What is the Jacobian matrix?

What are its applications? 

What is its physical and geometrical meaning?

Can someone please explain with examples?

Here is an example. Suppose you have two implicit differentiable functions

$$F(x,y,z,u,v)=0,\qquad G(x,y,z,u,v)=0$$

and the functions, also differentiable, $u=f(x,y,z)$ and $v=g(x,y,z)$ such
that 

$$F(x,y,z,f(x,y,z),g(x,y,z))=0,\qquad G(x,y,z,f(x,y,z),g(x,y,z))=0.$$

If you differentiate $F$ and $G$, you get

\begin{eqnarray*}
\frac{\partial F}{\partial x}+\frac{\partial F}{\partial u}\frac{\partial u}{
\partial x}+\frac{\partial F}{\partial v}\frac{\partial v}{\partial x} &=&0\qquad
\\
\frac{\partial G}{\partial x}+\frac{\partial G}{\partial u}\frac{\partial u}{
\partial x}+\frac{\partial G}{\partial v}\frac{\partial v}{\partial x} &=&0.
\end{eqnarray*}

Solving this system you obtain

$$\frac{\partial u}{\partial x}=-\frac{\det \begin{pmatrix}
\frac{\partial F}{\partial x} &  \frac{\partial F}{\partial v} \\   \frac{\partial G}{\partial x} &  \frac{\partial G}{\partial v}
\end{pmatrix}}{\det \begin{pmatrix}
\frac{\partial F}{\partial u} &  \frac{\partial F}{\partial v} \\ 
 \frac{\partial G}{\partial u} &  \frac{\partial G}{\partial v}
\end{pmatrix}}$$

and similar for $\dfrac{\partial u}{\partial y}$, $\dfrac{\partial u}{\partial
z}$, $\dfrac{\partial v}{\partial x}$, $\dfrac{\partial v}{\partial y}$, $%
\dfrac{\partial v}{\partial z}$. The compact notation for the denominator is 

$$\frac{\partial (F,G)}{\partial (u,v)}=\det 
\begin{pmatrix}
\frac{\partial F}{\partial u} & \frac{\partial F}{\partial v} \\ 
\frac{\partial G}{\partial u}  & \frac{\partial G}{\partial v}
\end{pmatrix}$$

and similar for the numerator. Then

$$\dfrac{\partial u}{\partial x}=-\dfrac{\dfrac{\partial (F,G)}{\partial (x,v)}}{%
\dfrac{\partial (F,G)}{\partial (u,v)}}$$

where $\dfrac{\partial (F,G)}{\partial (x,y)},\dfrac{\partial (F,G)}{\partial
(u,v)}$ are Jacobians (after the 19th century German mathematician Carl Jacobi).



The absolute value of the Jacobian of a coordinate system transformation is also used to convert a multiple integral from one  system into another. In $\mathbb{R}^2$ it measures how much the unit area is distorted by the given transformation, and in $\mathbb{R}^3$ this factor measures the unit volume distortion, etc. 



Another example: the following coordinate transformation (due to Beukers, Calabi and Kolk)

$$x=\frac{\sin u}{\cos v}$$

$$y=\frac{\sin v}{\cos u}$$

transforms (see this question of mine) the square domain $0\lt x\lt 1$ and $0\lt y\lt 1$ into the triangle domain $u,v>0,u+v<\pi /2$ (in Proofs from the BOOK by M. Aigner and G. Ziegler).

For this transformation you get (see Proof 2 in this collection of proofs by Robin Chapman) 

$$\dfrac{\partial (x,y)}{\partial (u,v)}=1-x^2y^{2}.$$



Jacobian sign and orientation of closed curves. Assume you have two small closed curves, one around $(x_0,y_0)$ and another around $u_0,v_0$, this one being the image of the first under the mapping $u=f(x,y),v=g(x,y)$. If the sign of $\dfrac{\partial (x,y)}{\partial (u,v)}$ is positive, then both curves will be travelled in the same sense. If the sign is negative, they will have opposite senses. (See Oriented Regions and their Orientation.) 
