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Vector calculus

The Taylor series expansion of the scalar field around some reference point x0 shows that the first-order variations are governed by the gradient, this column vector of first partial derivatives, and the second-order variations are governed by the Hessian, this 3x3 matrix of second partial derivatives.

But we care about changes in the gradient vector, so when we differentiate this, we see that the change in the gradient vector around x0 is found by multiplication by the Hessian matrix.

So the Hessian is the basis of our curvature computations.

To see exactly how we get curvature from the Hessian, ...