System Configuration
Define the number of functions (m) and variables (n).
Function Definitions
Enter each function f(x,y...) and the evaluation point.
Evaluation Point (x,y...)
Resulting Jacobian
The matrix of partial derivatives.
Symbolic Jacobian (∂fᵢ/∂xⱼ)
Evaluated Jacobian
How to Use the Calculator
Get your Jacobian matrix in three simple steps.
Configure System
Use the steppers to set the number of functions (m) and variables (n) in your system. The input fields will update automatically.
Define Functions & Point
Enter each of your functions (f₁, f₂, ...) and the specific point (x, y, ...) at which you want to evaluate the Jacobian.
Analyze the Results
The calculator instantly computes and displays both the symbolic matrix of partial derivatives and the final numerical Jacobian evaluated at your point.
What is the Jacobian Matrix?
The generalization of the derivative for multivariable functions.
For a vector-valued function f: ℝⁿ → ℝᵐ, the Jacobian matrix is the matrix of all its first-order partial derivatives. If you have 'm' functions and 'n' variables, the Jacobian will be an 'm x n' matrix.
Each entry Jij in the matrix represents the partial derivative of the i-th function with respect to the j-th variable:
In essence, the Jacobian matrix represents the best linear approximation of the function at a given point. It describes how the output of the function changes as you make infinitesimal changes to the input variables. This is a fundamental concept in vector calculus and our Jacobi matrix calculator makes computing it effortless.
An Analytical Workbench
Features designed for flexibility and deep insight.
Dynamic System Size
Easily configure your system by adding or removing functions and variables on the fly. The calculator's interface adapts instantly to any m x n system.
Symbolic Differentiation
Powered by a robust math engine, the tool first computes the symbolic partial derivatives, showing you the general form of the Jacobian before evaluation.
Instant Numerical Evaluation
Alongside the symbolic form, the calculator provides the final numerical matrix evaluated at the precise point you specify, giving you the concrete data you need.
Applications of the Jacobian
Where this powerful matrix is used in practice.
Optimization
The Jacobian is a core component of optimization algorithms like the Newton-Raphson method for finding roots of systems of nonlinear equations.
Calculus & Physics
Used for change of variables in multiple integrals. The determinant of the Jacobian (the "Jacobian determinant") is the scaling factor for area or volume.
Robotics & Machine Learning
In robotics, it relates joint velocities to end-effector velocities. In machine learning, it's fundamental to backpropagation and understanding neural network sensitivity.
