1. Definition
The gradient ∇f=(f_x,f_y,f_z) points in the direction of steepest ascent; its magnitude |∇f| is the largest directional derivative.
2. Symbols
| ∇f | gradient vector |
| |∇f| | magnitude |
| ∇f | steepest ascent |
| −∇f | steepest descent |
3. How it works
- Assemble partials into a vector;
- Compute its norm;
- That direction steepest ascent.
4. Steps
- Enter f (add z for 3D);
- Enter the point;
- Click Calculate.
5. Example
Example: f=x²+y² at (1,1).
Solution: ∇f=(2,2), |∇f|=√8≈2.828, pointing northeast.
6. Pitfalls
The gradient is a vector;
It is perpendicular to level curves/surfaces;
Leave z₀ blank for 2D.