Hessian Matrix Calculator

Calculate the Hessian matrix of second partial derivatives for multivariable functions.

Function Input

Enter a multivariable function f(x,y) or f(x,y,z):

Use * for multiplication, ^ for exponents, and standard math functions

Current Function:

Enter a function to see it displayed here

Evaluate at Point (Optional)

Enter coordinates to evaluate the Hessian matrix at a specific point:

About Hessian Matrices

What is a Hessian Matrix?

The Hessian matrix is a square matrix of second-order partial derivatives of a scalar-valued function. It describes the local curvature of a multivariable function and is fundamental in optimization, machine learning, and mathematical analysis.

Mathematical Definition

For a function f(x₁, x₂, ..., xₙ), the Hessian matrix H is defined as:

How to Use This Calculator

  1. Select the number of variables (2 or 3)
  2. Enter your function using standard mathematical notation
  3. Optionally, enter coordinates to evaluate the Hessian at a specific point
  4. Click "Calculate Hessian Matrix" to see the results
  5. View the symbolic matrix, partial derivatives, and analysis

Input Format

Enter expressions using standard mathematical notation:

  • Use * for multiplication: x*y
  • Use ^ for exponents: x^2
  • Use standard functions: sin(x), cos(y), exp(x), log(x)
  • Use parentheses for grouping: (x+y)^2

Example Functions

2D Examples

  • x^2 + y^2 (paraboloid)
  • x^2 - y^2 (saddle)
  • x^2 + x*y + y^2
  • sin(x) + cos(y)
  • exp(x^2 + y^2)

3D Examples

  • x^2 + y^2 + z^2
  • x*y + y*z + x*z
  • x^3 + y^3 + z^3
  • sin(x)*cos(y)*exp(z)
  • x^2*y + y^2*z + z^2*x

Applications

  • Optimization: Finding critical points and determining their nature
  • Machine Learning: Second-order optimization methods (Newton's method)
  • Physics: Analyzing potential energy surfaces and stability
  • Economics: Studying utility functions and market equilibria
  • Computer Graphics: Surface curvature analysis