Jacobian Calculator
Compute Jacobian Matrices
Calculate the Jacobian matrix of a vector-valued function - the matrix of all first-order partial derivatives. Essential for multivariable calculus, transformations, and analyzing systems of equations.
Compute the Jacobian Matrix
Enter your functions and variables to calculate the Jacobian matrix.
Note: Use * for multiplication (e.g., 2*x instead of 2x).
Understanding the Jacobian Matrix
The Jacobian matrix represents all first-order partial derivatives of a vector-valued function. It's fundamental in vector calculus, especially in transformations and optimization problems.
Example
Consider the functions:
The Jacobian matrix is:
J = \egin{bmatrix} 2x & 2y \\ e^x * \sin(y) & e^x * \cos(y) \end{bmatrix}
Understanding the Jacobian
The Jacobian matrix contains all first-order partial derivatives of a vector function F: ℝⁿ → ℝᵐ. For F(x₁,...,xₙ) = [f₁,...,fₘ], Jacobian J[i,j] = ∂fᵢ/∂xⱼ. It generalizes the derivative to multiple variables. The Jacobian determinant measures local volume scaling under transformation.
Applications
Jacobians are crucial for: change of variables in multiple integrals, Newton's method for systems of equations, analyzing stability of dynamical systems, robotics and kinematics (relating joint to endpoint velocities), optimization and constraint problems, and implicit function theorem applications.
Special Cases
When n = m (square Jacobian): determinant is the Jacobian determinant, used in transformation of integrals. For single function f: ℝⁿ → ℝ, Jacobian is the gradient vector. For vector field in ℝ³, relates to divergence and curl operations in vector calculus.