastest increase of the scalar field. Its magnitude gives the local rate of change.
Magnitude is computed with: |∇f| = √[(∂f/∂x)² + (∂f/∂y)²]. Direction is computed with: θ = atan2(∂f/∂y, ∂f/∂x).
A gradient field shows the direction of fastest increase for a scalar function. Each vector points uphill, and its length reflects how quickly the value changes locally.
The gradient connects geometry, optimization, and multivariable calculus. It identifies steepest ascent, supports tangent approximations, and helps locate stationary points or rapid changes.
The x-component measures change along the x-direction. The y-component measures change along the y-direction. Together, they form the complete local change vector.
Gradient magnitude shows how strong the local change is at a point. Large values mean rapid variation, while values near zero suggest a flatter surface.
Angles often use the range from −180° to 180°. A negative angle simply means the vector points below the positive x-axis in standard position.
Yes. It includes quadratic, Gaussian, and trigonometric scalar fields. These options model bowls, peaks, ridges, oscillations, and mixed directional behaviors.
Use a smaller step for finer detail and smoother plots. Use a larger step for faster computation and simpler summaries when exploring wider regions.
A gradient field is a special vector field created from a scalar function. Not every vector field is a gradient field, but every gradient field is a vector 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.