Calculator Input
Example Data Table
This sample uses the default expressions and point shown when the page first loads.
| Function | Expression | Point | Function Value | Key Partial Derivatives |
|---|---|---|---|---|
| f1 | x^2*y + sin(z) |
(1, 2, 0.5) | 2.479426 | ∂f1/∂x = 4.000000, ∂f1/∂y = 1.000000, ∂f1/∂z = 0.877583 |
| f2 | x*y + z^2 |
(1, 2, 0.5) | 2.250000 | ∂f2/∂x = 2.000000, ∂f2/∂y = 1.000000, ∂f2/∂z = 1.000000 |
| f3 | exp(x) - y + z |
(1, 2, 0.5) | 1.218282 | ∂f3/∂x = 2.718282, ∂f3/∂y = -1.000000, ∂f3/∂z = 1.000000 |
| Sample determinant | 4.577600 | |||
Formula Used
The calculator evaluates a vector function F(x) at a chosen operating point and estimates the Jacobian matrix numerically with the central difference method.
Jij = ∂fi / ∂xj ∂fi / ∂xj ≈ [ fi(x + h ej) - fi(x - h ej) ] / (2h) J(x) Δ = -F(x) xnew = x + ΔHere, h is the derivative step size, ej is the unit direction vector, and Δ is the Newton correction step. The calculator also reports matrix rank, determinant for square systems, and a simple column-sum sensitivity measure.
How to Use This Calculator
- Select how many variables and functions your system contains.
- Enter each function using explicit multiplication such as
x*y. - Provide the operating point for each active variable.
- Adjust derivative step size and decimal precision when needed.
- Press Compute Jacobian Matrix to generate the matrix, diagnostics, and Newton correction.
- Review the heatmap and sensitivity chart to understand which variables dominate local behavior.
- Use the export buttons to save the result as CSV or PDF.
FAQs
1. What does this Jacobian calculator solve?
It evaluates multivariable functions at a chosen point, estimates the Jacobian matrix, checks rank and determinant, and solves one Newton correction step when the system is square and numerically stable.
2. Are derivatives computed symbolically?
No. This version uses central differences, which numerically approximate each partial derivative around the selected operating point. That makes it flexible for many custom expressions entered directly in the form.
3. Why is the determinant missing sometimes?
A determinant only exists for square matrices. If your number of functions differs from your number of variables, the Jacobian is rectangular, so the determinant is intentionally not shown.
4. Why might the Newton correction fail?
The Jacobian may be singular, nearly singular, or numerically unstable at the chosen point. In those cases, the linear system cannot be solved reliably for the correction vector.
5. Which functions can I enter?
You can use x, y, z, numbers, parentheses, powers, and functions like sin, cos, tan, exp, log, sqrt, abs, sinh, cosh, tanh, and pow. Enter multiplication explicitly.
6. What does the sensitivity chart mean?
It sums the absolute values in each Jacobian column. Larger totals indicate variables that have stronger local influence across the current set of functions near the chosen point.
7. How should I choose the step size?
A smaller step often improves local accuracy, but an extremely tiny step may amplify rounding noise. The default value is a practical starting point for most smooth functions.
8. Can this handle optimization or nonlinear systems practice?
Yes. It is useful for Newton method experiments, local sensitivity checks, linearization, and educational work with nonlinear vector functions in calculus, numerical analysis, and applied modeling.