Which multivariable derivative concept do I need?
Total derivative, Jacobian, directional derivative, Hessian -- four related but distinct objects.
Ask what you're actually differentiating and with respect to what. The TOTAL DERIVATIVE (as a linear map) and the JACOBIAN (its matrix in coordinates) are the same object seen two ways. The DIRECTIONAL DERIVATIVE is that same linear map applied to one specific vector. The HESSIAN is a second-order object entirely -- it classifies critical points, it doesn't approximate f linearly.
The first-order object, two views
The best linear approximation to f at a point -- a single matrix, Df(\mathbf{a}), not a list of separate partials.
The coordinate-free linear map -- best linear approximation to f at a point.
The matrix of all first partial derivatives -- the total derivative written out in coordinates.
The SAME object, written out as a matrix of partial derivatives.
Applied to one direction
The second-order object
The matrix of all second partial derivatives -- the quadratic term in a function's Taylor expansion.
Example
For a critical point of f(x,y): eigenvalues of H_f both positive is a local min, both negative a local max, mixed signs a saddle.
Point table
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