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In this Manifold notes in more informal way, there is an example that talks about a Jacobian which is something I do not understand.

In his answer, Lee Mosher, said "For a simple example in 1-dimension where the Jacobian is just the ordinary derivative from first semester Calculus, think of two coordinate patches on the unit circle in the $xy$ plane, the first being the open upper semicircle using the $x$ coordinate, and the second being the open right semicircle using the $y$ coordinate. These two coordinate patches overlap in the open first quadrant of the circle, giving one coordinate $0<x<1$ and another coordinate $1>y>0$ which are related by the change of coordinate mapping $$y(x)=\sqrt{1-x^2}$$ whose Jacobian $\frac{d}{dx}(\sqrt{1-x^2})$ is nonsingular on the interval $0<x<1$."

First what does he mean when he says "the first being the open upper semicircle using the $x$ coordinate"? It is a semi-circle, that is to say as you shade it for example, you are going to use te $x$ and $y$ coordinate. Then, what is a Jocobian and why do we care about it?

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Jacobians are used especially in multivariable calculus, when investigating functions which are differentiable in more than one dimension.

It is a matrix which stores the partial derivatives of a multivariable function. If the function has it's outputs stored as a colum vector, then each new column in the Jacobian is derivative with respect to one (new) of the in-variables.

Basically $${\bf J}_{j,k}(f) = \frac{\partial{f_j({\bf x})}}{\partial x_k}$$

Another way of looking at it you can say it is an outer product of a gradient operator and a multivalued function, performing a gradient for each output $f_j$ and storing them in the rows of the Jacobian.

A matrix being singular means it is not invertible, a 1x1 matrix (which this one is) is invertible whenever it's only element is not 0.

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