Calculator Inputs
Single column page with responsive 3/2/1 calculator gridChoose a two-variable or three-variable scalar field, enter coefficients, select a point, and inspect the gradient vector instantly.
Example Data Table
| Mode | Sample Field | Point | Gradient | Magnitude | Directional Example |
|---|---|---|---|---|---|
| 2D | f(x,y)=4+3x−2y+1.5x²+2y²+xy | (1,2) | ⟨8,7⟩ | 10.6301 | Along ⟨2,1⟩ → 10.2859 |
| 3D | f(x,y,z)=2+x+y+z+x²+y²+0.5z²+xy+0.5xz−0.25yz | (1,1,1) | ⟨3.5,2.75,2.25⟩ | 4.9844 | Along ⟨1,1,1⟩ → 4.9075 |
Formula Used
Scalar field model: \( f(x,y,z)=c+a_xx+a_yy+a_zz+a_{xx}x^2+a_{yy}y^2+a_{zz}z^2+a_{xy}xy+a_{xz}xz+a_{yz}yz \)
Gradient vector: \( \nabla f = \left\langle \frac{\partial f}{\partial x}, \frac{\partial f}{\partial y}, \frac{\partial f}{\partial z} \right\rangle \)
Partial derivatives:
- \( \frac{\partial f}{\partial x}=a_x+2a_{xx}x+a_{xy}y+a_{xz}z \)
- \( \frac{\partial f}{\partial y}=a_y+2a_{yy}y+a_{xy}x+a_{yz}z \)
- \( \frac{\partial f}{\partial z}=a_z+2a_{zz}z+a_{xz}x+a_{yz}y \)
Gradient magnitude: \( |\nabla f|=\sqrt{(\partial f/\partial x)^2+(\partial f/\partial y)^2+(\partial f/\partial z)^2} \)
Directional derivative: \( D_{\hat{u}}f=\nabla f \cdot \hat{u} \), where \( \hat{u} \) is the normalized direction vector.
How to Use This Calculator
- Select whether your scalar field uses two variables or three variables.
- Enter the constant, linear, squared, and cross-term coefficients.
- Type the evaluation point where you want the gradient vector.
- Provide a direction vector to compute a directional derivative.
- Set the graph range and grid density for the surface plot.
- Click Compute Gradient to show the result above the form.
- Review the value, gradient components, magnitude, angle, and steepest change rates.
- Use the export buttons to save the result summary as CSV or PDF.
FAQs
1) What does the gradient vector represent?
The gradient points toward the direction of fastest increase of a scalar field. Its magnitude tells how quickly the function rises in that direction at the chosen point.
2) Why is the directional derivative useful?
It measures how fast the field changes along a specific direction you choose. This is helpful in optimization, physics, engineering, and local sensitivity studies.
3) What happens if the gradient magnitude is zero?
A zero magnitude means the local rate of steepest ascent is zero. The point may be stationary, critical, or part of a flat region, depending on nearby behavior.
4) Can I use this for two-variable functions only?
Yes. Switch to 2D mode and the calculator ignores z-related terms. The graph then shows the surface over x and y using the entered range.
5) Why does the calculator normalize the direction vector?
Directional derivatives are usually defined using a unit direction. Normalization ensures the reported rate reflects direction only, not the original vector length.
6) Does the surface plot show the full three-variable field?
In 3D mode, the plot shows a slice with z fixed at your selected z₀. That keeps visualization practical while still matching the evaluation point.
7) What are cross terms like xy, xz, and yz?
Cross terms couple variables together. They change partial derivatives because each variable influences the slope of the field through interaction with another variable.
8) When should I increase the grid points?
Use more grid points when you want smoother plots. Keep the value moderate for faster loading, especially on mobile devices or slower hosting environments.