Tools / Calculus
Multivariable Calculus Lab
This lab computes numeric gradients, directional derivatives, Jacobians, and Hessians for simple user-entered functions. It is designed for local analysis, debugging intuition, and quick engineering checks rather than full symbolic derivation.
Local Results
Jacobian
Hessian
Gradient Field Snapshot
The arrows show the local direction of steepest increase on a sampled grid. The highlighted point is the one used for the numeric calculations above.
Section Curve
This slice fixes the current y value and shows how the scalar function changes as x moves through that horizontal cross-section.
How To Use This Lab
Enter a scalar function of x and y, choose a point, and press Update Analysis.
If you also provide vector outputs, the lab will build a Jacobian for that vector-valued mapping at the same point.
- The gradient reports the direction of steepest increase of the scalar function.
- The directional derivative measures change only along the direction vector you provide.
- The Hessian reports local second-order curvature information.
- The Jacobian reports first-order sensitivity of a vector-valued output.
Fundamental Mathematics
Multivariable calculus studies local change in several directions at once. The gradient is the vector of first partial derivatives. The directional derivative is the dot product of that gradient with a chosen unit direction. The Jacobian is the matrix of first derivatives for a vector-valued function, and the Hessian is the square matrix of second derivatives for a scalar-valued function.
Together, these objects describe local slope, sensitivity, and curvature, which is why they appear so often in optimization, control, estimation, machine learning, and scientific computing.