Hessian Calculator
Enter a function of x and y, pick an evaluation point, and get the full Hessian matrix plus determinant and critical-point classification.
Use * for multiplication, ^ or ** for power.
x =
y =
Syntax reference
x^2 or x**2power
sin(x), cos(x), tan(x)trig
exp(x)e^x
log(x)natural log
sqrt(x)square root
abs(x)absolute value
2*x*yexplicit multiply
PI, Econstants
Examples
Enter a function and click Compute Hessian
Hessian Matrix H(f) at ()
fxx =
fxy =
fyx =
fyy =
Determinant D = fxx·fyy − (fxy)²
Second Derivative Test
D > 0 and fxx > 0 → Local minimum
D > 0 and fxx < 0 → Local maximum
D < 0 → Saddle point
D = 0 → Inconclusive (higher-order test needed)
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Summary
Enter a function of x and y, pick an evaluation point, and get the full Hessian matrix plus determinant and critical-point classification.
How it works
- Enter a function of x and y using standard math notation (e.g. x^2 + y^2 or sin(x)*cos(y)).
- Enter the x and y coordinates of the evaluation point.
- Click "Compute Hessian" to compute all four second-order partial derivatives numerically.
- The 2×2 Hessian matrix is displayed along with each entry labeled.
- The determinant D and second derivative test conclusion are shown below the matrix.
Use cases
- Classify critical points found in multivariable calculus optimization problems.
- Verify hand-computed Hessian matrices for homework or exams.
- Check whether a stationary point of an engineering objective function is a minimum.
- Explore the curvature of surfaces in machine learning loss landscapes.
- Confirm saddle-point behavior in economic or game-theoretic models.
- Cross-check results when studying second-order conditions in optimization theory.
Frequently Asked Questions
Last updated: 2026-07-23 ·
Reviewed by Nham Vu