The Hessian is the Jacobian of the gradient of a function that maps from ND to 1D So the gradient, Jacobian and Hessian are different operations for different functions.
I'm currently reading about submersions and immersion's Lee's Introduction to Smooth Manifolds (p.77), and I'm slightly confused about what is meant when he says that the rank of $F$ and $p$ is "the rank of the Jacobian matrix of $F$ in any smooth chart."
I met this equation frequently in Guass-Newton optimizations. But I dont understand why the left and right side of the equation can be equal. Lets say the Jacobian is $2$ by $2$ and Hessian is $$\\
Could anyone explain in simple words (and maybe with an example) what the difference between the gradient and the Jacobian is? The gradient is a vector with the partial derivatives, right?
0 The Jacobian matrix is a listing of all the function's derivatives relative to the standard basis. It tells you how fast the function changes in each of its various dimensions, as the input coordinates change. It plays the same role that the derivative does in single variable calculus.
I'm researching a problem which suggests that progress could be achieved if the Jacobian of a vector function might be in some way considered in the manner of a covariance matrix. Specifically, and...
