Calculator
Use explicit multiplication like 2*x and y*z. Supported functions include sin, cos, tan, sqrt, abs, exp, ln, log, and pow(a,b).
Example data table
| Function | Point | Gradient | Magnitude |
|---|---|---|---|
x^2 + y^2 + z^2 |
(1, 2, 3) | <2.000000, 4.000000, 6.000000> | 7.483315 |
sin(x) + y*z |
(0.5, 2, 3) | <0.877583, 3.000000, 2.000000> | 3.706911 |
x*y + y*z + z*x |
(2, 1, 4) | <5.000000, 6.000000, 3.000000> | 8.366600 |
Formula used
Gradient definition
∇f(x,y,z) = < ∂f/∂x, ∂f/∂y, ∂f/∂z >
Central-difference partial derivative estimates
∂f/∂x ≈ [f(x+h,y,z) - f(x-h,y,z)] / (2h)
∂f/∂y ≈ [f(x,y+h,z) - f(x,y-h,z)] / (2h)
∂f/∂z ≈ [f(x,y,z+h) - f(x,y,z-h)] / (2h)
Gradient magnitude
|∇f| = sqrt[(∂f/∂x)^2 + (∂f/∂y)^2 + (∂f/∂z)^2]
Unit gradient vector
u∇ = ∇f / |∇f|
Directional derivative
Duf = ∇f · u
This calculator uses numerical differentiation. Smaller step sizes often improve local accuracy, but overly tiny values may increase rounding noise.
How to use this calculator
- Enter a scalar function using
x,y, andz. - Type the point where you want the gradient.
- Choose a step size
hfor central differences. - Select decimal places for the displayed output.
- Optionally enter a direction vector for directional derivative results.
- Click Find Gradient to calculate the vector and graph.
- Use the CSV or PDF buttons to save the result.
FAQs
1. What does the gradient represent?
The gradient points toward the steepest increase of a scalar field. Its size shows how fast the field changes at the chosen point.
2. Is this calculator exact or approximate?
It uses central-difference estimates, so the result is numerical. For many smooth functions, the approximation is highly accurate with a sensible step size.
3. Which functions can I enter?
You can use standard operators, powers, parentheses, and functions like sin, cos, tan, sinh, cosh, tanh, sqrt, abs, exp, ln, log, floor, ceil, round, and pow.
4. Why should I use explicit multiplication?
Enter 2*x instead of 2x. Explicit multiplication keeps the parser clear and reduces syntax mistakes during evaluation.
5. What step size should I choose?
A small positive value like 0.00001 often works well. Very large values reduce accuracy, while extremely tiny values can amplify floating-point noise.
6. What happens when the gradient magnitude is zero?
A zero magnitude means the local gradient is flat. In that case, the unit gradient direction is undefined, so the calculator displays zeros.
7. Can I calculate a directional derivative too?
Yes. Add an optional direction vector. The calculator normalizes it first, then computes the directional derivative using the dot product with the gradient.
8. Why offer CSV and PDF exports?
Exports help you save calculations, share results, document coursework, and compare gradient values from multiple functions or sample points.